\(\int \frac {\tan ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^2} \, dx\) [593]

Optimal result

Integrand size = 23, antiderivative size = 318 \[ \int \frac {\tan ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {a^{3/2} \left (a^2+5 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{3/2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2-2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2-2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {a^2 \sqrt {\tan (c+d x)}}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {3646, 3734, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \[ \int \frac {\tan ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}-\frac {a^2 \sqrt {\tan (c+d x)}}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {\left (a^2-2 a b-b^2\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}+\frac {\left (a^2-2 a b-b^2\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}+\frac {a^{3/2} \left (a^2+5 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{3/2} d \left (a^2+b^2\right )^2} \]

[In]

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

[Out]

((a^2 + 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 + 2*a*b - b^2)*
ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) + (a^(3/2)*(a^2 + 5*b^2)*ArcTan[(Sqrt[b]*Sqr
t[Tan[c + d*x]])/Sqrt[a]])/(b^(3/2)*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]
+ Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + ((a^2 - 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[
c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - (a^2*Sqrt[Tan[c + d*x]])/(b*(a^2 + b^2)*d*(a + b*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 3646

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

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 3734

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*
x])^n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Dist[(A*b^2 - a*b*B + 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, B, 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 {a^2 \sqrt {\tan (c+d x)}}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {\frac {a^2}{2}-a b \tan (c+d x)+\frac {1}{2} \left (a^2+2 b^2\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{b \left (a^2+b^2\right )} \\ & = -\frac {a^2 \sqrt {\tan (c+d x)}}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {-2 a b^2-b \left (a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{b \left (a^2+b^2\right )^2}+\frac {\left (a^2 \left (a^2+5 b^2\right )\right ) \int \frac {1+\tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{2 b \left (a^2+b^2\right )^2} \\ & = -\frac {a^2 \sqrt {\tan (c+d x)}}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {2 \text {Subst}\left (\int \frac {-2 a b^2-b \left (a^2-b^2\right ) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{b \left (a^2+b^2\right )^2 d}+\frac {\left (a^2 \left (a^2+5 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{2 b \left (a^2+b^2\right )^2 d} \\ & = -\frac {a^2 \sqrt {\tan (c+d x)}}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\left (a^2-2 a b-b^2\right ) \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 )^2 d}-\frac {\left (a^2+2 a b-b^2\right ) \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 )^2 d}+\frac {\left (a^2 \left (a^2+5 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{b \left (a^2+b^2\right )^2 d} \\ & = \frac {a^{3/2} \left (a^2+5 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{3/2} \left (a^2+b^2\right )^2 d}-\frac {a^2 \sqrt {\tan (c+d x)}}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {\left (a^2-2 a b-b^2\right ) \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 )^2 d}-\frac {\left (a^2-2 a b-b^2\right ) \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 )^2 d}-\frac {\left (a^2+2 a b-b^2\right ) \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 )^2 d}-\frac {\left (a^2+2 a b-b^2\right ) \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 )^2 d} \\ & = \frac {a^{3/2} \left (a^2+5 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{3/2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2-2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2-2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {a^2 \sqrt {\tan (c+d x)}}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {\left (a^2+2 a b-b^2\right ) \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 )^2 d}+\frac {\left (a^2+2 a b-b^2\right ) \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 )^2 d} \\ & = \frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {a^{3/2} \left (a^2+5 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{3/2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2-2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2-2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {a^2 \sqrt {\tan (c+d x)}}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.80 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.49 \[ \int \frac {\tan ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {(-1)^{3/4} (-i a+b)^2 \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+\frac {a^{3/2} \left (a^2+5 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{3/2}}+(-1)^{3/4} (a-i b)^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )-\frac {a^2 \left (a^2+b^2\right ) \sqrt {\tan (c+d x)}}{b (a+b \tan (c+d x))}}{\left (a^2+b^2\right )^2 d} \]

[In]

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

[Out]

((-1)^(3/4)*((-I)*a + b)^2*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + (a^(3/2)*(a^2 + 5*b^2)*ArcTan[(Sqrt[b]*Sqrt
[Tan[c + d*x]])/Sqrt[a]])/b^(3/2) + (-1)^(3/4)*(a - I*b)^2*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] - (a^2*(a^2
+ b^2)*Sqrt[Tan[c + d*x]])/(b*(a + b*Tan[c + d*x])))/((a^2 + b^2)^2*d)

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(\frac {\frac {-\frac {a b \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{2}+\frac {\left (-a^{2}+b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{2}}+\frac {2 a^{2} \left (-\frac {\left (a^{2}+b^{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{2 b \left (a +b \tan \left (d x +c \right )\right )}+\frac {\left (a^{2}+5 b^{2}\right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{2 b \sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(281\)
default	\(\frac {\frac {-\frac {a b \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{2}+\frac {\left (-a^{2}+b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{2}}+\frac {2 a^{2} \left (-\frac {\left (a^{2}+b^{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{2 b \left (a +b \tan \left (d x +c \right )\right )}+\frac {\left (a^{2}+5 b^{2}\right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{2 b \sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(281\)

[In]

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

[Out]

1/d*(2/(a^2+b^2)^2*(-1/4*a*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)+t
an(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*(-a^2+b^2)*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/(a^2+b^2)^2*(-1/2*(a^2+b^2)/b*tan(d*x+c)^(1/2)/(a+b
*tan(d*x+c))+1/2*(a^2+5*b^2)/b/(a*b)^(1/2)*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2664 vs. \(2 (278) = 556\).

Time = 0.48 (sec) , antiderivative size = 5353, normalized size of antiderivative = 16.83 \[ \int \frac {\tan ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.87 \[ \int \frac {\tan ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {4 \, a^{2} \sqrt {\tan \left (d x + c\right )}}{a^{3} b + a b^{3} + {\left (a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )} - \frac {4 \, {\left (a^{4} + 5 \, a^{2} b^{2}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \sqrt {a b}} + \frac {2 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}}}{4 \, d} \]

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 7.65 (sec) , antiderivative size = 11104, normalized size of antiderivative = 34.92 \[ \int \frac {\tan ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

- atan(((1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*((1i/(4*(a^4*d^2 + b^
4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*((1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b
*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(((8*(96*a^2*b^14*d^4 + 480*a^4*b^12*d^4 + 960*a^6*b^10*d^4 + 960*a^8*b^8*d^4
+ 480*a^10*b^6*d^4 + 96*a^12*b^4*d^4))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5)
- (16*tan(c + d*x)^(1/2)*(1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(32*
b^18*d^4 + 160*a^2*b^16*d^4 + 288*a^4*b^14*d^4 + 160*a^6*b^12*d^4 - 160*a^8*b^10*d^4 - 288*a^10*b^8*d^4 - 160*
a^12*b^6*d^4 - 32*a^14*b^4*d^4))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*(1i/(4
*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2) + (16*tan(c + d*x)^(1/2)*(60*a*b^13
*d^2 + 8*a^13*b*d^2 + 52*a^3*b^11*d^2 + 128*a^5*b^9*d^2 + 424*a^7*b^7*d^2 + 380*a^9*b^5*d^2 + 100*a^11*b^3*d^2
))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4)) + (8*(4*a*b^11*d^2 + 16*a^11*b*d^2 -
304*a^3*b^9*d^2 - 120*a^5*b^7*d^2 + 320*a^7*b^5*d^2 + 148*a^9*b^3*d^2))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5
+ 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5)) + (16*tan(c + d*x)^(1/2)*(a^10 - 2*b^10 - 4*a^2*b^8 - 27*a^4*b^6 + 15*a^6*b^
4 + 9*a^8*b^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*1i - (1i/(4*(a^4*d^2 +
b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*((1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3
*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*((1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))
^(1/2)*(((8*(96*a^2*b^14*d^4 + 480*a^4*b^12*d^4 + 960*a^6*b^10*d^4 + 960*a^8*b^8*d^4 + 480*a^10*b^6*d^4 + 96*a
^12*b^4*d^4))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) + (16*tan(c + d*x)^(1/2)*(
1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(32*b^18*d^4 + 160*a^2*b^16*d^
4 + 288*a^4*b^14*d^4 + 160*a^6*b^12*d^4 - 160*a^8*b^10*d^4 - 288*a^10*b^8*d^4 - 160*a^12*b^6*d^4 - 32*a^14*b^4
*d^4))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*(1i/(4*(a^4*d^2 + b^4*d^2 + a*b^
3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2) - (16*tan(c + d*x)^(1/2)*(60*a*b^13*d^2 + 8*a^13*b*d^2 + 52*a
^3*b^11*d^2 + 128*a^5*b^9*d^2 + 424*a^7*b^7*d^2 + 380*a^9*b^5*d^2 + 100*a^11*b^3*d^2))/(b^9*d^4 + a^8*b*d^4 +
4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4)) + (8*(4*a*b^11*d^2 + 16*a^11*b*d^2 - 304*a^3*b^9*d^2 - 120*a^5
*b^7*d^2 + 320*a^7*b^5*d^2 + 148*a^9*b^3*d^2))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^
3*d^5)) - (16*tan(c + d*x)^(1/2)*(a^10 - 2*b^10 - 4*a^2*b^8 - 27*a^4*b^6 + 15*a^6*b^4 + 9*a^8*b^2))/(b^9*d^4 +
a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*1i)/((1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a
^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*((1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)
))^(1/2)*((1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(((8*(96*a^2*b^14*d
^4 + 480*a^4*b^12*d^4 + 960*a^6*b^10*d^4 + 960*a^8*b^8*d^4 + 480*a^10*b^6*d^4 + 96*a^12*b^4*d^4))/(b^9*d^5 + a
^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) - (16*tan(c + d*x)^(1/2)*(1i/(4*(a^4*d^2 + b^4*d^2 +
a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(32*b^18*d^4 + 160*a^2*b^16*d^4 + 288*a^4*b^14*d^4 + 160
*a^6*b^12*d^4 - 160*a^8*b^10*d^4 - 288*a^10*b^8*d^4 - 160*a^12*b^6*d^4 - 32*a^14*b^4*d^4))/(b^9*d^4 + a^8*b*d^
4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*(1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i -
6*a^2*b^2*d^2)))^(1/2) + (16*tan(c + d*x)^(1/2)*(60*a*b^13*d^2 + 8*a^13*b*d^2 + 52*a^3*b^11*d^2 + 128*a^5*b^9*
d^2 + 424*a^7*b^7*d^2 + 380*a^9*b^5*d^2 + 100*a^11*b^3*d^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*
d^4 + 4*a^6*b^3*d^4)) + (8*(4*a*b^11*d^2 + 16*a^11*b*d^2 - 304*a^3*b^9*d^2 - 120*a^5*b^7*d^2 + 320*a^7*b^5*d^2
+ 148*a^9*b^3*d^2))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5)) + (16*tan(c + d*x)
^(1/2)*(a^10 - 2*b^10 - 4*a^2*b^8 - 27*a^4*b^6 + 15*a^6*b^4 + 9*a^8*b^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4
+ 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4)) + (1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2))
)^(1/2)*((1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*((1i/(4*(a^4*d^2 + b
^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(((8*(96*a^2*b^14*d^4 + 480*a^4*b^12*d^4 + 960*a
^6*b^10*d^4 + 960*a^8*b^8*d^4 + 480*a^10*b^6*d^4 + 96*a^12*b^4*d^4))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*
a^4*b^5*d^5 + 4*a^6*b^3*d^5) + (16*tan(c + d*x)^(1/2)*(1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i
- 6*a^2*b^2*d^2)))^(1/2)*(32*b^18*d^4 + 160*a^2*b^16*d^4 + 288*a^4*b^14*d^4 + 160*a^6*b^12*d^4 - 160*a^8*b^10*
d^4 - 288*a^10*b^8*d^4 - 160*a^12*b^6*d^4 - 32*a^14*b^4*d^4))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5
*d^4 + 4*a^6*b^3*d^4))*(1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2) - (16*
tan(c + d*x)^(1/2)*(60*a*b^13*d^2 + 8*a^13*b*d^2 + 52*a^3*b^11*d^2 + 128*a^5*b^9*d^2 + 424*a^7*b^7*d^2 + 380*a
^9*b^5*d^2 + 100*a^11*b^3*d^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4)) + (8*(4
*a*b^11*d^2 + 16*a^11*b*d^2 - 304*a^3*b^9*d^2 - 120*a^5*b^7*d^2 + 320*a^7*b^5*d^2 + 148*a^9*b^3*d^2))/(b^9*d^5
+ a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5)) - (16*tan(c + d*x)^(1/2)*(a^10 - 2*b^10 - 4*a^2
*b^8 - 27*a^4*b^6 + 15*a^6*b^4 + 9*a^8*b^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*
d^4)) + (16*(a^8 + 10*a^2*b^6 + 27*a^4*b^4 + 10*a^6*b^2))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5
+ 4*a^6*b^3*d^5)))*(1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*2i - atan
((((1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2)*((((((((8*(96*a^2*b^14*d^4
+ 480*a^4*b^12*d^4 + 960*a^6*b^10*d^4 + 960*a^8*b^8*d^4 + 480*a^10*b^6*d^4 + 96*a^12*b^4*d^4))/(b^9*d^5 + a^8
*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) - (8*tan(c + d*x)^(1/2)*(1/(a^4*d^2*1i + b^4*d^2*1i +
4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2)*(32*b^18*d^4 + 160*a^2*b^16*d^4 + 288*a^4*b^14*d^4 + 160*a^
6*b^12*d^4 - 160*a^8*b^10*d^4 - 288*a^10*b^8*d^4 - 160*a^12*b^6*d^4 - 32*a^14*b^4*d^4))/(b^9*d^4 + a^8*b*d^4 +
4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2
*b^2*d^2*6i))^(1/2))/2 + (16*tan(c + d*x)^(1/2)*(60*a*b^13*d^2 + 8*a^13*b*d^2 + 52*a^3*b^11*d^2 + 128*a^5*b^9*
d^2 + 424*a^7*b^7*d^2 + 380*a^9*b^5*d^2 + 100*a^11*b^3*d^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*
d^4 + 4*a^6*b^3*d^4))*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2))/2 + (8
*(4*a*b^11*d^2 + 16*a^11*b*d^2 - 304*a^3*b^9*d^2 - 120*a^5*b^7*d^2 + 320*a^7*b^5*d^2 + 148*a^9*b^3*d^2))/(b^9*
d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5))*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 -
4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2))/2 + (16*tan(c + d*x)^(1/2)*(a^10 - 2*b^10 - 4*a^2*b^8 - 27*a^4*b^6 + 15*
a^6*b^4 + 9*a^8*b^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*1i)/2 - ((1/(a^4*
d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2)*((((((((8*(96*a^2*b^14*d^4 + 480*a^4*
b^12*d^4 + 960*a^6*b^10*d^4 + 960*a^8*b^8*d^4 + 480*a^10*b^6*d^4 + 96*a^12*b^4*d^4))/(b^9*d^5 + a^8*b*d^5 + 4*
a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) + (8*tan(c + d*x)^(1/2)*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2
- 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2)*(32*b^18*d^4 + 160*a^2*b^16*d^4 + 288*a^4*b^14*d^4 + 160*a^6*b^12*d^4
- 160*a^8*b^10*d^4 - 288*a^10*b^8*d^4 - 160*a^12*b^6*d^4 - 32*a^14*b^4*d^4))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*
d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i
))^(1/2))/2 - (16*tan(c + d*x)^(1/2)*(60*a*b^13*d^2 + 8*a^13*b*d^2 + 52*a^3*b^11*d^2 + 128*a^5*b^9*d^2 + 424*a
^7*b^7*d^2 + 380*a^9*b^5*d^2 + 100*a^11*b^3*d^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6
*b^3*d^4))*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2))/2 + (8*(4*a*b^11*
d^2 + 16*a^11*b*d^2 - 304*a^3*b^9*d^2 - 120*a^5*b^7*d^2 + 320*a^7*b^5*d^2 + 148*a^9*b^3*d^2))/(b^9*d^5 + a^8*b
*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5))*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2
- a^2*b^2*d^2*6i))^(1/2))/2 - (16*tan(c + d*x)^(1/2)*(a^10 - 2*b^10 - 4*a^2*b^8 - 27*a^4*b^6 + 15*a^6*b^4 + 9
*a^8*b^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*1i)/2)/(((1/(a^4*d^2*1i + b^
4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2)*((((((((8*(96*a^2*b^14*d^4 + 480*a^4*b^12*d^4 +
960*a^6*b^10*d^4 + 960*a^8*b^8*d^4 + 480*a^10*b^6*d^4 + 96*a^12*b^4*d^4))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5
+ 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) - (8*tan(c + d*x)^(1/2)*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*
d^2 - a^2*b^2*d^2*6i))^(1/2)*(32*b^18*d^4 + 160*a^2*b^16*d^4 + 288*a^4*b^14*d^4 + 160*a^6*b^12*d^4 - 160*a^8*b
^10*d^4 - 288*a^10*b^8*d^4 - 160*a^12*b^6*d^4 - 32*a^14*b^4*d^4))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4
*b^5*d^4 + 4*a^6*b^3*d^4))*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2))/2
+ (16*tan(c + d*x)^(1/2)*(60*a*b^13*d^2 + 8*a^13*b*d^2 + 52*a^3*b^11*d^2 + 128*a^5*b^9*d^2 + 424*a^7*b^7*d^2
+ 380*a^9*b^5*d^2 + 100*a^11*b^3*d^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*
(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2))/2 + (8*(4*a*b^11*d^2 + 16*a^
11*b*d^2 - 304*a^3*b^9*d^2 - 120*a^5*b^7*d^2 + 320*a^7*b^5*d^2 + 148*a^9*b^3*d^2))/(b^9*d^5 + a^8*b*d^5 + 4*a^
2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5))*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*
d^2*6i))^(1/2))/2 + (16*tan(c + d*x)^(1/2)*(a^10 - 2*b^10 - 4*a^2*b^8 - 27*a^4*b^6 + 15*a^6*b^4 + 9*a^8*b^2))/
(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4)))/2 + ((1/(a^4*d^2*1i + b^4*d^2*1i + 4*a
*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2)*((((((((8*(96*a^2*b^14*d^4 + 480*a^4*b^12*d^4 + 960*a^6*b^10*d
^4 + 960*a^8*b^8*d^4 + 480*a^10*b^6*d^4 + 96*a^12*b^4*d^4))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d
^5 + 4*a^6*b^3*d^5) + (8*tan(c + d*x)^(1/2)*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*
d^2*6i))^(1/2)*(32*b^18*d^4 + 160*a^2*b^16*d^4 + 288*a^4*b^14*d^4 + 160*a^6*b^12*d^4 - 160*a^8*b^10*d^4 - 288*
a^10*b^8*d^4 - 160*a^12*b^6*d^4 - 32*a^14*b^4*d^4))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a
^6*b^3*d^4))*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2))/2 - (16*tan(c +
d*x)^(1/2)*(60*a*b^13*d^2 + 8*a^13*b*d^2 + 52*a^3*b^11*d^2 + 128*a^5*b^9*d^2 + 424*a^7*b^7*d^2 + 380*a^9*b^5*
d^2 + 100*a^11*b^3*d^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*(1/(a^4*d^2*1i
+ b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2))/2 + (8*(4*a*b^11*d^2 + 16*a^11*b*d^2 - 304
*a^3*b^9*d^2 - 120*a^5*b^7*d^2 + 320*a^7*b^5*d^2 + 148*a^9*b^3*d^2))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*
a^4*b^5*d^5 + 4*a^6*b^3*d^5))*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2)
)/2 - (16*tan(c + d*x)^(1/2)*(a^10 - 2*b^10 - 4*a^2*b^8 - 27*a^4*b^6 + 15*a^6*b^4 + 9*a^8*b^2))/(b^9*d^4 + a^8
*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4)))/2 + (16*(a^8 + 10*a^2*b^6 + 27*a^4*b^4 + 10*a^6*b^2)
)/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5)))*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^
3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2)*1i - (atan((((a^2 + 5*b^2)*((16*tan(c + d*x)^(1/2)*(a^10 - 2*b^10
- 4*a^2*b^8 - 27*a^4*b^6 + 15*a^6*b^4 + 9*a^8*b^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*
a^6*b^3*d^4) + ((a^2 + 5*b^2)*(-a^3*b^3)^(1/2)*((8*(4*a*b^11*d^2 + 16*a^11*b*d^2 - 304*a^3*b^9*d^2 - 120*a^5*b
^7*d^2 + 320*a^7*b^5*d^2 + 148*a^9*b^3*d^2))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*
d^5) + (((16*tan(c + d*x)^(1/2)*(60*a*b^13*d^2 + 8*a^13*b*d^2 + 52*a^3*b^11*d^2 + 128*a^5*b^9*d^2 + 424*a^7*b^
7*d^2 + 380*a^9*b^5*d^2 + 100*a^11*b^3*d^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*
d^4) + ((a^2 + 5*b^2)*(-a^3*b^3)^(1/2)*((8*(96*a^2*b^14*d^4 + 480*a^4*b^12*d^4 + 960*a^6*b^10*d^4 + 960*a^8*b^
8*d^4 + 480*a^10*b^6*d^4 + 96*a^12*b^4*d^4))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*
d^5) - (8*tan(c + d*x)^(1/2)*(a^2 + 5*b^2)*(-a^3*b^3)^(1/2)*(32*b^18*d^4 + 160*a^2*b^16*d^4 + 288*a^4*b^14*d^4
+ 160*a^6*b^12*d^4 - 160*a^8*b^10*d^4 - 288*a^10*b^8*d^4 - 160*a^12*b^6*d^4 - 32*a^14*b^4*d^4))/((b^7*d + 2*a
^2*b^5*d + a^4*b^3*d)*(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))))/(2*(b^7*d + 2*a
^2*b^5*d + a^4*b^3*d)))*(a^2 + 5*b^2)*(-a^3*b^3)^(1/2))/(2*(b^7*d + 2*a^2*b^5*d + a^4*b^3*d))))/(2*(b^7*d + 2*
a^2*b^5*d + a^4*b^3*d)))*(-a^3*b^3)^(1/2)*1i)/(2*(b^7*d + 2*a^2*b^5*d + a^4*b^3*d)) + ((a^2 + 5*b^2)*((16*tan(
c + d*x)^(1/2)*(a^10 - 2*b^10 - 4*a^2*b^8 - 27*a^4*b^6 + 15*a^6*b^4 + 9*a^8*b^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2
*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4) - ((a^2 + 5*b^2)*(-a^3*b^3)^(1/2)*((8*(4*a*b^11*d^2 + 16*a^11*b*d^2
- 304*a^3*b^9*d^2 - 120*a^5*b^7*d^2 + 320*a^7*b^5*d^2 + 148*a^9*b^3*d^2))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5
+ 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) - (((16*tan(c + d*x)^(1/2)*(60*a*b^13*d^2 + 8*a^13*b*d^2 + 52*a^3*b^11*d^2 +
128*a^5*b^9*d^2 + 424*a^7*b^7*d^2 + 380*a^9*b^5*d^2 + 100*a^11*b^3*d^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4
+ 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4) - ((a^2 + 5*b^2)*(-a^3*b^3)^(1/2)*((8*(96*a^2*b^14*d^4 + 480*a^4*b^12*d^4 +
960*a^6*b^10*d^4 + 960*a^8*b^8*d^4 + 480*a^10*b^6*d^4 + 96*a^12*b^4*d^4))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5
+ 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) + (8*tan(c + d*x)^(1/2)*(a^2 + 5*b^2)*(-a^3*b^3)^(1/2)*(32*b^18*d^4 + 160*a^
2*b^16*d^4 + 288*a^4*b^14*d^4 + 160*a^6*b^12*d^4 - 160*a^8*b^10*d^4 - 288*a^10*b^8*d^4 - 160*a^12*b^6*d^4 - 32
*a^14*b^4*d^4))/((b^7*d + 2*a^2*b^5*d + a^4*b^3*d)*(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^
6*b^3*d^4))))/(2*(b^7*d + 2*a^2*b^5*d + a^4*b^3*d)))*(a^2 + 5*b^2)*(-a^3*b^3)^(1/2))/(2*(b^7*d + 2*a^2*b^5*d +
a^4*b^3*d))))/(2*(b^7*d + 2*a^2*b^5*d + a^4*b^3*d)))*(-a^3*b^3)^(1/2)*1i)/(2*(b^7*d + 2*a^2*b^5*d + a^4*b^3*d
)))/((16*(a^8 + 10*a^2*b^6 + 27*a^4*b^4 + 10*a^6*b^2))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 +
4*a^6*b^3*d^5) + ((a^2 + 5*b^2)*((16*tan(c + d*x)^(1/2)*(a^10 - 2*b^10 - 4*a^2*b^8 - 27*a^4*b^6 + 15*a^6*b^4 +
9*a^8*b^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4) + ((a^2 + 5*b^2)*(-a^3*b^3)
^(1/2)*((8*(4*a*b^11*d^2 + 16*a^11*b*d^2 - 304*a^3*b^9*d^2 - 120*a^5*b^7*d^2 + 320*a^7*b^5*d^2 + 148*a^9*b^3*d
^2))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) + (((16*tan(c + d*x)^(1/2)*(60*a*b^
13*d^2 + 8*a^13*b*d^2 + 52*a^3*b^11*d^2 + 128*a^5*b^9*d^2 + 424*a^7*b^7*d^2 + 380*a^9*b^5*d^2 + 100*a^11*b^3*d
^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4) + ((a^2 + 5*b^2)*(-a^3*b^3)^(1/2)*(
(8*(96*a^2*b^14*d^4 + 480*a^4*b^12*d^4 + 960*a^6*b^10*d^4 + 960*a^8*b^8*d^4 + 480*a^10*b^6*d^4 + 96*a^12*b^4*d
^4))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) - (8*tan(c + d*x)^(1/2)*(a^2 + 5*b^
2)*(-a^3*b^3)^(1/2)*(32*b^18*d^4 + 160*a^2*b^16*d^4 + 288*a^4*b^14*d^4 + 160*a^6*b^12*d^4 - 160*a^8*b^10*d^4 -
288*a^10*b^8*d^4 - 160*a^12*b^6*d^4 - 32*a^14*b^4*d^4))/((b^7*d + 2*a^2*b^5*d + a^4*b^3*d)*(b^9*d^4 + a^8*b*d
^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))))/(2*(b^7*d + 2*a^2*b^5*d + a^4*b^3*d)))*(a^2 + 5*b^2)*(-
a^3*b^3)^(1/2))/(2*(b^7*d + 2*a^2*b^5*d + a^4*b^3*d))))/(2*(b^7*d + 2*a^2*b^5*d + a^4*b^3*d)))*(-a^3*b^3)^(1/2
))/(2*(b^7*d + 2*a^2*b^5*d + a^4*b^3*d)) - ((a^2 + 5*b^2)*((16*tan(c + d*x)^(1/2)*(a^10 - 2*b^10 - 4*a^2*b^8 -
27*a^4*b^6 + 15*a^6*b^4 + 9*a^8*b^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4) -
((a^2 + 5*b^2)*(-a^3*b^3)^(1/2)*((8*(4*a*b^11*d^2 + 16*a^11*b*d^2 - 304*a^3*b^9*d^2 - 120*a^5*b^7*d^2 + 320*a
^7*b^5*d^2 + 148*a^9*b^3*d^2))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) - (((16*t
an(c + d*x)^(1/2)*(60*a*b^13*d^2 + 8*a^13*b*d^2 + 52*a^3*b^11*d^2 + 128*a^5*b^9*d^2 + 424*a^7*b^7*d^2 + 380*a^
9*b^5*d^2 + 100*a^11*b^3*d^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4) - ((a^2 +
5*b^2)*(-a^3*b^3)^(1/2)*((8*(96*a^2*b^14*d^4 + 480*a^4*b^12*d^4 + 960*a^6*b^10*d^4 + 960*a^8*b^8*d^4 + 480*a^
10*b^6*d^4 + 96*a^12*b^4*d^4))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) + (8*tan(
c + d*x)^(1/2)*(a^2 + 5*b^2)*(-a^3*b^3)^(1/2)*(32*b^18*d^4 + 160*a^2*b^16*d^4 + 288*a^4*b^14*d^4 + 160*a^6*b^1
2*d^4 - 160*a^8*b^10*d^4 - 288*a^10*b^8*d^4 - 160*a^12*b^6*d^4 - 32*a^14*b^4*d^4))/((b^7*d + 2*a^2*b^5*d + a^4
*b^3*d)*(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))))/(2*(b^7*d + 2*a^2*b^5*d + a^4
*b^3*d)))*(a^2 + 5*b^2)*(-a^3*b^3)^(1/2))/(2*(b^7*d + 2*a^2*b^5*d + a^4*b^3*d))))/(2*(b^7*d + 2*a^2*b^5*d + a^
4*b^3*d)))*(-a^3*b^3)^(1/2))/(2*(b^7*d + 2*a^2*b^5*d + a^4*b^3*d))))*(a^2 + 5*b^2)*(-a^3*b^3)^(1/2)*1i)/(b^7*d
+ 2*a^2*b^5*d + a^4*b^3*d) - (a^2*tan(c + d*x)^(1/2))/(b*(a*d + b*d*tan(c + d*x))*(a^2 + b^2))