3.1124 \(\int \frac{1}{(a+i a \tan (e+f x))^3 \sqrt{c+d \tan (e+f x)}} \, dx\)

Optimal. Leaf size=298 \[ \frac{\left (2 c^2+7 i c d-10 d^2\right ) \sqrt{c+d \tan (e+f x)}}{16 f (-d+i c)^3 \left (a^3+i a^3 \tan (e+f x)\right )}+\frac{\left (-8 c^2 d+2 i c^3-13 i c d^2+12 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{16 a^3 f (c+i d)^{7/2}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{8 a^3 f \sqrt{c-i d}}+\frac{(-8 d+3 i c) \sqrt{c+d \tan (e+f x)}}{24 a f (c+i d)^2 (a+i a \tan (e+f x))^2}-\frac{\sqrt{c+d \tan (e+f x)}}{6 f (-d+i c) (a+i a \tan (e+f x))^3} \]

[Out]

((-I/8)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^3*Sqrt[c - I*d]*f) + (((2*I)*c^3 - 8*c^2*d - (13*I
)*c*d^2 + 12*d^3)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(16*a^3*(c + I*d)^(7/2)*f) - Sqrt[c + d*Tan
[e + f*x]]/(6*(I*c - d)*f*(a + I*a*Tan[e + f*x])^3) + (((3*I)*c - 8*d)*Sqrt[c + d*Tan[e + f*x]])/(24*a*(c + I*
d)^2*f*(a + I*a*Tan[e + f*x])^2) + ((2*c^2 + (7*I)*c*d - 10*d^2)*Sqrt[c + d*Tan[e + f*x]])/(16*(I*c - d)^3*f*(
a^3 + I*a^3*Tan[e + f*x]))

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Rubi [A]  time = 0.956889, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3559, 3596, 3539, 3537, 63, 208} \[ \frac{\left (2 c^2+7 i c d-10 d^2\right ) \sqrt{c+d \tan (e+f x)}}{16 f (-d+i c)^3 \left (a^3+i a^3 \tan (e+f x)\right )}+\frac{\left (-8 c^2 d+2 i c^3-13 i c d^2+12 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{16 a^3 f (c+i d)^{7/2}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{8 a^3 f \sqrt{c-i d}}+\frac{(-8 d+3 i c) \sqrt{c+d \tan (e+f x)}}{24 a f (c+i d)^2 (a+i a \tan (e+f x))^2}-\frac{\sqrt{c+d \tan (e+f x)}}{6 f (-d+i c) (a+i a \tan (e+f x))^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + I*a*Tan[e + f*x])^3*Sqrt[c + d*Tan[e + f*x]]),x]

[Out]

((-I/8)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^3*Sqrt[c - I*d]*f) + (((2*I)*c^3 - 8*c^2*d - (13*I
)*c*d^2 + 12*d^3)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(16*a^3*(c + I*d)^(7/2)*f) - Sqrt[c + d*Tan
[e + f*x]]/(6*(I*c - d)*f*(a + I*a*Tan[e + f*x])^3) + (((3*I)*c - 8*d)*Sqrt[c + d*Tan[e + f*x]])/(24*a*(c + I*
d)^2*f*(a + I*a*Tan[e + f*x])^2) + ((2*c^2 + (7*I)*c*d - 10*d^2)*Sqrt[c + d*Tan[e + f*x]])/(16*(I*c - d)^3*f*(
a^3 + I*a^3*Tan[e + f*x]))

Rule 3559

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

Rule 3596

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

Rule 3539

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c
+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x]
)^m*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]
&& NeQ[c^2 + d^2, 0] &&  !IntegerQ[m]

Rule 3537

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

Rule 63

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 208

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

Rubi steps

\begin{align*} \int \frac{1}{(a+i a \tan (e+f x))^3 \sqrt{c+d \tan (e+f x)}} \, dx &=-\frac{\sqrt{c+d \tan (e+f x)}}{6 (i c-d) f (a+i a \tan (e+f x))^3}-\frac{\int \frac{-\frac{1}{2} a (6 i c-11 d)-\frac{5}{2} i a d \tan (e+f x)}{(a+i a \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)}} \, dx}{6 a^2 (i c-d)}\\ &=-\frac{\sqrt{c+d \tan (e+f x)}}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac{(3 i c-8 d) \sqrt{c+d \tan (e+f x)}}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2}-\frac{\int \frac{-\frac{3}{2} a^2 \left (4 c^2+11 i c d-12 d^2\right )-\frac{3}{2} a^2 (3 c+8 i d) d \tan (e+f x)}{(a+i a \tan (e+f x)) \sqrt{c+d \tan (e+f x)}} \, dx}{24 a^4 (c+i d)^2}\\ &=-\frac{\sqrt{c+d \tan (e+f x)}}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac{(3 i c-8 d) \sqrt{c+d \tan (e+f x)}}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2}+\frac{\left (2 c^2+7 i c d-10 d^2\right ) \sqrt{c+d \tan (e+f x)}}{16 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right )}-\frac{\int \frac{\frac{3}{2} a^3 (i c-2 d) \left (4 c^2+6 i c d-7 d^2\right )+\frac{3}{2} a^3 d \left (2 i c^2-7 c d-10 i d^2\right ) \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx}{48 a^6 (i c-d)^3}\\ &=-\frac{\sqrt{c+d \tan (e+f x)}}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac{(3 i c-8 d) \sqrt{c+d \tan (e+f x)}}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2}+\frac{\left (2 c^2+7 i c d-10 d^2\right ) \sqrt{c+d \tan (e+f x)}}{16 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac{\int \frac{1+i \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx}{16 a^3}+\frac{\left (2 c^3+8 i c^2 d-13 c d^2-12 i d^3\right ) \int \frac{1-i \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx}{32 a^3 (c+i d)^3}\\ &=-\frac{\sqrt{c+d \tan (e+f x)}}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac{(3 i c-8 d) \sqrt{c+d \tan (e+f x)}}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2}+\frac{\left (2 c^2+7 i c d-10 d^2\right ) \sqrt{c+d \tan (e+f x)}}{16 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac{i \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{c-i d x}} \, dx,x,i \tan (e+f x)\right )}{16 a^3 f}-\frac{\left (2 i c^3-8 c^2 d-13 i c d^2+12 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{32 a^3 (c+i d)^3 f}\\ &=-\frac{\sqrt{c+d \tan (e+f x)}}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac{(3 i c-8 d) \sqrt{c+d \tan (e+f x)}}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2}+\frac{\left (2 c^2+7 i c d-10 d^2\right ) \sqrt{c+d \tan (e+f x)}}{16 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-\frac{i c}{d}+\frac{i x^2}{d}} \, dx,x,\sqrt{c+d \tan (e+f x)}\right )}{8 a^3 d f}-\frac{\left (2 c^3+8 i c^2 d-13 c d^2-12 i d^3\right ) \operatorname{Subst}\left (\int \frac{1}{-1+\frac{i c}{d}-\frac{i x^2}{d}} \, dx,x,\sqrt{c+d \tan (e+f x)}\right )}{16 a^3 (c+i d)^3 d f}\\ &=-\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{8 a^3 \sqrt{c-i d} f}+\frac{\left (2 i c^3-8 c^2 d-13 i c d^2+12 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{16 a^3 (c+i d)^{7/2} f}-\frac{\sqrt{c+d \tan (e+f x)}}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac{(3 i c-8 d) \sqrt{c+d \tan (e+f x)}}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2}+\frac{\left (2 c^2+7 i c d-10 d^2\right ) \sqrt{c+d \tan (e+f x)}}{16 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right )}\\ \end{align*}

Mathematica [A]  time = 2.98672, size = 324, normalized size = 1.09 \[ \frac{\sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \left (\frac{2 \cos (e+f x) (\sin (3 f x)+i \cos (3 f x)) \sqrt{c+d \tan (e+f x)} \left (i \left (9 c^2+32 i c d-38 d^2\right ) \sin (2 (e+f x))+\left (13 c^2+40 i c d-42 d^2\right ) \cos (2 (e+f x))+7 c^2+19 i c d-12 d^2\right )}{3 (c+i d)^3}-\frac{2 (\cos (3 e)+i \sin (3 e)) \left (\sqrt{-c+i d} \left (8 c^2 d-2 i c^3+13 i c d^2-12 d^3\right ) \tan ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c-i d}}\right )-2 i (-c-i d)^{7/2} \tan ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c+i d}}\right )\right )}{(-c-i d)^{7/2} \sqrt{-c+i d}}\right )}{32 f (a+i a \tan (e+f x))^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + I*a*Tan[e + f*x])^3*Sqrt[c + d*Tan[e + f*x]]),x]

[Out]

(Sec[e + f*x]^3*(Cos[f*x] + I*Sin[f*x])^3*((-2*(Sqrt[-c + I*d]*((-2*I)*c^3 + 8*c^2*d + (13*I)*c*d^2 - 12*d^3)*
ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]] - (2*I)*(-c - I*d)^(7/2)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[
-c + I*d]])*(Cos[3*e] + I*Sin[3*e]))/((-c - I*d)^(7/2)*Sqrt[-c + I*d]) + (2*Cos[e + f*x]*(I*Cos[3*f*x] + Sin[3
*f*x])*(7*c^2 + (19*I)*c*d - 12*d^2 + (13*c^2 + (40*I)*c*d - 42*d^2)*Cos[2*(e + f*x)] + I*(9*c^2 + (32*I)*c*d
- 38*d^2)*Sin[2*(e + f*x)])*Sqrt[c + d*Tan[e + f*x]])/(3*(c + I*d)^3)))/(32*f*(a + I*a*Tan[e + f*x])^3)

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Maple [B]  time = 0.092, size = 1105, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^3,x)

[Out]

13/16*I/f/a^3*d^2/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*
c+19/12*I/f/a^3*d^4/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(3/2)+1/8/f/a^3*d/(-
I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(5/2)*c^2-5/8/f/a^3*d^3/(-I*d+d*tan(f*x+e)
)^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(5/2)+13/16*I/f/a^3*d^2/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*
d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(1/2)*c^3-1/8*I/f/a^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)/(-I*d-c)^(1/2)*arctan
((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c^3-1/4/f/a^3*d/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(
c+d*tan(f*x+e))^(3/2)*c^3+31/12/f/a^3*d^3/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e)
)^(3/2)*c-45/16*I/f/a^3*d^4/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(1/2)*c-5/4*
I/f/a^3*d^2/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(3/2)*c^2+1/8/f/a^3*d/(-I*d+
d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(1/2)*c^4-19/8/f/a^3*d^3/(-I*d+d*tan(f*x+e))^3
/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(1/2)*c^2+9/8/f/a^3*d^5/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2
+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(1/2)+1/8*I/f/a^3/(I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))+
7/16*I/f/a^3*d^2/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(5/2)*c+1/2/f/a^3*d/(-I
*d^3-3*c*d^2+3*I*c^2*d+c^3)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c^2-3/4/f/a^3*d^3/(-I
*d^3-3*c*d^2+3*I*c^2*d+c^3)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [B]  time = 6.31359, size = 4361, normalized size = 14.63 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

((-24*I*a^3*c^3 + 72*a^3*c^2*d + 72*I*a^3*c*d^2 - 24*a^3*d^3)*f*sqrt(1/64*I/((-I*a^6*c - a^6*d)*f^2))*e^(6*I*f
*x + 6*I*e)*log(-2*(8*((I*a^3*c + a^3*d)*f*e^(2*I*f*x + 2*I*e) + (I*a^3*c + a^3*d)*f)*sqrt(((c - I*d)*e^(2*I*f
*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(1/64*I/((-I*a^6*c - a^6*d)*f^2)) - (c - I*d)*e^(2*I*f*x
+ 2*I*e) - c)*e^(-2*I*f*x - 2*I*e)) + (24*I*a^3*c^3 - 72*a^3*c^2*d - 72*I*a^3*c*d^2 + 24*a^3*d^3)*f*sqrt(1/64
*I/((-I*a^6*c - a^6*d)*f^2))*e^(6*I*f*x + 6*I*e)*log(-2*(8*((-I*a^3*c - a^3*d)*f*e^(2*I*f*x + 2*I*e) + (-I*a^3
*c - a^3*d)*f)*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(1/64*I/((-I*a^6*
c - a^6*d)*f^2)) - (c - I*d)*e^(2*I*f*x + 2*I*e) - c)*e^(-2*I*f*x - 2*I*e)) + (24*I*a^3*c^3 - 72*a^3*c^2*d - 7
2*I*a^3*c*d^2 + 24*a^3*d^3)*f*sqrt((-4*I*c^6 + 32*c^5*d + 116*I*c^4*d^2 - 256*c^3*d^3 - 361*I*c^2*d^4 + 312*c*
d^5 + 144*I*d^6)/((256*I*a^6*c^7 - 1792*a^6*c^6*d - 5376*I*a^6*c^5*d^2 + 8960*a^6*c^4*d^3 + 8960*I*a^6*c^3*d^4
- 5376*a^6*c^2*d^5 - 1792*I*a^6*c*d^6 + 256*a^6*d^7)*f^2))*e^(6*I*f*x + 6*I*e)*log((2*I*c^4 - 10*c^3*d - 21*I
*c^2*d^2 + 25*c*d^3 + 12*I*d^4 + ((16*a^3*c^4 + 64*I*a^3*c^3*d - 96*a^3*c^2*d^2 - 64*I*a^3*c*d^3 + 16*a^3*d^4)
*f*e^(2*I*f*x + 2*I*e) + (16*a^3*c^4 + 64*I*a^3*c^3*d - 96*a^3*c^2*d^2 - 64*I*a^3*c*d^3 + 16*a^3*d^4)*f)*sqrt(
((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt((-4*I*c^6 + 32*c^5*d + 116*I*c^4*d^2
- 256*c^3*d^3 - 361*I*c^2*d^4 + 312*c*d^5 + 144*I*d^6)/((256*I*a^6*c^7 - 1792*a^6*c^6*d - 5376*I*a^6*c^5*d^2
+ 8960*a^6*c^4*d^3 + 8960*I*a^6*c^3*d^4 - 5376*a^6*c^2*d^5 - 1792*I*a^6*c*d^6 + 256*a^6*d^7)*f^2)) + (2*I*c^4
- 8*c^3*d - 13*I*c^2*d^2 + 12*c*d^3)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/((16*a^3*c^4 + 64*I*a^3*c^3*d -
96*a^3*c^2*d^2 - 64*I*a^3*c*d^3 + 16*a^3*d^4)*f)) + (-24*I*a^3*c^3 + 72*a^3*c^2*d + 72*I*a^3*c*d^2 - 24*a^3*d
^3)*f*sqrt((-4*I*c^6 + 32*c^5*d + 116*I*c^4*d^2 - 256*c^3*d^3 - 361*I*c^2*d^4 + 312*c*d^5 + 144*I*d^6)/((256*I
*a^6*c^7 - 1792*a^6*c^6*d - 5376*I*a^6*c^5*d^2 + 8960*a^6*c^4*d^3 + 8960*I*a^6*c^3*d^4 - 5376*a^6*c^2*d^5 - 17
92*I*a^6*c*d^6 + 256*a^6*d^7)*f^2))*e^(6*I*f*x + 6*I*e)*log((2*I*c^4 - 10*c^3*d - 21*I*c^2*d^2 + 25*c*d^3 + 12
*I*d^4 - ((16*a^3*c^4 + 64*I*a^3*c^3*d - 96*a^3*c^2*d^2 - 64*I*a^3*c*d^3 + 16*a^3*d^4)*f*e^(2*I*f*x + 2*I*e) +
(16*a^3*c^4 + 64*I*a^3*c^3*d - 96*a^3*c^2*d^2 - 64*I*a^3*c*d^3 + 16*a^3*d^4)*f)*sqrt(((c - I*d)*e^(2*I*f*x +
2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt((-4*I*c^6 + 32*c^5*d + 116*I*c^4*d^2 - 256*c^3*d^3 - 361*I*c
^2*d^4 + 312*c*d^5 + 144*I*d^6)/((256*I*a^6*c^7 - 1792*a^6*c^6*d - 5376*I*a^6*c^5*d^2 + 8960*a^6*c^4*d^3 + 896
0*I*a^6*c^3*d^4 - 5376*a^6*c^2*d^5 - 1792*I*a^6*c*d^6 + 256*a^6*d^7)*f^2)) + (2*I*c^4 - 8*c^3*d - 13*I*c^2*d^2
+ 12*c*d^3)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/((16*a^3*c^4 + 64*I*a^3*c^3*d - 96*a^3*c^2*d^2 - 64*I*a
^3*c*d^3 + 16*a^3*d^4)*f)) - (2*c^2 + 4*I*c*d - 2*d^2 + (11*c^2 + 36*I*c*d - 40*d^2)*e^(6*I*f*x + 6*I*e) + (18
*c^2 + 55*I*c*d - 52*d^2)*e^(4*I*f*x + 4*I*e) + (9*c^2 + 23*I*c*d - 14*d^2)*e^(2*I*f*x + 2*I*e))*sqrt(((c - I*
d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1)))*e^(-6*I*f*x - 6*I*e)/((96*I*a^3*c^3 - 288*a^3*c^
2*d - 288*I*a^3*c*d^2 + 96*a^3*d^3)*f)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+d*tan(f*x+e))**(1/2)/(a+I*a*tan(f*x+e))**3,x)

[Out]

Exception raised: AttributeError

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Giac [B]  time = 1.67949, size = 915, normalized size = 3.07 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^3,x, algorithm="giac")

[Out]

-1/2*d^4*(8*(2*c^3 + 8*I*c^2*d - 13*c*d^2 - 12*I*d^3)*arctan(4*(sqrt(d*tan(f*x + e) + c)*c - sqrt(c^2 + d^2)*s
qrt(d*tan(f*x + e) + c))/(c*sqrt(-8*c + 8*sqrt(c^2 + d^2)) + I*sqrt(-8*c + 8*sqrt(c^2 + d^2))*d - sqrt(c^2 + d
^2)*sqrt(-8*c + 8*sqrt(c^2 + d^2))))/((-16*I*a^3*c^3*d^4*f + 48*a^3*c^2*d^5*f + 48*I*a^3*c*d^6*f - 16*a^3*d^7*
f)*sqrt(-8*c + 8*sqrt(c^2 + d^2))*(I*d/(c - sqrt(c^2 + d^2)) + 1)) - 4*(-6*I*(d*tan(f*x + e) + c)^(5/2)*c^2 +
12*I*(d*tan(f*x + e) + c)^(3/2)*c^3 - 6*I*sqrt(d*tan(f*x + e) + c)*c^4 + 21*(d*tan(f*x + e) + c)^(5/2)*c*d - 6
0*(d*tan(f*x + e) + c)^(3/2)*c^2*d + 39*sqrt(d*tan(f*x + e) + c)*c^3*d + 30*I*(d*tan(f*x + e) + c)^(5/2)*d^2 -
124*I*(d*tan(f*x + e) + c)^(3/2)*c*d^2 + 114*I*sqrt(d*tan(f*x + e) + c)*c^2*d^2 + 76*(d*tan(f*x + e) + c)^(3/
2)*d^3 - 135*sqrt(d*tan(f*x + e) + c)*c*d^3 - 54*I*sqrt(d*tan(f*x + e) + c)*d^4)/((-96*I*a^3*c^3*d^3*f + 288*a
^3*c^2*d^4*f + 288*I*a^3*c*d^5*f - 96*a^3*d^6*f)*(d*tan(f*x + e) - I*d)^3) - I*arctan(4*(sqrt(d*tan(f*x + e) +
c)*c - sqrt(c^2 + d^2)*sqrt(d*tan(f*x + e) + c))/(c*sqrt(-8*c + 8*sqrt(c^2 + d^2)) - I*sqrt(-8*c + 8*sqrt(c^2
+ d^2))*d - sqrt(c^2 + d^2)*sqrt(-8*c + 8*sqrt(c^2 + d^2))))/(a^3*sqrt(-8*c + 8*sqrt(c^2 + d^2))*d^4*f*(-I*d/
(c - sqrt(c^2 + d^2)) + 1)))