\(\int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx\) [589]

Optimal result

Integrand size = 23, antiderivative size = 271 \[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx=\frac {(a-b) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a-b) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}+\frac {2 b^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} \left (a^2+b^2\right ) d}+\frac {(a+b) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a+b) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 b}{a^2 d \sqrt {\tan (c+d x)}} \]

[Out]

2*b^(7/2)*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^(1/2))/a^(5/2)/(a^2+b^2)/d-1/2*(a-b)*arctan(-1+2^(1/2)*tan(d*x+c)^
(1/2))/(a^2+b^2)/d*2^(1/2)-1/2*(a-b)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)/d*2^(1/2)+1/4*(a+b)*ln(1-2^(
1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)/d*2^(1/2)-1/4*(a+b)*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2
+b^2)/d*2^(1/2)+2*b/a^2/d/tan(d*x+c)^(1/2)-2/3/a/d/tan(d*x+c)^(3/2)

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3650, 3730, 3735, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx=\frac {(a-b) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )}-\frac {(a-b) \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )}+\frac {(a+b) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}-\frac {(a+b) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}+\frac {2 b}{a^2 d \sqrt {\tan (c+d x)}}+\frac {2 b^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d \left (a^2+b^2\right )}-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x)} \]

[In]

Int[1/(Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])),x]

[Out]

((a - b)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - ((a - b)*ArcTan[1 + Sqrt[2]*Sqrt[Ta
n[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) + (2*b^(7/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(5/2)*(a^2
+ b^2)*d) + ((a + b)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((a + b)
*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - 2/(3*a*d*Tan[c + d*x]^(3/2))
+ (2*b)/(a^2*d*Sqrt[Tan[c + d*x]])

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 631

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 642

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 1176

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 1179

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 1182

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 3615

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2/f, Subst[I
nt[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2,
0] && NeQ[c^2 + d^2, 0]

Rule 3650

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[b^2*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + D
ist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c -
a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && I
ntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] &&
NeQ[a, 0])))

Rule 3715

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*
tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]
/; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]

Rule 3730

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Ta
n[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3735

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*ta
n[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*x])^n*Simp[a*(A - C) - (A*b - b*
C)*Tan[e + f*x], x], x], x] + Dist[(A*b^2 + a^2*C)/(a^2 + b^2), Int[(c + d*Tan[e + f*x])^n*((1 + Tan[e + f*x]^
2)/(a + b*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^
2, 0] && NeQ[c^2 + d^2, 0] &&  !GtQ[n, 0] &&  !LeQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 \int \frac {\frac {3 b}{2}+\frac {3}{2} a \tan (c+d x)+\frac {3}{2} b \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))} \, dx}{3 a} \\ & = -\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 b}{a^2 d \sqrt {\tan (c+d x)}}+\frac {4 \int \frac {-\frac {3}{4} \left (a^2-b^2\right )+\frac {3}{4} b^2 \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{3 a^2} \\ & = -\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 b}{a^2 d \sqrt {\tan (c+d x)}}+\frac {4 \int \frac {-\frac {3 a^3}{4}+\frac {3}{4} a^2 b \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{3 a^2 \left (a^2+b^2\right )}+\frac {b^4 \int \frac {1+\tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{a^2 \left (a^2+b^2\right )} \\ & = -\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 b}{a^2 d \sqrt {\tan (c+d x)}}+\frac {8 \text {Subst}\left (\int \frac {-\frac {3 a^3}{4}+\frac {3}{4} a^2 b x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{3 a^2 \left (a^2+b^2\right ) d}+\frac {b^4 \text {Subst}\left (\int \frac {1}{\sqrt {x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{a^2 \left (a^2+b^2\right ) d} \\ & = -\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 b}{a^2 d \sqrt {\tan (c+d x)}}-\frac {(a-b) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}+\frac {\left (2 b^4\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {(a+b) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d} \\ & = \frac {2 b^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} \left (a^2+b^2\right ) d}-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 b}{a^2 d \sqrt {\tan (c+d x)}}-\frac {(a-b) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}-\frac {(a-b) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}+\frac {(a+b) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a+b) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d} \\ & = \frac {2 b^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} \left (a^2+b^2\right ) d}+\frac {(a+b) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a+b) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 b}{a^2 d \sqrt {\tan (c+d x)}}-\frac {(a-b) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a-b) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d} \\ & = \frac {(a-b) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a-b) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}+\frac {2 b^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} \left (a^2+b^2\right ) d}+\frac {(a+b) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a+b) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 b}{a^2 d \sqrt {\tan (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.95 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx=\frac {\frac {6 \sqrt {2} a^2 (a-b) \left (\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )\right )}{a^2+b^2}+\frac {24 b^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} \left (a^2+b^2\right )}+\frac {3 \sqrt {2} a^2 (a+b) \left (\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )}{a^2+b^2}-\frac {8 a}{\tan ^{\frac {3}{2}}(c+d x)}+\frac {24 b}{\sqrt {\tan (c+d x)}}}{12 a^2 d} \]

[In]

Integrate[1/(Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])),x]

[Out]

((6*Sqrt[2]*a^2*(a - b)*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/(a^
2 + b^2) + (24*b^(7/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*(a^2 + b^2)) + (3*Sqrt[2]*a^2*(a
+ b)*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]
))/(a^2 + b^2) - (8*a)/Tan[c + d*x]^(3/2) + (24*b)/Sqrt[Tan[c + d*x]])/(12*a^2*d)

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.94

method	result	size
derivativedivides	\(\frac {\frac {-\frac {a \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {b \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{a^{2}+b^{2}}+\frac {2 b^{4} \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{a^{2} \left (a^{2}+b^{2}\right ) \sqrt {a b}}-\frac {2}{3 a \tan \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 b}{a^{2} \sqrt {\tan \left (d x +c \right )}}}{d}\)	\(255\)
default	\(\frac {\frac {-\frac {a \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {b \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{a^{2}+b^{2}}+\frac {2 b^{4} \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{a^{2} \left (a^{2}+b^{2}\right ) \sqrt {a b}}-\frac {2}{3 a \tan \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 b}{a^{2} \sqrt {\tan \left (d x +c \right )}}}{d}\)	\(255\)

[In]

int(1/tan(d*x+c)^(5/2)/(a+b*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(2/(a^2+b^2)*(-1/8*a*2^(1/2)*(ln((1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d
*x+c)))+2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2)))+1/8*b*2^(1/2)*(ln((1-2^(1/
2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))+2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+
2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))))+2/a^2*b^4/(a^2+b^2)/(a*b)^(1/2)*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))
-2/3/a/tan(d*x+c)^(3/2)+2/a^2*b/tan(d*x+c)^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1460 vs. \(2 (227) = 454\).

Time = 0.38 (sec) , antiderivative size = 2946, normalized size of antiderivative = 10.87 \[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx=\text {Too large to display} \]

[In]

integrate(1/tan(d*x+c)^(5/2)/(a+b*tan(d*x+c)),x, algorithm="fricas")

[Out]

[1/6*(6*b^3*sqrt(-b/a)*log((2*a*sqrt(-b/a)*sqrt(tan(d*x + c)) + b*tan(d*x + c) - a)/(b*tan(d*x + c) + a))*tan(
d*x + c)^2 + 3*(a^4 + a^2*b^2)*d*sqrt(((a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6
*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2))*log(((a^4*b + 2*a^2*b^3 + b^
5)*d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + (a^3 - a*b^2)*d)
*sqrt(((a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b
^8)*d^4)) + 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2)) - (a^2 - b^2)*sqrt(tan(d*x + c)))*tan(d*x + c)^2 - 3*(a^4 +
a^2*b^2)*d*sqrt(((a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a
^2*b^6 + b^8)*d^4)) + 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2))*log(-((a^4*b + 2*a^2*b^3 + b^5)*d^3*sqrt(-(a^4 - 2
*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + (a^3 - a*b^2)*d)*sqrt(((a^4 + 2*a^2*b
^2 + b^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + 2*a*b)/((
a^4 + 2*a^2*b^2 + b^4)*d^2)) - (a^2 - b^2)*sqrt(tan(d*x + c)))*tan(d*x + c)^2 - 3*(a^4 + a^2*b^2)*d*sqrt(-((a^
4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))
- 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2))*log(((a^4*b + 2*a^2*b^3 + b^5)*d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8
+ 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - (a^3 - a*b^2)*d)*sqrt(-((a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-
(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - 2*a*b)/((a^4 + 2*a^2*b^2 + b^
4)*d^2)) - (a^2 - b^2)*sqrt(tan(d*x + c)))*tan(d*x + c)^2 + 3*(a^4 + a^2*b^2)*d*sqrt(-((a^4 + 2*a^2*b^2 + b^4)
*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - 2*a*b)/((a^4 + 2*a
^2*b^2 + b^4)*d^2))*log(-((a^4*b + 2*a^2*b^3 + b^5)*d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^
4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - (a^3 - a*b^2)*d)*sqrt(-((a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 +
b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2)) - (a^2 - b
^2)*sqrt(tan(d*x + c)))*tan(d*x + c)^2 - 4*(a^3 + a*b^2 - 3*(a^2*b + b^3)*tan(d*x + c))*sqrt(tan(d*x + c)))/((
a^4 + a^2*b^2)*d*tan(d*x + c)^2), -1/6*(12*b^3*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(tan(d*x + c))))*tan(d*x +
c)^2 - 3*(a^4 + a^2*b^2)*d*sqrt(((a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 +
6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2))*log(((a^4*b + 2*a^2*b^3 + b^5)*d^3
*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + (a^3 - a*b^2)*d)*sqrt(
((a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^
4)) + 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2)) - (a^2 - b^2)*sqrt(tan(d*x + c)))*tan(d*x + c)^2 + 3*(a^4 + a^2*b^
2)*d*sqrt(((a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6
+ b^8)*d^4)) + 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2))*log(-((a^4*b + 2*a^2*b^3 + b^5)*d^3*sqrt(-(a^4 - 2*a^2*b
^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + (a^3 - a*b^2)*d)*sqrt(((a^4 + 2*a^2*b^2 + b
^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + 2*a*b)/((a^4 +
2*a^2*b^2 + b^4)*d^2)) - (a^2 - b^2)*sqrt(tan(d*x + c)))*tan(d*x + c)^2 + 3*(a^4 + a^2*b^2)*d*sqrt(-((a^4 + 2*
a^2*b^2 + b^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - 2*a*
b)/((a^4 + 2*a^2*b^2 + b^4)*d^2))*log(((a^4*b + 2*a^2*b^3 + b^5)*d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a
^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - (a^3 - a*b^2)*d)*sqrt(-((a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(a^4 -
2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2
)) - (a^2 - b^2)*sqrt(tan(d*x + c)))*tan(d*x + c)^2 - 3*(a^4 + a^2*b^2)*d*sqrt(-((a^4 + 2*a^2*b^2 + b^4)*d^2*s
qrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - 2*a*b)/((a^4 + 2*a^2*b^2
+ b^4)*d^2))*log(-((a^4*b + 2*a^2*b^3 + b^5)*d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4
+ 4*a^2*b^6 + b^8)*d^4)) - (a^3 - a*b^2)*d)*sqrt(-((a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/(
(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2)) - (a^2 - b^2)*sq
rt(tan(d*x + c)))*tan(d*x + c)^2 + 4*(a^3 + a*b^2 - 3*(a^2*b + b^3)*tan(d*x + c))*sqrt(tan(d*x + c)))/((a^4 +
a^2*b^2)*d*tan(d*x + c)^2)]

Sympy [F]

\[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx=\int \frac {1}{\left (a + b \tan {\left (c + d x \right )}\right ) \tan ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx \]

[In]

integrate(1/tan(d*x+c)**(5/2)/(a+b*tan(d*x+c)),x)

[Out]

Integral(1/((a + b*tan(c + d*x))*tan(c + d*x)**(5/2)), x)

Maxima [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx=\frac {\frac {24 \, b^{4} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{4} + a^{2} b^{2}\right )} \sqrt {a b}} - \frac {3 \, {\left (2 \, \sqrt {2} {\left (a - b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a - b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left (a + b\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left (a + b\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )}}{a^{2} + b^{2}} + \frac {8 \, {\left (3 \, b \tan \left (d x + c\right ) - a\right )}}{a^{2} \tan \left (d x + c\right )^{\frac {3}{2}}}}{12 \, d} \]

[In]

integrate(1/tan(d*x+c)^(5/2)/(a+b*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/12*(24*b^4*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^4 + a^2*b^2)*sqrt(a*b)) - 3*(2*sqrt(2)*(a - b)*arctan(
1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*(a - b)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*
x + c)))) + sqrt(2)*(a + b)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) - sqrt(2)*(a + b)*log(-sqrt(2)*
sqrt(tan(d*x + c)) + tan(d*x + c) + 1))/(a^2 + b^2) + 8*(3*b*tan(d*x + c) - a)/(a^2*tan(d*x + c)^(3/2)))/d

Giac [F(-1)]

Timed out. \[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx=\text {Timed out} \]

[In]

integrate(1/tan(d*x+c)^(5/2)/(a+b*tan(d*x+c)),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 6.65 (sec) , antiderivative size = 4806, normalized size of antiderivative = 17.73 \[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx=\text {Too large to display} \]

[In]

int(1/(tan(c + d*x)^(5/2)*(a + b*tan(c + d*x))),x)

[Out]

(atan((((-a^5*b^7)^(1/2)*(tan(c + d*x)^(1/2)*(64*a^14*b^9*d^5 + 32*a^18*b^5*d^5) - ((-a^5*b^7)^(1/2)*(32*a^19*
b^5*d^6 - 384*a^15*b^9*d^6 + 32*a^21*b^3*d^6 + ((tan(c + d*x)^(1/2)*(512*a^15*b^10*d^7 + 448*a^19*b^6*d^7 - 12
8*a^21*b^4*d^7 - 64*a^23*b^2*d^7) + ((-a^5*b^7)^(1/2)*(512*a^16*b^10*d^8 + 512*a^18*b^8*d^8 - 384*a^20*b^6*d^8
- 256*a^22*b^4*d^8 + 128*a^24*b^2*d^8 - (tan(c + d*x)^(1/2)*(-a^5*b^7)^(1/2)*(512*a^18*b^9*d^9 + 512*a^20*b^7
*d^9 - 512*a^22*b^5*d^9 - 512*a^24*b^3*d^9))/(a^5*d*(a^2 + b^2))))/(a^5*d*(a^2 + b^2)))*(-a^5*b^7)^(1/2))/(a^5
*d*(a^2 + b^2))))/(a^5*d*(a^2 + b^2)))*1i)/(a^5*d*(a^2 + b^2)) + ((-a^5*b^7)^(1/2)*(tan(c + d*x)^(1/2)*(64*a^1
4*b^9*d^5 + 32*a^18*b^5*d^5) - ((-a^5*b^7)^(1/2)*(384*a^15*b^9*d^6 - 32*a^19*b^5*d^6 - 32*a^21*b^3*d^6 + ((tan
(c + d*x)^(1/2)*(512*a^15*b^10*d^7 + 448*a^19*b^6*d^7 - 128*a^21*b^4*d^7 - 64*a^23*b^2*d^7) - ((-a^5*b^7)^(1/2
)*(512*a^16*b^10*d^8 + 512*a^18*b^8*d^8 - 384*a^20*b^6*d^8 - 256*a^22*b^4*d^8 + 128*a^24*b^2*d^8 + (tan(c + d*
x)^(1/2)*(-a^5*b^7)^(1/2)*(512*a^18*b^9*d^9 + 512*a^20*b^7*d^9 - 512*a^22*b^5*d^9 - 512*a^24*b^3*d^9))/(a^5*d*
(a^2 + b^2))))/(a^5*d*(a^2 + b^2)))*(-a^5*b^7)^(1/2))/(a^5*d*(a^2 + b^2))))/(a^5*d*(a^2 + b^2)))*1i)/(a^5*d*(a
^2 + b^2)))/(64*a^14*b^8*d^4 - ((-a^5*b^7)^(1/2)*(tan(c + d*x)^(1/2)*(64*a^14*b^9*d^5 + 32*a^18*b^5*d^5) - ((-
a^5*b^7)^(1/2)*(32*a^19*b^5*d^6 - 384*a^15*b^9*d^6 + 32*a^21*b^3*d^6 + ((tan(c + d*x)^(1/2)*(512*a^15*b^10*d^7
+ 448*a^19*b^6*d^7 - 128*a^21*b^4*d^7 - 64*a^23*b^2*d^7) + ((-a^5*b^7)^(1/2)*(512*a^16*b^10*d^8 + 512*a^18*b^
8*d^8 - 384*a^20*b^6*d^8 - 256*a^22*b^4*d^8 + 128*a^24*b^2*d^8 - (tan(c + d*x)^(1/2)*(-a^5*b^7)^(1/2)*(512*a^1
8*b^9*d^9 + 512*a^20*b^7*d^9 - 512*a^22*b^5*d^9 - 512*a^24*b^3*d^9))/(a^5*d*(a^2 + b^2))))/(a^5*d*(a^2 + b^2))
)*(-a^5*b^7)^(1/2))/(a^5*d*(a^2 + b^2))))/(a^5*d*(a^2 + b^2))))/(a^5*d*(a^2 + b^2)) + ((-a^5*b^7)^(1/2)*(tan(c
+ d*x)^(1/2)*(64*a^14*b^9*d^5 + 32*a^18*b^5*d^5) - ((-a^5*b^7)^(1/2)*(384*a^15*b^9*d^6 - 32*a^19*b^5*d^6 - 32
*a^21*b^3*d^6 + ((tan(c + d*x)^(1/2)*(512*a^15*b^10*d^7 + 448*a^19*b^6*d^7 - 128*a^21*b^4*d^7 - 64*a^23*b^2*d^
7) - ((-a^5*b^7)^(1/2)*(512*a^16*b^10*d^8 + 512*a^18*b^8*d^8 - 384*a^20*b^6*d^8 - 256*a^22*b^4*d^8 + 128*a^24*
b^2*d^8 + (tan(c + d*x)^(1/2)*(-a^5*b^7)^(1/2)*(512*a^18*b^9*d^9 + 512*a^20*b^7*d^9 - 512*a^22*b^5*d^9 - 512*a
^24*b^3*d^9))/(a^5*d*(a^2 + b^2))))/(a^5*d*(a^2 + b^2)))*(-a^5*b^7)^(1/2))/(a^5*d*(a^2 + b^2))))/(a^5*d*(a^2 +
b^2))))/(a^5*d*(a^2 + b^2))))*(-a^5*b^7)^(1/2)*2i)/(a^5*d*(a^2 + b^2)) - atan(((((1/(b^2*d^2*1i - a^2*d^2*1i
+ 2*a*b*d^2))^(1/2)*((((tan(c + d*x)^(1/2)*(512*a^15*b^10*d^7 + 448*a^19*b^6*d^7 - 128*a^21*b^4*d^7 - 64*a^23*
b^2*d^7))/2 + ((1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*(256*a^16*b^10*d^8 + 256*a^18*b^8*d^8 - 192*a^2
0*b^6*d^8 - 128*a^22*b^4*d^8 + 64*a^24*b^2*d^8 - (tan(c + d*x)^(1/2)*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))
^(1/2)*(512*a^18*b^9*d^9 + 512*a^20*b^7*d^9 - 512*a^22*b^5*d^9 - 512*a^24*b^3*d^9))/4))/2)*(1/(b^2*d^2*1i - a^
2*d^2*1i + 2*a*b*d^2))^(1/2))/2 - 192*a^15*b^9*d^6 + 16*a^19*b^5*d^6 + 16*a^21*b^3*d^6))/2 - (tan(c + d*x)^(1/
2)*(64*a^14*b^9*d^5 + 32*a^18*b^5*d^5))/2)*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*1i + (((1/(b^2*d^2*
1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*((((tan(c + d*x)^(1/2)*(512*a^15*b^10*d^7 + 448*a^19*b^6*d^7 - 128*a^21*b^
4*d^7 - 64*a^23*b^2*d^7))/2 - ((1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*(256*a^16*b^10*d^8 + 256*a^18*b
^8*d^8 - 192*a^20*b^6*d^8 - 128*a^22*b^4*d^8 + 64*a^24*b^2*d^8 + (tan(c + d*x)^(1/2)*(1/(b^2*d^2*1i - a^2*d^2*
1i + 2*a*b*d^2))^(1/2)*(512*a^18*b^9*d^9 + 512*a^20*b^7*d^9 - 512*a^22*b^5*d^9 - 512*a^24*b^3*d^9))/4))/2)*(1/
(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2))/2 + 192*a^15*b^9*d^6 - 16*a^19*b^5*d^6 - 16*a^21*b^3*d^6))/2 - (
tan(c + d*x)^(1/2)*(64*a^14*b^9*d^5 + 32*a^18*b^5*d^5))/2)*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*1i)
/((((1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*((((tan(c + d*x)^(1/2)*(512*a^15*b^10*d^7 + 448*a^19*b^6*d
^7 - 128*a^21*b^4*d^7 - 64*a^23*b^2*d^7))/2 + ((1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*(256*a^16*b^10*
d^8 + 256*a^18*b^8*d^8 - 192*a^20*b^6*d^8 - 128*a^22*b^4*d^8 + 64*a^24*b^2*d^8 - (tan(c + d*x)^(1/2)*(1/(b^2*d
^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*(512*a^18*b^9*d^9 + 512*a^20*b^7*d^9 - 512*a^22*b^5*d^9 - 512*a^24*b^3*
d^9))/4))/2)*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2))/2 - 192*a^15*b^9*d^6 + 16*a^19*b^5*d^6 + 16*a^21
*b^3*d^6))/2 - (tan(c + d*x)^(1/2)*(64*a^14*b^9*d^5 + 32*a^18*b^5*d^5))/2)*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b
*d^2))^(1/2) - (((1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*((((tan(c + d*x)^(1/2)*(512*a^15*b^10*d^7 + 4
48*a^19*b^6*d^7 - 128*a^21*b^4*d^7 - 64*a^23*b^2*d^7))/2 - ((1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*(2
56*a^16*b^10*d^8 + 256*a^18*b^8*d^8 - 192*a^20*b^6*d^8 - 128*a^22*b^4*d^8 + 64*a^24*b^2*d^8 + (tan(c + d*x)^(1
/2)*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*(512*a^18*b^9*d^9 + 512*a^20*b^7*d^9 - 512*a^22*b^5*d^9 -
512*a^24*b^3*d^9))/4))/2)*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2))/2 + 192*a^15*b^9*d^6 - 16*a^19*b^5*
d^6 - 16*a^21*b^3*d^6))/2 - (tan(c + d*x)^(1/2)*(64*a^14*b^9*d^5 + 32*a^18*b^5*d^5))/2)*(1/(b^2*d^2*1i - a^2*d
^2*1i + 2*a*b*d^2))^(1/2) + 64*a^14*b^8*d^4))*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*1i - (2/(3*a) -
(2*b*tan(c + d*x))/a^2)/(d*tan(c + d*x)^(3/2)) - atan((((1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*((1i/(
4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*(tan(c + d*x)^(1/2)*(512*a^15*b^10*d^7 + 448*a^19*b^6*d^7 - 128*a^2
1*b^4*d^7 - 64*a^23*b^2*d^7) - (1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*(tan(c + d*x)^(1/2)*(1i/(4*(b^2
*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*(512*a^18*b^9*d^9 + 512*a^20*b^7*d^9 - 512*a^22*b^5*d^9 - 512*a^24*b^3*d^
9) - 512*a^16*b^10*d^8 - 512*a^18*b^8*d^8 + 384*a^20*b^6*d^8 + 256*a^22*b^4*d^8 - 128*a^24*b^2*d^8)) - 384*a^1
5*b^9*d^6 + 32*a^19*b^5*d^6 + 32*a^21*b^3*d^6) - tan(c + d*x)^(1/2)*(64*a^14*b^9*d^5 + 32*a^18*b^5*d^5))*(1i/(
4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*1i + ((1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*((1i/(4*(b^2*
d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*(tan(c + d*x)^(1/2)*(512*a^15*b^10*d^7 + 448*a^19*b^6*d^7 - 128*a^21*b^4*d
^7 - 64*a^23*b^2*d^7) - (1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*(tan(c + d*x)^(1/2)*(1i/(4*(b^2*d^2 -
a^2*d^2 + a*b*d^2*2i)))^(1/2)*(512*a^18*b^9*d^9 + 512*a^20*b^7*d^9 - 512*a^22*b^5*d^9 - 512*a^24*b^3*d^9) + 51
2*a^16*b^10*d^8 + 512*a^18*b^8*d^8 - 384*a^20*b^6*d^8 - 256*a^22*b^4*d^8 + 128*a^24*b^2*d^8)) + 384*a^15*b^9*d
^6 - 32*a^19*b^5*d^6 - 32*a^21*b^3*d^6) - tan(c + d*x)^(1/2)*(64*a^14*b^9*d^5 + 32*a^18*b^5*d^5))*(1i/(4*(b^2*
d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*1i)/(((1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*((1i/(4*(b^2*d^2 - a
^2*d^2 + a*b*d^2*2i)))^(1/2)*(tan(c + d*x)^(1/2)*(512*a^15*b^10*d^7 + 448*a^19*b^6*d^7 - 128*a^21*b^4*d^7 - 64
*a^23*b^2*d^7) - (1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*(tan(c + d*x)^(1/2)*(1i/(4*(b^2*d^2 - a^2*d^2
+ a*b*d^2*2i)))^(1/2)*(512*a^18*b^9*d^9 + 512*a^20*b^7*d^9 - 512*a^22*b^5*d^9 - 512*a^24*b^3*d^9) - 512*a^16*
b^10*d^8 - 512*a^18*b^8*d^8 + 384*a^20*b^6*d^8 + 256*a^22*b^4*d^8 - 128*a^24*b^2*d^8)) - 384*a^15*b^9*d^6 + 32
*a^19*b^5*d^6 + 32*a^21*b^3*d^6) - tan(c + d*x)^(1/2)*(64*a^14*b^9*d^5 + 32*a^18*b^5*d^5))*(1i/(4*(b^2*d^2 - a
^2*d^2 + a*b*d^2*2i)))^(1/2) - ((1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*((1i/(4*(b^2*d^2 - a^2*d^2 + a
*b*d^2*2i)))^(1/2)*(tan(c + d*x)^(1/2)*(512*a^15*b^10*d^7 + 448*a^19*b^6*d^7 - 128*a^21*b^4*d^7 - 64*a^23*b^2*
d^7) - (1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*(tan(c + d*x)^(1/2)*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2
*2i)))^(1/2)*(512*a^18*b^9*d^9 + 512*a^20*b^7*d^9 - 512*a^22*b^5*d^9 - 512*a^24*b^3*d^9) + 512*a^16*b^10*d^8 +
512*a^18*b^8*d^8 - 384*a^20*b^6*d^8 - 256*a^22*b^4*d^8 + 128*a^24*b^2*d^8)) + 384*a^15*b^9*d^6 - 32*a^19*b^5*
d^6 - 32*a^21*b^3*d^6) - tan(c + d*x)^(1/2)*(64*a^14*b^9*d^5 + 32*a^18*b^5*d^5))*(1i/(4*(b^2*d^2 - a^2*d^2 + a
*b*d^2*2i)))^(1/2) + 64*a^14*b^8*d^4))*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*2i