3.386 \(\int \frac {1}{(a+b \tan ^4(c+d x))^2} \, dx\)
Optimal. Leaf size=648 \[ \frac {\sqrt [4]{b} \left (\sqrt {a}-3 \sqrt {b}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} d (a+b)}+\frac {\sqrt [4]{b} \left (\sqrt {a}-\sqrt {b}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} d (a+b)^2}-\frac {\sqrt [4]{b} \left (\sqrt {a}-3 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}+1\right )}{8 \sqrt {2} a^{7/4} d (a+b)}-\frac {\sqrt [4]{b} \left (\sqrt {a}-\sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} d (a+b)^2}-\frac {\sqrt [4]{b} \left (\sqrt {a}+3 \sqrt {b}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )}{16 \sqrt {2} a^{7/4} d (a+b)}-\frac {\sqrt [4]{b} \left (\sqrt {a}+\sqrt {b}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )}{4 \sqrt {2} a^{3/4} d (a+b)^2}+\frac {\sqrt [4]{b} \left (\sqrt {a}+3 \sqrt {b}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )}{16 \sqrt {2} a^{7/4} d (a+b)}+\frac {\sqrt [4]{b} \left (\sqrt {a}+\sqrt {b}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )}{4 \sqrt {2} a^{3/4} d (a+b)^2}+\frac {b \tan (c+d x) \left (1-\tan ^2(c+d x)\right )}{4 a d (a+b) \left (a+b \tan ^4(c+d x)\right )}+\frac {x}{(a+b)^2} \]
[Out]
x/(a+b)^2+1/16*b^(1/4)*arctan(1-b^(1/4)*2^(1/2)*tan(d*x+c)/a^(1/4))*(a^(1/2)-3*b^(1/2))/a^(7/4)/(a+b)/d*2^(1/2
)-1/16*b^(1/4)*arctan(1+b^(1/4)*2^(1/2)*tan(d*x+c)/a^(1/4))*(a^(1/2)-3*b^(1/2))/a^(7/4)/(a+b)/d*2^(1/2)+1/4*b^
(1/4)*arctan(1-b^(1/4)*2^(1/2)*tan(d*x+c)/a^(1/4))*(a^(1/2)-b^(1/2))/a^(3/4)/(a+b)^2/d*2^(1/2)-1/4*b^(1/4)*arc
tan(1+b^(1/4)*2^(1/2)*tan(d*x+c)/a^(1/4))*(a^(1/2)-b^(1/2))/a^(3/4)/(a+b)^2/d*2^(1/2)-1/8*b^(1/4)*ln(a^(1/2)-a
^(1/4)*b^(1/4)*2^(1/2)*tan(d*x+c)+b^(1/2)*tan(d*x+c)^2)*(a^(1/2)+b^(1/2))/a^(3/4)/(a+b)^2/d*2^(1/2)+1/8*b^(1/4
)*ln(a^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*tan(d*x+c)+b^(1/2)*tan(d*x+c)^2)*(a^(1/2)+b^(1/2))/a^(3/4)/(a+b)^2/d*2^(1
/2)-1/32*b^(1/4)*ln(a^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*tan(d*x+c)+b^(1/2)*tan(d*x+c)^2)*(a^(1/2)+3*b^(1/2))/a^(7/
4)/(a+b)/d*2^(1/2)+1/32*b^(1/4)*ln(a^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*tan(d*x+c)+b^(1/2)*tan(d*x+c)^2)*(a^(1/2)+3
*b^(1/2))/a^(7/4)/(a+b)/d*2^(1/2)+1/4*b*tan(d*x+c)*(1-tan(d*x+c)^2)/a/(a+b)/d/(a+b*tan(d*x+c)^4)
________________________________________________________________________________________
Rubi [A] time = 0.66, antiderivative size = 648, normalized size of antiderivative = 1.00,
number of steps used = 23, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used
= {3661, 1239, 203, 1179, 1168, 1162, 617, 204, 1165, 628} \[ \frac {\sqrt [4]{b} \left (\sqrt {a}-3 \sqrt {b}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} d (a+b)}+\frac {\sqrt [4]{b} \left (\sqrt {a}-\sqrt {b}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} d (a+b)^2}-\frac {\sqrt [4]{b} \left (\sqrt {a}-3 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}+1\right )}{8 \sqrt {2} a^{7/4} d (a+b)}-\frac {\sqrt [4]{b} \left (\sqrt {a}-\sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} d (a+b)^2}-\frac {\sqrt [4]{b} \left (\sqrt {a}+3 \sqrt {b}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )}{16 \sqrt {2} a^{7/4} d (a+b)}-\frac {\sqrt [4]{b} \left (\sqrt {a}+\sqrt {b}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )}{4 \sqrt {2} a^{3/4} d (a+b)^2}+\frac {\sqrt [4]{b} \left (\sqrt {a}+3 \sqrt {b}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )}{16 \sqrt {2} a^{7/4} d (a+b)}+\frac {\sqrt [4]{b} \left (\sqrt {a}+\sqrt {b}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )}{4 \sqrt {2} a^{3/4} d (a+b)^2}+\frac {b \tan (c+d x) \left (1-\tan ^2(c+d x)\right )}{4 a d (a+b) \left (a+b \tan ^4(c+d x)\right )}+\frac {x}{(a+b)^2} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*Tan[c + d*x]^4)^(-2),x]
[Out]
x/(a + b)^2 + ((Sqrt[a] - Sqrt[b])*b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[2]*a^(3
/4)*(a + b)^2*d) + ((Sqrt[a] - 3*Sqrt[b])*b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/(8*Sqrt[
2]*a^(7/4)*(a + b)*d) - ((Sqrt[a] - Sqrt[b])*b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/(2*Sq
rt[2]*a^(3/4)*(a + b)^2*d) - ((Sqrt[a] - 3*Sqrt[b])*b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)]
)/(8*Sqrt[2]*a^(7/4)*(a + b)*d) - ((Sqrt[a] + Sqrt[b])*b^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d
*x] + Sqrt[b]*Tan[c + d*x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)^2*d) - ((Sqrt[a] + 3*Sqrt[b])*b^(1/4)*Log[Sqrt[a] -
Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])/(16*Sqrt[2]*a^(7/4)*(a + b)*d) + ((Sqrt[a] + S
qrt[b])*b^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])/(4*Sqrt[2]*a^(3/
4)*(a + b)^2*d) + ((Sqrt[a] + 3*Sqrt[b])*b^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*
Tan[c + d*x]^2])/(16*Sqrt[2]*a^(7/4)*(a + b)*d) + (b*Tan[c + d*x]*(1 - Tan[c + d*x]^2))/(4*a*(a + b)*d*(a + b*
Tan[c + d*x]^4))
Rule 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rule 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 617
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 628
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Rule 1162
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Rule 1165
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Rule 1168
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[-(a*c)]
Rule 1179
Int[((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> -Simp[(x*(d + e*x^2)*(a + c*x^4)^(p + 1))/(
4*a*(p + 1)), x] + Dist[1/(4*a*(p + 1)), Int[Simp[d*(4*p + 5) + e*(4*p + 7)*x^2, x]*(a + c*x^4)^(p + 1), x], x
] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
Rule 1239
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a +
c*x^4)^p, x], x] /; FreeQ[{a, c, d, e, p, q}, x] && ((IntegerQ[p] && IntegerQ[q]) || IGtQ[p, 0])
Rule 3661
Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x]
, x]}, Dist[(c*ff)/f, Subst[Int[(a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ
[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \tan ^4(c+d x)\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (a+b x^4\right )^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{(a+b)^2 \left (1+x^2\right )}+\frac {b-b x^2}{(a+b) \left (a+b x^4\right )^2}+\frac {b-b x^2}{(a+b)^2 \left (a+b x^4\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{(a+b)^2 d}+\frac {\operatorname {Subst}\left (\int \frac {b-b x^2}{a+b x^4} \, dx,x,\tan (c+d x)\right )}{(a+b)^2 d}+\frac {\operatorname {Subst}\left (\int \frac {b-b x^2}{\left (a+b x^4\right )^2} \, dx,x,\tan (c+d x)\right )}{(a+b) d}\\ &=\frac {x}{(a+b)^2}+\frac {b \tan (c+d x) \left (1-\tan ^2(c+d x)\right )}{4 a (a+b) d \left (a+b \tan ^4(c+d x)\right )}-\frac {\left (1-\frac {\sqrt {b}}{\sqrt {a}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx,x,\tan (c+d x)\right )}{2 (a+b)^2 d}+\frac {\left (1+\frac {\sqrt {b}}{\sqrt {a}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx,x,\tan (c+d x)\right )}{2 (a+b)^2 d}-\frac {\operatorname {Subst}\left (\int \frac {-3 b+b x^2}{a+b x^4} \, dx,x,\tan (c+d x)\right )}{4 a (a+b) d}\\ &=\frac {x}{(a+b)^2}+\frac {b \tan (c+d x) \left (1-\tan ^2(c+d x)\right )}{4 a (a+b) d \left (a+b \tan ^4(c+d x)\right )}-\frac {\left (1-\frac {\sqrt {b}}{\sqrt {a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\tan (c+d x)\right )}{4 (a+b)^2 d}-\frac {\left (1-\frac {\sqrt {b}}{\sqrt {a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\tan (c+d x)\right )}{4 (a+b)^2 d}-\frac {\left (\left (\sqrt {a}+\sqrt {b}\right ) \sqrt [4]{b}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\tan (c+d x)\right )}{4 \sqrt {2} a^{3/4} (a+b)^2 d}-\frac {\left (\left (\sqrt {a}+\sqrt {b}\right ) \sqrt [4]{b}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\tan (c+d x)\right )}{4 \sqrt {2} a^{3/4} (a+b)^2 d}-\frac {\left (1-\frac {3 \sqrt {b}}{\sqrt {a}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx,x,\tan (c+d x)\right )}{8 a (a+b) d}+\frac {\left (1+\frac {3 \sqrt {b}}{\sqrt {a}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx,x,\tan (c+d x)\right )}{8 a (a+b) d}\\ &=\frac {x}{(a+b)^2}-\frac {\left (\sqrt {a}+\sqrt {b}\right ) \sqrt [4]{b} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {b} \tan ^2(c+d x)\right )}{4 \sqrt {2} a^{3/4} (a+b)^2 d}+\frac {\left (\sqrt {a}+\sqrt {b}\right ) \sqrt [4]{b} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {b} \tan ^2(c+d x)\right )}{4 \sqrt {2} a^{3/4} (a+b)^2 d}+\frac {b \tan (c+d x) \left (1-\tan ^2(c+d x)\right )}{4 a (a+b) d \left (a+b \tan ^4(c+d x)\right )}-\frac {\left (\left (\sqrt {a}-\sqrt {b}\right ) \sqrt [4]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} (a+b)^2 d}+\frac {\left (\left (\sqrt {a}-\sqrt {b}\right ) \sqrt [4]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} (a+b)^2 d}-\frac {\left (1-\frac {3 \sqrt {b}}{\sqrt {a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\tan (c+d x)\right )}{16 a (a+b) d}-\frac {\left (1-\frac {3 \sqrt {b}}{\sqrt {a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\tan (c+d x)\right )}{16 a (a+b) d}-\frac {\left (\left (\sqrt {a}+3 \sqrt {b}\right ) \sqrt [4]{b}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\tan (c+d x)\right )}{16 \sqrt {2} a^{7/4} (a+b) d}-\frac {\left (\left (\sqrt {a}+3 \sqrt {b}\right ) \sqrt [4]{b}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\tan (c+d x)\right )}{16 \sqrt {2} a^{7/4} (a+b) d}\\ &=\frac {x}{(a+b)^2}+\frac {\left (\sqrt {a}-\sqrt {b}\right ) \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} (a+b)^2 d}-\frac {\left (\sqrt {a}-\sqrt {b}\right ) \sqrt [4]{b} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} (a+b)^2 d}-\frac {\left (\sqrt {a}+\sqrt {b}\right ) \sqrt [4]{b} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {b} \tan ^2(c+d x)\right )}{4 \sqrt {2} a^{3/4} (a+b)^2 d}-\frac {\left (\sqrt {a}+3 \sqrt {b}\right ) \sqrt [4]{b} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {b} \tan ^2(c+d x)\right )}{16 \sqrt {2} a^{7/4} (a+b) d}+\frac {\left (\sqrt {a}+\sqrt {b}\right ) \sqrt [4]{b} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {b} \tan ^2(c+d x)\right )}{4 \sqrt {2} a^{3/4} (a+b)^2 d}+\frac {\left (\sqrt {a}+3 \sqrt {b}\right ) \sqrt [4]{b} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {b} \tan ^2(c+d x)\right )}{16 \sqrt {2} a^{7/4} (a+b) d}+\frac {b \tan (c+d x) \left (1-\tan ^2(c+d x)\right )}{4 a (a+b) d \left (a+b \tan ^4(c+d x)\right )}-\frac {\left (\left (\sqrt {a}-3 \sqrt {b}\right ) \sqrt [4]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} (a+b) d}+\frac {\left (\left (\sqrt {a}-3 \sqrt {b}\right ) \sqrt [4]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} (a+b) d}\\ &=\frac {x}{(a+b)^2}+\frac {\left (\sqrt {a}-\sqrt {b}\right ) \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} (a+b)^2 d}+\frac {\left (\sqrt {a}-3 \sqrt {b}\right ) \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} (a+b) d}-\frac {\left (\sqrt {a}-\sqrt {b}\right ) \sqrt [4]{b} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} (a+b)^2 d}-\frac {\left (\sqrt {a}-3 \sqrt {b}\right ) \sqrt [4]{b} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} (a+b) d}-\frac {\left (\sqrt {a}+\sqrt {b}\right ) \sqrt [4]{b} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {b} \tan ^2(c+d x)\right )}{4 \sqrt {2} a^{3/4} (a+b)^2 d}-\frac {\left (\sqrt {a}+3 \sqrt {b}\right ) \sqrt [4]{b} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {b} \tan ^2(c+d x)\right )}{16 \sqrt {2} a^{7/4} (a+b) d}+\frac {\left (\sqrt {a}+\sqrt {b}\right ) \sqrt [4]{b} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {b} \tan ^2(c+d x)\right )}{4 \sqrt {2} a^{3/4} (a+b)^2 d}+\frac {\left (\sqrt {a}+3 \sqrt {b}\right ) \sqrt [4]{b} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {b} \tan ^2(c+d x)\right )}{16 \sqrt {2} a^{7/4} (a+b) d}+\frac {b \tan (c+d x) \left (1-\tan ^2(c+d x)\right )}{4 a (a+b) d \left (a+b \tan ^4(c+d x)\right )}\\ \end {align*}
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Mathematica [C] time = 6.29, size = 598, normalized size = 0.92 \[ -\frac {b \tan ^3(c+d x) \, _2F_1\left (\frac {3}{4},2;\frac {7}{4};-\frac {b \tan ^4(c+d x)}{a}\right )}{3 a^2 d (a+b)}-\frac {3 \left (\frac {2 \left (\frac {\sqrt {2} b^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{\sqrt [4]{a}}-\frac {\sqrt {2} b^{3/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{a}}\right )}{\sqrt {a}}+\frac {\frac {\sqrt {2} b^{3/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )}{\sqrt [4]{a}}-\frac {\sqrt {2} b^{3/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )}{\sqrt [4]{a}}}{\sqrt {a}}\right )}{32 a d (a+b)}+\frac {\tan ^{-1}(\tan (c+d x))}{d (a+b)^2}+\frac {\left (\sqrt {a}-\sqrt {b}\right ) \left (\frac {\sqrt {2} \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{\sqrt [4]{a}}-\frac {\sqrt {2} \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{a}}\right )}{4 \sqrt {a} d (a+b)^2}+\frac {b \tan (c+d x)}{4 a d (a+b) \left (a+b \tan ^4(c+d x)\right )}-\frac {\left (\sqrt {a}+\sqrt {b}\right ) \left (\frac {\sqrt {2} \sqrt [4]{b} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )}{\sqrt [4]{a}}-\frac {\sqrt {2} \sqrt [4]{b} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )}{\sqrt [4]{a}}\right )}{8 \sqrt {a} d (a+b)^2} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Tan[c + d*x]^4)^(-2),x]
[Out]
ArcTan[Tan[c + d*x]]/((a + b)^2*d) + ((Sqrt[a] - Sqrt[b])*((Sqrt[2]*b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Tan[c
+ d*x])/a^(1/4)])/a^(1/4) - (Sqrt[2]*b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/a^(1/4)))/(4*
Sqrt[a]*(a + b)^2*d) - ((Sqrt[a] + Sqrt[b])*((Sqrt[2]*b^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*
x] + Sqrt[b]*Tan[c + d*x]^2])/a^(1/4) - (Sqrt[2]*b^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] +
Sqrt[b]*Tan[c + d*x]^2])/a^(1/4)))/(8*Sqrt[a]*(a + b)^2*d) - (3*((2*((Sqrt[2]*b^(3/4)*ArcTan[1 - (Sqrt[2]*b^(1
/4)*Tan[c + d*x])/a^(1/4)])/a^(1/4) - (Sqrt[2]*b^(3/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/a^(
1/4)))/Sqrt[a] + ((Sqrt[2]*b^(3/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2
])/a^(1/4) - (Sqrt[2]*b^(3/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])/a^
(1/4))/Sqrt[a]))/(32*a*(a + b)*d) - (b*Hypergeometric2F1[3/4, 2, 7/4, -((b*Tan[c + d*x]^4)/a)]*Tan[c + d*x]^3)
/(3*a^2*(a + b)*d) + (b*Tan[c + d*x])/(4*a*(a + b)*d*(a + b*Tan[c + d*x]^4))
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fricas [B] time = 1.15, size = 4291, normalized size = 6.62 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+tan(d*x+c)^4*b)^2,x, algorithm="fricas")
[Out]
1/32*(32*a*b*d*x*tan(d*x + c)^4 + 32*a^2*d*x - 8*(a*b + b^2)*tan(d*x + c)^3 + ((a^3*b + 2*a^2*b^2 + a*b^3)*d*t
an(d*x + c)^4 + (a^4 + 2*a^3*b + a^2*b^2)*d)*sqrt(((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2*sqrt(
-(625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b
+ 28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4)) + 70*a^2*b
+ 44*a*b^2 + 6*b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2))*log((625*a^5 - 750*a^4*b - 1376*a
^3*b^2 - 594*a^2*b^3 - 81*a*b^4 + (625*a^4*b - 750*a^3*b^2 - 1376*a^2*b^3 - 594*a*b^4 - 81*b^5)*tan(d*x + c)^2
+ 2*(2*(a^11 + 5*a^10*b + 10*a^9*b^2 + 10*a^8*b^3 + 5*a^7*b^4 + a^6*b^5)*d^3*sqrt(-(625*a^6*b - 1950*a^5*b^2
- 529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^
3 + 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4))*tan(d*x + c) + (125*a^7 + 5*a^6*b - 44
2*a^5*b^2 - 490*a^4*b^3 - 195*a^3*b^4 - 27*a^2*b^5)*d*tan(d*x + c))*sqrt(((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b
^3 + a^3*b^4)*d^2*sqrt(-(625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81
*b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7
*b^8)*d^4)) + 70*a^2*b + 44*a*b^2 + 6*b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2)) + ((25*a^9
+ 109*a^8*b + 186*a^7*b^2 + 154*a^6*b^3 + 61*a^5*b^4 + 9*a^4*b^5)*d^2*tan(d*x + c)^2 - (25*a^9 + 109*a^8*b +
186*a^7*b^2 + 154*a^6*b^3 + 61*a^5*b^4 + 9*a^4*b^5)*d^2)*sqrt(-(625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*
a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*
a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4)))/(tan(d*x + c)^2 + 1)) - ((a^3*b + 2*a^2*b^2 + a*b^3)*d*tan
(d*x + c)^4 + (a^4 + 2*a^3*b + a^2*b^2)*d)*sqrt(((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2*sqrt(-(
625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b +
28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4)) + 70*a^2*b +
44*a*b^2 + 6*b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2))*log((625*a^5 - 750*a^4*b - 1376*a^3
*b^2 - 594*a^2*b^3 - 81*a*b^4 + (625*a^4*b - 750*a^3*b^2 - 1376*a^2*b^3 - 594*a*b^4 - 81*b^5)*tan(d*x + c)^2 -
2*(2*(a^11 + 5*a^10*b + 10*a^9*b^2 + 10*a^8*b^3 + 5*a^7*b^4 + a^6*b^5)*d^3*sqrt(-(625*a^6*b - 1950*a^5*b^2 -
529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3
+ 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4))*tan(d*x + c) + (125*a^7 + 5*a^6*b - 442*
a^5*b^2 - 490*a^4*b^3 - 195*a^3*b^4 - 27*a^2*b^5)*d*tan(d*x + c))*sqrt(((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3
+ a^3*b^4)*d^2*sqrt(-(625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b
^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b
^8)*d^4)) + 70*a^2*b + 44*a*b^2 + 6*b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2)) + ((25*a^9 +
109*a^8*b + 186*a^7*b^2 + 154*a^6*b^3 + 61*a^5*b^4 + 9*a^4*b^5)*d^2*tan(d*x + c)^2 - (25*a^9 + 109*a^8*b + 18
6*a^7*b^2 + 154*a^6*b^3 + 61*a^5*b^4 + 9*a^4*b^5)*d^2)*sqrt(-(625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*a^
3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*a^
10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4)))/(tan(d*x + c)^2 + 1)) - ((a^3*b + 2*a^2*b^2 + a*b^3)*d*tan(d
*x + c)^4 + (a^4 + 2*a^3*b + a^2*b^2)*d)*sqrt(-((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2*sqrt(-(6
25*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b +
28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4)) - 70*a^2*b - 4
4*a*b^2 - 6*b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2))*log(-(625*a^5 - 750*a^4*b - 1376*a^3
*b^2 - 594*a^2*b^3 - 81*a*b^4 + (625*a^4*b - 750*a^3*b^2 - 1376*a^2*b^3 - 594*a*b^4 - 81*b^5)*tan(d*x + c)^2 +
2*(2*(a^11 + 5*a^10*b + 10*a^9*b^2 + 10*a^8*b^3 + 5*a^7*b^4 + a^6*b^5)*d^3*sqrt(-(625*a^6*b - 1950*a^5*b^2 -
529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3
+ 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4))*tan(d*x + c) - (125*a^7 + 5*a^6*b - 442*
a^5*b^2 - 490*a^4*b^3 - 195*a^3*b^4 - 27*a^2*b^5)*d*tan(d*x + c))*sqrt(-((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^
3 + a^3*b^4)*d^2*sqrt(-(625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*
b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*
b^8)*d^4)) - 70*a^2*b - 44*a*b^2 - 6*b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2)) - ((25*a^9
+ 109*a^8*b + 186*a^7*b^2 + 154*a^6*b^3 + 61*a^5*b^4 + 9*a^4*b^5)*d^2*tan(d*x + c)^2 - (25*a^9 + 109*a^8*b + 1
86*a^7*b^2 + 154*a^6*b^3 + 61*a^5*b^4 + 9*a^4*b^5)*d^2)*sqrt(-(625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*a
^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*a
^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4)))/(tan(d*x + c)^2 + 1)) + ((a^3*b + 2*a^2*b^2 + a*b^3)*d*tan(
d*x + c)^4 + (a^4 + 2*a^3*b + a^2*b^2)*d)*sqrt(-((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2*sqrt(-(
625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b +
28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4)) - 70*a^2*b -
44*a*b^2 - 6*b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2))*log(-(625*a^5 - 750*a^4*b - 1376*a^
3*b^2 - 594*a^2*b^3 - 81*a*b^4 + (625*a^4*b - 750*a^3*b^2 - 1376*a^2*b^3 - 594*a*b^4 - 81*b^5)*tan(d*x + c)^2
- 2*(2*(a^11 + 5*a^10*b + 10*a^9*b^2 + 10*a^8*b^3 + 5*a^7*b^4 + a^6*b^5)*d^3*sqrt(-(625*a^6*b - 1950*a^5*b^2 -
529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3
+ 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4))*tan(d*x + c) - (125*a^7 + 5*a^6*b - 442
*a^5*b^2 - 490*a^4*b^3 - 195*a^3*b^4 - 27*a^2*b^5)*d*tan(d*x + c))*sqrt(-((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b
^3 + a^3*b^4)*d^2*sqrt(-(625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81
*b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7
*b^8)*d^4)) - 70*a^2*b - 44*a*b^2 - 6*b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2)) - ((25*a^9
+ 109*a^8*b + 186*a^7*b^2 + 154*a^6*b^3 + 61*a^5*b^4 + 9*a^4*b^5)*d^2*tan(d*x + c)^2 - (25*a^9 + 109*a^8*b +
186*a^7*b^2 + 154*a^6*b^3 + 61*a^5*b^4 + 9*a^4*b^5)*d^2)*sqrt(-(625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*
a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*
a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4)))/(tan(d*x + c)^2 + 1)) + 8*(a*b + b^2)*tan(d*x + c))/((a^3*
b + 2*a^2*b^2 + a*b^3)*d*tan(d*x + c)^4 + (a^4 + 2*a^3*b + a^2*b^2)*d)
________________________________________________________________________________________
giac [A] time = 3.41, size = 517, normalized size = 0.80 \[ -\frac {\frac {2 \, {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \tan \left (d x + c\right )\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )} {\left (\left (a b^{3}\right )^{\frac {3}{4}} {\left (5 \, a + b\right )} - \left (a b^{3}\right )^{\frac {1}{4}} {\left (7 \, a b^{2} + 3 \, b^{3}\right )}\right )}}{\sqrt {2} a^{4} b^{2} + 2 \, \sqrt {2} a^{3} b^{3} + \sqrt {2} a^{2} b^{4}} + \frac {2 \, {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \tan \left (d x + c\right )\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )} {\left (\left (a b^{3}\right )^{\frac {3}{4}} {\left (5 \, a + b\right )} - \left (a b^{3}\right )^{\frac {1}{4}} {\left (7 \, a b^{2} + 3 \, b^{3}\right )}\right )}}{\sqrt {2} a^{4} b^{2} + 2 \, \sqrt {2} a^{3} b^{3} + \sqrt {2} a^{2} b^{4}} - \frac {{\left (\left (a b^{3}\right )^{\frac {3}{4}} {\left (5 \, a + b\right )} + \left (a b^{3}\right )^{\frac {1}{4}} {\left (7 \, a b^{2} + 3 \, b^{3}\right )}\right )} \log \left (\tan \left (d x + c\right )^{2} + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} \tan \left (d x + c\right ) + \sqrt {\frac {a}{b}}\right )}{\sqrt {2} a^{4} b^{2} + 2 \, \sqrt {2} a^{3} b^{3} + \sqrt {2} a^{2} b^{4}} + \frac {{\left (\left (a b^{3}\right )^{\frac {3}{4}} {\left (5 \, a + b\right )} + \left (a b^{3}\right )^{\frac {1}{4}} {\left (7 \, a b^{2} + 3 \, b^{3}\right )}\right )} \log \left (\tan \left (d x + c\right )^{2} - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} \tan \left (d x + c\right ) + \sqrt {\frac {a}{b}}\right )}{\sqrt {2} a^{4} b^{2} + 2 \, \sqrt {2} a^{3} b^{3} + \sqrt {2} a^{2} b^{4}} - \frac {16 \, {\left (d x + c\right )}}{a^{2} + 2 \, a b + b^{2}} + \frac {4 \, {\left (b \tan \left (d x + c\right )^{3} - b \tan \left (d x + c\right )\right )}}{{\left (b \tan \left (d x + c\right )^{4} + a\right )} {\left (a^{2} + a b\right )}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+tan(d*x+c)^4*b)^2,x, algorithm="giac")
[Out]
-1/16*(2*(pi*floor((d*x + c)/pi + 1/2) + arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*tan(d*x + c))/(a/b)^(1/4)
))*((a*b^3)^(3/4)*(5*a + b) - (a*b^3)^(1/4)*(7*a*b^2 + 3*b^3))/(sqrt(2)*a^4*b^2 + 2*sqrt(2)*a^3*b^3 + sqrt(2)*
a^2*b^4) + 2*(pi*floor((d*x + c)/pi + 1/2) + arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*tan(d*x + c))/(a/b)^
(1/4)))*((a*b^3)^(3/4)*(5*a + b) - (a*b^3)^(1/4)*(7*a*b^2 + 3*b^3))/(sqrt(2)*a^4*b^2 + 2*sqrt(2)*a^3*b^3 + sqr
t(2)*a^2*b^4) - ((a*b^3)^(3/4)*(5*a + b) + (a*b^3)^(1/4)*(7*a*b^2 + 3*b^3))*log(tan(d*x + c)^2 + sqrt(2)*(a/b)
^(1/4)*tan(d*x + c) + sqrt(a/b))/(sqrt(2)*a^4*b^2 + 2*sqrt(2)*a^3*b^3 + sqrt(2)*a^2*b^4) + ((a*b^3)^(3/4)*(5*a
+ b) + (a*b^3)^(1/4)*(7*a*b^2 + 3*b^3))*log(tan(d*x + c)^2 - sqrt(2)*(a/b)^(1/4)*tan(d*x + c) + sqrt(a/b))/(s
qrt(2)*a^4*b^2 + 2*sqrt(2)*a^3*b^3 + sqrt(2)*a^2*b^4) - 16*(d*x + c)/(a^2 + 2*a*b + b^2) + 4*(b*tan(d*x + c)^3
- b*tan(d*x + c))/((b*tan(d*x + c)^4 + a)*(a^2 + a*b)))/d
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maple [A] time = 0.21, size = 886, normalized size = 1.37 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+b*tan(d*x+c)^4)^2,x)
[Out]
-1/4/d*b/(a+b)^2/(a+b*tan(d*x+c)^4)*tan(d*x+c)^3-1/4/d*b^2/(a+b)^2/(a+b*tan(d*x+c)^4)/a*tan(d*x+c)^3+1/4/d*b/(
a+b)^2/(a+b*tan(d*x+c)^4)*tan(d*x+c)+1/4/d*b^2/(a+b)^2/(a+b*tan(d*x+c)^4)/a*tan(d*x+c)+7/32/d*b/(a+b)^2/a*(1/b
*a)^(1/4)*2^(1/2)*ln((tan(d*x+c)^2+(1/b*a)^(1/4)*tan(d*x+c)*2^(1/2)+(1/b*a)^(1/2))/(tan(d*x+c)^2-(1/b*a)^(1/4)
*tan(d*x+c)*2^(1/2)+(1/b*a)^(1/2)))+3/32/d*b^2/(a+b)^2/a^2*(1/b*a)^(1/4)*2^(1/2)*ln((tan(d*x+c)^2+(1/b*a)^(1/4
)*tan(d*x+c)*2^(1/2)+(1/b*a)^(1/2))/(tan(d*x+c)^2-(1/b*a)^(1/4)*tan(d*x+c)*2^(1/2)+(1/b*a)^(1/2)))-7/16/d*b/(a
+b)^2/a*(1/b*a)^(1/4)*2^(1/2)*arctan(-2^(1/2)/(1/b*a)^(1/4)*tan(d*x+c)+1)-3/16/d*b^2/(a+b)^2/a^2*(1/b*a)^(1/4)
*2^(1/2)*arctan(-2^(1/2)/(1/b*a)^(1/4)*tan(d*x+c)+1)+7/16/d*b/(a+b)^2/a*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(
1/b*a)^(1/4)*tan(d*x+c)+1)+3/16/d*b^2/(a+b)^2/a^2*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*tan(d*x+c
)+1)-5/32/d/(a+b)^2/(1/b*a)^(1/4)*2^(1/2)*ln((tan(d*x+c)^2-(1/b*a)^(1/4)*tan(d*x+c)*2^(1/2)+(1/b*a)^(1/2))/(ta
n(d*x+c)^2+(1/b*a)^(1/4)*tan(d*x+c)*2^(1/2)+(1/b*a)^(1/2)))-1/32/d*b/(a+b)^2/a/(1/b*a)^(1/4)*2^(1/2)*ln((tan(d
*x+c)^2-(1/b*a)^(1/4)*tan(d*x+c)*2^(1/2)+(1/b*a)^(1/2))/(tan(d*x+c)^2+(1/b*a)^(1/4)*tan(d*x+c)*2^(1/2)+(1/b*a)
^(1/2)))+5/16/d/(a+b)^2/(1/b*a)^(1/4)*2^(1/2)*arctan(-2^(1/2)/(1/b*a)^(1/4)*tan(d*x+c)+1)+1/16/d*b/(a+b)^2/a/(
1/b*a)^(1/4)*2^(1/2)*arctan(-2^(1/2)/(1/b*a)^(1/4)*tan(d*x+c)+1)-5/16/d/(a+b)^2/(1/b*a)^(1/4)*2^(1/2)*arctan(2
^(1/2)/(1/b*a)^(1/4)*tan(d*x+c)+1)-1/16/d*b/(a+b)^2/a/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*tan(d
*x+c)+1)+1/d/(a+b)^2*arctan(tan(d*x+c))
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maxima [A] time = 0.80, size = 394, normalized size = 0.61 \[ -\frac {\frac {b {\left (\frac {2 \, \sqrt {2} {\left (b {\left (\sqrt {a} - 3 \, \sqrt {b}\right )} + 5 \, a^{\frac {3}{2}} - 7 \, a \sqrt {b}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} \tan \left (d x + c\right ) + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} {\left (b {\left (\sqrt {a} - 3 \, \sqrt {b}\right )} + 5 \, a^{\frac {3}{2}} - 7 \, a \sqrt {b}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} \tan \left (d x + c\right ) - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} {\left (b {\left (\sqrt {a} + 3 \, \sqrt {b}\right )} + 5 \, a^{\frac {3}{2}} + 7 \, a \sqrt {b}\right )} \log \left (\sqrt {b} \tan \left (d x + c\right )^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \tan \left (d x + c\right ) + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (b {\left (\sqrt {a} + 3 \, \sqrt {b}\right )} + 5 \, a^{\frac {3}{2}} + 7 \, a \sqrt {b}\right )} \log \left (\sqrt {b} \tan \left (d x + c\right )^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \tan \left (d x + c\right ) + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}}\right )}}{a^{3} + 2 \, a^{2} b + a b^{2}} + \frac {8 \, {\left (b \tan \left (d x + c\right )^{3} - b \tan \left (d x + c\right )\right )}}{{\left (a^{2} b + a b^{2}\right )} \tan \left (d x + c\right )^{4} + a^{3} + a^{2} b} - \frac {32 \, {\left (d x + c\right )}}{a^{2} + 2 \, a b + b^{2}}}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+tan(d*x+c)^4*b)^2,x, algorithm="maxima")
[Out]
-1/32*(b*(2*sqrt(2)*(b*(sqrt(a) - 3*sqrt(b)) + 5*a^(3/2) - 7*a*sqrt(b))*arctan(1/2*sqrt(2)*(2*sqrt(b)*tan(d*x
+ c) + sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*(b*
(sqrt(a) - 3*sqrt(b)) + 5*a^(3/2) - 7*a*sqrt(b))*arctan(1/2*sqrt(2)*(2*sqrt(b)*tan(d*x + c) - sqrt(2)*a^(1/4)*
b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - sqrt(2)*(b*(sqrt(a) + 3*sqrt(b)) + 5
*a^(3/2) + 7*a*sqrt(b))*log(sqrt(b)*tan(d*x + c)^2 + sqrt(2)*a^(1/4)*b^(1/4)*tan(d*x + c) + sqrt(a))/(a^(3/4)*
b^(3/4)) + sqrt(2)*(b*(sqrt(a) + 3*sqrt(b)) + 5*a^(3/2) + 7*a*sqrt(b))*log(sqrt(b)*tan(d*x + c)^2 - sqrt(2)*a^
(1/4)*b^(1/4)*tan(d*x + c) + sqrt(a))/(a^(3/4)*b^(3/4)))/(a^3 + 2*a^2*b + a*b^2) + 8*(b*tan(d*x + c)^3 - b*tan
(d*x + c))/((a^2*b + a*b^2)*tan(d*x + c)^4 + a^3 + a^2*b) - 32*(d*x + c)/(a^2 + 2*a*b + b^2))/d
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mupad [B] time = 15.69, size = 11516, normalized size = 17.77 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a + b*tan(c + d*x)^4)^2,x)
[Out]
((b*tan(c + d*x))/(4*a*(a + b)) - (b*tan(c + d*x)^3)/(4*a*(a + b)))/(d*(a + b*tan(c + d*x)^4)) - (2*atan((((((
((((960*a^7*b^8 - 224*a^5*b^10 - 144*a^6*b^9 - 48*a^4*b^11 + 2480*a^8*b^7 + 2592*a^9*b^6 + 1296*a^10*b^5 + 256
*a^11*b^4)*1i)/(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2) - (tan(c + d*x)*(65536*a^6*b^11 + 327680*a^7*
b^10 + 589824*a^8*b^9 + 327680*a^9*b^8 - 327680*a^10*b^7 - 589824*a^11*b^6 - 327680*a^12*b^5 - 65536*a^13*b^4)
)/(128*(4*a*b + 2*a^2 + 2*b^2)*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*1i)/(4*a*b + 2*a^2 + 2*b^2)
- (tan(c + d*x)*(1152*a^2*b^11 + 7936*a^3*b^10 + 20352*a^4*b^9 + 8704*a^5*b^8 - 66688*a^6*b^7 - 110848*a^7*b^
6 - 49024*a^8*b^5)*1i)/(128*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*1i)/(4*a*b + 2*a^2 + 2*b^2) -
(((45*a*b^10)/16 + (305*a^2*b^9)/16 + (385*a^3*b^8)/8 + (657*a^4*b^7)/8 + (2081*a^5*b^6)/16 + (1277*a^6*b^5)/1
6)*1i)/(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2))/(4*a*b + 2*a^2 + 2*b^2) - (tan(c + d*x)*(612*a*b^8 +
81*b^9 + 1894*a^2*b^7 + 2532*a^3*b^6 + 1425*a^4*b^5))/(128*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2))
)/(4*a*b + 2*a^2 + 2*b^2) - ((((((((960*a^7*b^8 - 224*a^5*b^10 - 144*a^6*b^9 - 48*a^4*b^11 + 2480*a^8*b^7 + 25
92*a^9*b^6 + 1296*a^10*b^5 + 256*a^11*b^4)*1i)/(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2) + (tan(c + d*
x)*(65536*a^6*b^11 + 327680*a^7*b^10 + 589824*a^8*b^9 + 327680*a^9*b^8 - 327680*a^10*b^7 - 589824*a^11*b^6 - 3
27680*a^12*b^5 - 65536*a^13*b^4))/(128*(4*a*b + 2*a^2 + 2*b^2)*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^
2)))*1i)/(4*a*b + 2*a^2 + 2*b^2) + (tan(c + d*x)*(1152*a^2*b^11 + 7936*a^3*b^10 + 20352*a^4*b^9 + 8704*a^5*b^8
- 66688*a^6*b^7 - 110848*a^7*b^6 - 49024*a^8*b^5)*1i)/(128*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2))
)*1i)/(4*a*b + 2*a^2 + 2*b^2) - (((45*a*b^10)/16 + (305*a^2*b^9)/16 + (385*a^3*b^8)/8 + (657*a^4*b^7)/8 + (208
1*a^5*b^6)/16 + (1277*a^6*b^5)/16)*1i)/(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2))/(4*a*b + 2*a^2 + 2*b
^2) + (tan(c + d*x)*(612*a*b^8 + 81*b^9 + 1894*a^2*b^7 + 2532*a^3*b^6 + 1425*a^4*b^5))/(128*(4*a^7*b + a^8 + a
^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))/(4*a*b + 2*a^2 + 2*b^2))/((((((((((960*a^7*b^8 - 224*a^5*b^10 - 144*a^6*b^9
- 48*a^4*b^11 + 2480*a^8*b^7 + 2592*a^9*b^6 + 1296*a^10*b^5 + 256*a^11*b^4)*1i)/(4*a^7*b + a^8 + a^4*b^4 + 4*a
^5*b^3 + 6*a^6*b^2) - (tan(c + d*x)*(65536*a^6*b^11 + 327680*a^7*b^10 + 589824*a^8*b^9 + 327680*a^9*b^8 - 3276
80*a^10*b^7 - 589824*a^11*b^6 - 327680*a^12*b^5 - 65536*a^13*b^4))/(128*(4*a*b + 2*a^2 + 2*b^2)*(4*a^7*b + a^8
+ a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*1i)/(4*a*b + 2*a^2 + 2*b^2) - (tan(c + d*x)*(1152*a^2*b^11 + 7936*a^3*b^
10 + 20352*a^4*b^9 + 8704*a^5*b^8 - 66688*a^6*b^7 - 110848*a^7*b^6 - 49024*a^8*b^5)*1i)/(128*(4*a^7*b + a^8 +
a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*1i)/(4*a*b + 2*a^2 + 2*b^2) - (((45*a*b^10)/16 + (305*a^2*b^9)/16 + (385*a^
3*b^8)/8 + (657*a^4*b^7)/8 + (2081*a^5*b^6)/16 + (1277*a^6*b^5)/16)*1i)/(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 +
6*a^6*b^2))*1i)/(4*a*b + 2*a^2 + 2*b^2) - (tan(c + d*x)*(612*a*b^8 + 81*b^9 + 1894*a^2*b^7 + 2532*a^3*b^6 + 1
425*a^4*b^5)*1i)/(128*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))/(4*a*b + 2*a^2 + 2*b^2) + (((((((((9
60*a^7*b^8 - 224*a^5*b^10 - 144*a^6*b^9 - 48*a^4*b^11 + 2480*a^8*b^7 + 2592*a^9*b^6 + 1296*a^10*b^5 + 256*a^11
*b^4)*1i)/(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2) + (tan(c + d*x)*(65536*a^6*b^11 + 327680*a^7*b^10
+ 589824*a^8*b^9 + 327680*a^9*b^8 - 327680*a^10*b^7 - 589824*a^11*b^6 - 327680*a^12*b^5 - 65536*a^13*b^4))/(12
8*(4*a*b + 2*a^2 + 2*b^2)*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*1i)/(4*a*b + 2*a^2 + 2*b^2) + (t
an(c + d*x)*(1152*a^2*b^11 + 7936*a^3*b^10 + 20352*a^4*b^9 + 8704*a^5*b^8 - 66688*a^6*b^7 - 110848*a^7*b^6 - 4
9024*a^8*b^5)*1i)/(128*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*1i)/(4*a*b + 2*a^2 + 2*b^2) - (((45
*a*b^10)/16 + (305*a^2*b^9)/16 + (385*a^3*b^8)/8 + (657*a^4*b^7)/8 + (2081*a^5*b^6)/16 + (1277*a^6*b^5)/16)*1i
)/(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2))*1i)/(4*a*b + 2*a^2 + 2*b^2) + (tan(c + d*x)*(612*a*b^8 +
81*b^9 + 1894*a^2*b^7 + 2532*a^3*b^6 + 1425*a^4*b^5)*1i)/(128*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2
)))/(4*a*b + 2*a^2 + 2*b^2) + ((135*a*b^6)/64 + (81*b^7)/128 + (125*a^2*b^5)/128)/(4*a^7*b + a^8 + a^4*b^4 + 4
*a^5*b^3 + 6*a^6*b^2))))/(d*(4*a*b + 2*a^2 + 2*b^2)) + (atan(((((((245760*a^7*b^8 - 57344*a^5*b^10 - 36864*a^6
*b^9 - 12288*a^4*b^11 + 634880*a^8*b^7 + 663552*a^9*b^6 + 331776*a^10*b^5 + 65536*a^11*b^4)/(256*(4*a^7*b + a^
8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)) - (tan(c + d*x)*((9*b^3*(-a^7*b)^(1/2) - 25*a^3*(-a^7*b)^(1/2) + 70*a^6*
b + 6*a^4*b^3 + 44*a^5*b^2 + 41*a*b^2*(-a^7*b)^(1/2) + 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^
4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2)*(65536*a^6*b^11 + 327680*a^7*b^10 + 589824*a^8*b^9 + 327680*a^9*b^8 - 32768
0*a^10*b^7 - 589824*a^11*b^6 - 327680*a^12*b^5 - 65536*a^13*b^4))/(128*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 +
6*a^6*b^2)))*((9*b^3*(-a^7*b)^(1/2) - 25*a^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 + 41*a*b^2*(-a
^7*b)^(1/2) + 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2) + (tan
(c + d*x)*(1152*a^2*b^11 + 7936*a^3*b^10 + 20352*a^4*b^9 + 8704*a^5*b^8 - 66688*a^6*b^7 - 110848*a^7*b^6 - 490
24*a^8*b^5))/(128*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*((9*b^3*(-a^7*b)^(1/2) - 25*a^3*(-a^7*b)
^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 + 41*a*b^2*(-a^7*b)^(1/2) + 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b
+ a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2) - (720*a*b^10 + 4880*a^2*b^9 + 12320*a^3*b^8 + 21024*a^4*b^
7 + 33296*a^5*b^6 + 20432*a^6*b^5)/(256*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*((9*b^3*(-a^7*b)^(
1/2) - 25*a^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 + 41*a*b^2*(-a^7*b)^(1/2) + 39*a^2*b*(-a^7*b)
^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2) + (tan(c + d*x)*(612*a*b^8 + 81*b^9 +
1894*a^2*b^7 + 2532*a^3*b^6 + 1425*a^4*b^5))/(128*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*((9*b^3
*(-a^7*b)^(1/2) - 25*a^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 + 41*a*b^2*(-a^7*b)^(1/2) + 39*a^2
*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2)*1i - (((((245760*a^7*b^8 -
57344*a^5*b^10 - 36864*a^6*b^9 - 12288*a^4*b^11 + 634880*a^8*b^7 + 663552*a^9*b^6 + 331776*a^10*b^5 + 65536*a
^11*b^4)/(256*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)) + (tan(c + d*x)*((9*b^3*(-a^7*b)^(1/2) - 25*a
^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 + 41*a*b^2*(-a^7*b)^(1/2) + 39*a^2*b*(-a^7*b)^(1/2))/(25
6*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2)*(65536*a^6*b^11 + 327680*a^7*b^10 + 589824*a^8*b
^9 + 327680*a^9*b^8 - 327680*a^10*b^7 - 589824*a^11*b^6 - 327680*a^12*b^5 - 65536*a^13*b^4))/(128*(4*a^7*b + a
^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*((9*b^3*(-a^7*b)^(1/2) - 25*a^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3
+ 44*a^5*b^2 + 41*a*b^2*(-a^7*b)^(1/2) + 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3
+ 6*a^9*b^2)))^(1/2) - (tan(c + d*x)*(1152*a^2*b^11 + 7936*a^3*b^10 + 20352*a^4*b^9 + 8704*a^5*b^8 - 66688*a^6
*b^7 - 110848*a^7*b^6 - 49024*a^8*b^5))/(128*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*((9*b^3*(-a^7
*b)^(1/2) - 25*a^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 + 41*a*b^2*(-a^7*b)^(1/2) + 39*a^2*b*(-a
^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2) - (720*a*b^10 + 4880*a^2*b^9 + 1
2320*a^3*b^8 + 21024*a^4*b^7 + 33296*a^5*b^6 + 20432*a^6*b^5)/(256*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^
6*b^2)))*((9*b^3*(-a^7*b)^(1/2) - 25*a^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 + 41*a*b^2*(-a^7*b
)^(1/2) + 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2) - (tan(c +
d*x)*(612*a*b^8 + 81*b^9 + 1894*a^2*b^7 + 2532*a^3*b^6 + 1425*a^4*b^5))/(128*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5
*b^3 + 6*a^6*b^2)))*((9*b^3*(-a^7*b)^(1/2) - 25*a^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 + 41*a*
b^2*(-a^7*b)^(1/2) + 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2)
*1i)/((((((245760*a^7*b^8 - 57344*a^5*b^10 - 36864*a^6*b^9 - 12288*a^4*b^11 + 634880*a^8*b^7 + 663552*a^9*b^6
+ 331776*a^10*b^5 + 65536*a^11*b^4)/(256*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)) - (tan(c + d*x)*((
9*b^3*(-a^7*b)^(1/2) - 25*a^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 + 41*a*b^2*(-a^7*b)^(1/2) + 3
9*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2)*(65536*a^6*b^11 + 327
680*a^7*b^10 + 589824*a^8*b^9 + 327680*a^9*b^8 - 327680*a^10*b^7 - 589824*a^11*b^6 - 327680*a^12*b^5 - 65536*a
^13*b^4))/(128*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*((9*b^3*(-a^7*b)^(1/2) - 25*a^3*(-a^7*b)^(1
/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 + 41*a*b^2*(-a^7*b)^(1/2) + 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b +
a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2) + (tan(c + d*x)*(1152*a^2*b^11 + 7936*a^3*b^10 + 20352*a^4*b^9
+ 8704*a^5*b^8 - 66688*a^6*b^7 - 110848*a^7*b^6 - 49024*a^8*b^5))/(128*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 +
6*a^6*b^2)))*((9*b^3*(-a^7*b)^(1/2) - 25*a^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 + 41*a*b^2*(-
a^7*b)^(1/2) + 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2) - (72
0*a*b^10 + 4880*a^2*b^9 + 12320*a^3*b^8 + 21024*a^4*b^7 + 33296*a^5*b^6 + 20432*a^6*b^5)/(256*(4*a^7*b + a^8 +
a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*((9*b^3*(-a^7*b)^(1/2) - 25*a^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44
*a^5*b^2 + 41*a*b^2*(-a^7*b)^(1/2) + 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*
a^9*b^2)))^(1/2) + (tan(c + d*x)*(612*a*b^8 + 81*b^9 + 1894*a^2*b^7 + 2532*a^3*b^6 + 1425*a^4*b^5))/(128*(4*a^
7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*((9*b^3*(-a^7*b)^(1/2) - 25*a^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a
^4*b^3 + 44*a^5*b^2 + 41*a*b^2*(-a^7*b)^(1/2) + 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a
^8*b^3 + 6*a^9*b^2)))^(1/2) + (((((245760*a^7*b^8 - 57344*a^5*b^10 - 36864*a^6*b^9 - 12288*a^4*b^11 + 634880*a
^8*b^7 + 663552*a^9*b^6 + 331776*a^10*b^5 + 65536*a^11*b^4)/(256*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*
b^2)) + (tan(c + d*x)*((9*b^3*(-a^7*b)^(1/2) - 25*a^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 + 41*
a*b^2*(-a^7*b)^(1/2) + 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/
2)*(65536*a^6*b^11 + 327680*a^7*b^10 + 589824*a^8*b^9 + 327680*a^9*b^8 - 327680*a^10*b^7 - 589824*a^11*b^6 - 3
27680*a^12*b^5 - 65536*a^13*b^4))/(128*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*((9*b^3*(-a^7*b)^(1
/2) - 25*a^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 + 41*a*b^2*(-a^7*b)^(1/2) + 39*a^2*b*(-a^7*b)^
(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2) - (tan(c + d*x)*(1152*a^2*b^11 + 7936*
a^3*b^10 + 20352*a^4*b^9 + 8704*a^5*b^8 - 66688*a^6*b^7 - 110848*a^7*b^6 - 49024*a^8*b^5))/(128*(4*a^7*b + a^8
+ a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*((9*b^3*(-a^7*b)^(1/2) - 25*a^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 +
44*a^5*b^2 + 41*a*b^2*(-a^7*b)^(1/2) + 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 +
6*a^9*b^2)))^(1/2) - (720*a*b^10 + 4880*a^2*b^9 + 12320*a^3*b^8 + 21024*a^4*b^7 + 33296*a^5*b^6 + 20432*a^6*b^
5)/(256*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*((9*b^3*(-a^7*b)^(1/2) - 25*a^3*(-a^7*b)^(1/2) + 7
0*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 + 41*a*b^2*(-a^7*b)^(1/2) + 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 +
a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2) - (tan(c + d*x)*(612*a*b^8 + 81*b^9 + 1894*a^2*b^7 + 2532*a^3*b^6 + 1
425*a^4*b^5))/(128*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*((9*b^3*(-a^7*b)^(1/2) - 25*a^3*(-a^7*b
)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 + 41*a*b^2*(-a^7*b)^(1/2) + 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*
b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2) + (270*a*b^6 + 81*b^7 + 125*a^2*b^5)/(128*(4*a^7*b + a^8 +
a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2))))*((9*b^3*(-a^7*b)^(1/2) - 25*a^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 4
4*a^5*b^2 + 41*a*b^2*(-a^7*b)^(1/2) + 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6
*a^9*b^2)))^(1/2)*2i)/d + (atan(((((((245760*a^7*b^8 - 57344*a^5*b^10 - 36864*a^6*b^9 - 12288*a^4*b^11 + 63488
0*a^8*b^7 + 663552*a^9*b^6 + 331776*a^10*b^5 + 65536*a^11*b^4)/(256*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a
^6*b^2)) - (tan(c + d*x)*((25*a^3*(-a^7*b)^(1/2) - 9*b^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 -
41*a*b^2*(-a^7*b)^(1/2) - 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^
(1/2)*(65536*a^6*b^11 + 327680*a^7*b^10 + 589824*a^8*b^9 + 327680*a^9*b^8 - 327680*a^10*b^7 - 589824*a^11*b^6
- 327680*a^12*b^5 - 65536*a^13*b^4))/(128*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*((25*a^3*(-a^7*b
)^(1/2) - 9*b^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 - 41*a*b^2*(-a^7*b)^(1/2) - 39*a^2*b*(-a^7*
b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2) + (tan(c + d*x)*(1152*a^2*b^11 + 79
36*a^3*b^10 + 20352*a^4*b^9 + 8704*a^5*b^8 - 66688*a^6*b^7 - 110848*a^7*b^6 - 49024*a^8*b^5))/(128*(4*a^7*b +
a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*((25*a^3*(-a^7*b)^(1/2) - 9*b^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3
+ 44*a^5*b^2 - 41*a*b^2*(-a^7*b)^(1/2) - 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3
+ 6*a^9*b^2)))^(1/2) - (720*a*b^10 + 4880*a^2*b^9 + 12320*a^3*b^8 + 21024*a^4*b^7 + 33296*a^5*b^6 + 20432*a^6
*b^5)/(256*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*((25*a^3*(-a^7*b)^(1/2) - 9*b^3*(-a^7*b)^(1/2)
+ 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 - 41*a*b^2*(-a^7*b)^(1/2) - 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11
+ a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2) + (tan(c + d*x)*(612*a*b^8 + 81*b^9 + 1894*a^2*b^7 + 2532*a^3*b^6
+ 1425*a^4*b^5))/(128*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*((25*a^3*(-a^7*b)^(1/2) - 9*b^3*(-a^
7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 - 41*a*b^2*(-a^7*b)^(1/2) - 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^
10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2)*1i - (((((245760*a^7*b^8 - 57344*a^5*b^10 - 36864*a^6*b
^9 - 12288*a^4*b^11 + 634880*a^8*b^7 + 663552*a^9*b^6 + 331776*a^10*b^5 + 65536*a^11*b^4)/(256*(4*a^7*b + a^8
+ a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)) + (tan(c + d*x)*((25*a^3*(-a^7*b)^(1/2) - 9*b^3*(-a^7*b)^(1/2) + 70*a^6*b
+ 6*a^4*b^3 + 44*a^5*b^2 - 41*a*b^2*(-a^7*b)^(1/2) - 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4
+ 4*a^8*b^3 + 6*a^9*b^2)))^(1/2)*(65536*a^6*b^11 + 327680*a^7*b^10 + 589824*a^8*b^9 + 327680*a^9*b^8 - 327680*
a^10*b^7 - 589824*a^11*b^6 - 327680*a^12*b^5 - 65536*a^13*b^4))/(128*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*
a^6*b^2)))*((25*a^3*(-a^7*b)^(1/2) - 9*b^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 - 41*a*b^2*(-a^7
*b)^(1/2) - 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2) - (tan(c
+ d*x)*(1152*a^2*b^11 + 7936*a^3*b^10 + 20352*a^4*b^9 + 8704*a^5*b^8 - 66688*a^6*b^7 - 110848*a^7*b^6 - 49024
*a^8*b^5))/(128*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*((25*a^3*(-a^7*b)^(1/2) - 9*b^3*(-a^7*b)^(
1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 - 41*a*b^2*(-a^7*b)^(1/2) - 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b +
a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2) - (720*a*b^10 + 4880*a^2*b^9 + 12320*a^3*b^8 + 21024*a^4*b^7
+ 33296*a^5*b^6 + 20432*a^6*b^5)/(256*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*((25*a^3*(-a^7*b)^(1
/2) - 9*b^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 - 41*a*b^2*(-a^7*b)^(1/2) - 39*a^2*b*(-a^7*b)^(
1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2) - (tan(c + d*x)*(612*a*b^8 + 81*b^9 + 1
894*a^2*b^7 + 2532*a^3*b^6 + 1425*a^4*b^5))/(128*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*((25*a^3*
(-a^7*b)^(1/2) - 9*b^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 - 41*a*b^2*(-a^7*b)^(1/2) - 39*a^2*b
*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2)*1i)/((((((245760*a^7*b^8 - 5
7344*a^5*b^10 - 36864*a^6*b^9 - 12288*a^4*b^11 + 634880*a^8*b^7 + 663552*a^9*b^6 + 331776*a^10*b^5 + 65536*a^1
1*b^4)/(256*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)) - (tan(c + d*x)*((25*a^3*(-a^7*b)^(1/2) - 9*b^3
*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 - 41*a*b^2*(-a^7*b)^(1/2) - 39*a^2*b*(-a^7*b)^(1/2))/(256*
(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2)*(65536*a^6*b^11 + 327680*a^7*b^10 + 589824*a^8*b^9
+ 327680*a^9*b^8 - 327680*a^10*b^7 - 589824*a^11*b^6 - 327680*a^12*b^5 - 65536*a^13*b^4))/(128*(4*a^7*b + a^8
+ a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*((25*a^3*(-a^7*b)^(1/2) - 9*b^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 +
44*a^5*b^2 - 41*a*b^2*(-a^7*b)^(1/2) - 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 +
6*a^9*b^2)))^(1/2) + (tan(c + d*x)*(1152*a^2*b^11 + 7936*a^3*b^10 + 20352*a^4*b^9 + 8704*a^5*b^8 - 66688*a^6*b
^7 - 110848*a^7*b^6 - 49024*a^8*b^5))/(128*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*((25*a^3*(-a^7*
b)^(1/2) - 9*b^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 - 41*a*b^2*(-a^7*b)^(1/2) - 39*a^2*b*(-a^7
*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2) - (720*a*b^10 + 4880*a^2*b^9 + 123
20*a^3*b^8 + 21024*a^4*b^7 + 33296*a^5*b^6 + 20432*a^6*b^5)/(256*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*
b^2)))*((25*a^3*(-a^7*b)^(1/2) - 9*b^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 - 41*a*b^2*(-a^7*b)^
(1/2) - 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2) + (tan(c + d
*x)*(612*a*b^8 + 81*b^9 + 1894*a^2*b^7 + 2532*a^3*b^6 + 1425*a^4*b^5))/(128*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b
^3 + 6*a^6*b^2)))*((25*a^3*(-a^7*b)^(1/2) - 9*b^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 - 41*a*b^
2*(-a^7*b)^(1/2) - 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2) +
(((((245760*a^7*b^8 - 57344*a^5*b^10 - 36864*a^6*b^9 - 12288*a^4*b^11 + 634880*a^8*b^7 + 663552*a^9*b^6 + 331
776*a^10*b^5 + 65536*a^11*b^4)/(256*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)) + (tan(c + d*x)*((25*a^
3*(-a^7*b)^(1/2) - 9*b^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 - 41*a*b^2*(-a^7*b)^(1/2) - 39*a^2
*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2)*(65536*a^6*b^11 + 327680*a
^7*b^10 + 589824*a^8*b^9 + 327680*a^9*b^8 - 327680*a^10*b^7 - 589824*a^11*b^6 - 327680*a^12*b^5 - 65536*a^13*b
^4))/(128*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*((25*a^3*(-a^7*b)^(1/2) - 9*b^3*(-a^7*b)^(1/2) +
70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 - 41*a*b^2*(-a^7*b)^(1/2) - 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11
+ a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2) - (tan(c + d*x)*(1152*a^2*b^11 + 7936*a^3*b^10 + 20352*a^4*b^9 + 87
04*a^5*b^8 - 66688*a^6*b^7 - 110848*a^7*b^6 - 49024*a^8*b^5))/(128*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^
6*b^2)))*((25*a^3*(-a^7*b)^(1/2) - 9*b^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 - 41*a*b^2*(-a^7*b
)^(1/2) - 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2) - (720*a*b
^10 + 4880*a^2*b^9 + 12320*a^3*b^8 + 21024*a^4*b^7 + 33296*a^5*b^6 + 20432*a^6*b^5)/(256*(4*a^7*b + a^8 + a^4*
b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*((25*a^3*(-a^7*b)^(1/2) - 9*b^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*
b^2 - 41*a*b^2*(-a^7*b)^(1/2) - 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b
^2)))^(1/2) - (tan(c + d*x)*(612*a*b^8 + 81*b^9 + 1894*a^2*b^7 + 2532*a^3*b^6 + 1425*a^4*b^5))/(128*(4*a^7*b +
a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*b^2)))*((25*a^3*(-a^7*b)^(1/2) - 9*b^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^
3 + 44*a^5*b^2 - 41*a*b^2*(-a^7*b)^(1/2) - 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^
3 + 6*a^9*b^2)))^(1/2) + (270*a*b^6 + 81*b^7 + 125*a^2*b^5)/(128*(4*a^7*b + a^8 + a^4*b^4 + 4*a^5*b^3 + 6*a^6*
b^2))))*((25*a^3*(-a^7*b)^(1/2) - 9*b^3*(-a^7*b)^(1/2) + 70*a^6*b + 6*a^4*b^3 + 44*a^5*b^2 - 41*a*b^2*(-a^7*b)
^(1/2) - 39*a^2*b*(-a^7*b)^(1/2))/(256*(4*a^10*b + a^11 + a^7*b^4 + 4*a^8*b^3 + 6*a^9*b^2)))^(1/2)*2i)/d
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \tan ^{4}{\left (c + d x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+tan(d*x+c)**4*b)**2,x)
[Out]
Integral((a + b*tan(c + d*x)**4)**(-2), x)
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