3.1071 \(\int (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^2 \, dx\)

Optimal. Leaf size=153 \[ -\frac{2 a^3 (c-i d)^2 \tan (e+f x)}{f}-\frac{4 i a^3 (c-i d)^2 \log (\cos (e+f x))}{f}+4 a^3 x (c-i d)^2+\frac{2 c d (a+i a \tan (e+f x))^3}{3 f}+\frac{i a (c-i d)^2 (a+i a \tan (e+f x))^2}{2 f}-\frac{i d^2 (a+i a \tan (e+f x))^4}{4 a f} \]

[Out]

4*a^3*(c - I*d)^2*x - ((4*I)*a^3*(c - I*d)^2*Log[Cos[e + f*x]])/f - (2*a^3*(c - I*d)^2*Tan[e + f*x])/f + ((I/2
)*a*(c - I*d)^2*(a + I*a*Tan[e + f*x])^2)/f + (2*c*d*(a + I*a*Tan[e + f*x])^3)/(3*f) - ((I/4)*d^2*(a + I*a*Tan
[e + f*x])^4)/(a*f)

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Rubi [A]  time = 0.196024, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {3543, 3527, 3478, 3477, 3475} \[ -\frac{2 a^3 (c-i d)^2 \tan (e+f x)}{f}-\frac{4 i a^3 (c-i d)^2 \log (\cos (e+f x))}{f}+4 a^3 x (c-i d)^2+\frac{2 c d (a+i a \tan (e+f x))^3}{3 f}+\frac{i a (c-i d)^2 (a+i a \tan (e+f x))^2}{2 f}-\frac{i d^2 (a+i a \tan (e+f x))^4}{4 a f} \]

Antiderivative was successfully verified.

[In]

Int[(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^2,x]

[Out]

4*a^3*(c - I*d)^2*x - ((4*I)*a^3*(c - I*d)^2*Log[Cos[e + f*x]])/f - (2*a^3*(c - I*d)^2*Tan[e + f*x])/f + ((I/2
)*a*(c - I*d)^2*(a + I*a*Tan[e + f*x])^2)/f + (2*c*d*(a + I*a*Tan[e + f*x])^3)/(3*f) - ((I/4)*d^2*(a + I*a*Tan
[e + f*x])^4)/(a*f)

Rule 3543

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

Rule 3527

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

Rule 3478

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a + b*Tan[c + d*x])^(n - 1))/(d*(n - 1)
), x] + Dist[2*a, Int[(a + b*Tan[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && G
tQ[n, 1]

Rule 3477

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^2, x_Symbol] :> Simp[(a^2 - b^2)*x, x] + (Dist[2*a*b, Int[Tan[c + d
*x], x], x] + Simp[(b^2*Tan[c + d*x])/d, x]) /; FreeQ[{a, b, c, d}, x]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^2 \, dx &=-\frac{i d^2 (a+i a \tan (e+f x))^4}{4 a f}+\int (a+i a \tan (e+f x))^3 \left (c^2-d^2+2 c d \tan (e+f x)\right ) \, dx\\ &=\frac{2 c d (a+i a \tan (e+f x))^3}{3 f}-\frac{i d^2 (a+i a \tan (e+f x))^4}{4 a f}+(c-i d)^2 \int (a+i a \tan (e+f x))^3 \, dx\\ &=\frac{i a (c-i d)^2 (a+i a \tan (e+f x))^2}{2 f}+\frac{2 c d (a+i a \tan (e+f x))^3}{3 f}-\frac{i d^2 (a+i a \tan (e+f x))^4}{4 a f}+\left (2 a (c-i d)^2\right ) \int (a+i a \tan (e+f x))^2 \, dx\\ &=4 a^3 (c-i d)^2 x-\frac{2 a^3 (c-i d)^2 \tan (e+f x)}{f}+\frac{i a (c-i d)^2 (a+i a \tan (e+f x))^2}{2 f}+\frac{2 c d (a+i a \tan (e+f x))^3}{3 f}-\frac{i d^2 (a+i a \tan (e+f x))^4}{4 a f}+\left (4 i a^3 (c-i d)^2\right ) \int \tan (e+f x) \, dx\\ &=4 a^3 (c-i d)^2 x-\frac{4 i a^3 (c-i d)^2 \log (\cos (e+f x))}{f}-\frac{2 a^3 (c-i d)^2 \tan (e+f x)}{f}+\frac{i a (c-i d)^2 (a+i a \tan (e+f x))^2}{2 f}+\frac{2 c d (a+i a \tan (e+f x))^3}{3 f}-\frac{i d^2 (a+i a \tan (e+f x))^4}{4 a f}\\ \end{align*}

Mathematica [B]  time = 8.76334, size = 948, normalized size = 6.2 \[ \frac{\left (\frac{1}{3} \cos (3 e)-\frac{1}{3} i \sin (3 e)\right ) \left (-3 \sin (f x) d^2-2 i c \sin (f x) d\right ) (i \tan (e+f x) a+a)^3}{f \left (\cos \left (\frac{e}{2}\right )-\sin \left (\frac{e}{2}\right )\right ) \left (\cos \left (\frac{e}{2}\right )+\sin \left (\frac{e}{2}\right )\right ) (\cos (f x)+i \sin (f x))^3}+\frac{\cos ^2(e+f x) \left (\frac{1}{3} \cos (3 e)-\frac{1}{3} i \sin (3 e)\right ) \left (-9 \sin (f x) c^2+26 i d \sin (f x) c+15 d^2 \sin (f x)\right ) (i \tan (e+f x) a+a)^3}{f \left (\cos \left (\frac{e}{2}\right )-\sin \left (\frac{e}{2}\right )\right ) \left (\cos \left (\frac{e}{2}\right )+\sin \left (\frac{e}{2}\right )\right ) (\cos (f x)+i \sin (f x))^3}+\frac{x \cos ^3(e+f x) \left (2 c^2 \cos ^3(e)-2 d^2 \cos ^3(e)-4 i c d \cos ^3(e)-8 i c^2 \sin (e) \cos ^2(e)+8 i d^2 \sin (e) \cos ^2(e)-16 c d \sin (e) \cos ^2(e)-2 c^2 \cos (e)+2 d^2 \cos (e)-12 c^2 \sin ^2(e) \cos (e)+12 d^2 \sin ^2(e) \cos (e)+24 i c d \sin ^2(e) \cos (e)+4 i c d \cos (e)+8 i c^2 \sin ^3(e)-8 i d^2 \sin ^3(e)+16 c d \sin ^3(e)+4 i c^2 \sin (e)-4 i d^2 \sin (e)+8 c d \sin (e)+2 c^2 \sin ^3(e) \tan (e)-2 d^2 \sin ^3(e) \tan (e)-4 i c d \sin ^3(e) \tan (e)+2 c^2 \sin (e) \tan (e)-2 d^2 \sin (e) \tan (e)-4 i c d \sin (e) \tan (e)+i (c-i d)^2 (4 \cos (3 e)-4 i \sin (3 e)) \tan (e)\right ) (i \tan (e+f x) a+a)^3}{(\cos (f x)+i \sin (f x))^3}+\frac{\cos ^3(e+f x) \left (\cos \left (\frac{3 e}{2}\right ) c^2-i \sin \left (\frac{3 e}{2}\right ) c^2-2 i d \cos \left (\frac{3 e}{2}\right ) c-2 d \sin \left (\frac{3 e}{2}\right ) c-d^2 \cos \left (\frac{3 e}{2}\right )+i d^2 \sin \left (\frac{3 e}{2}\right )\right ) \left (-2 i \cos \left (\frac{3 e}{2}\right ) \log \left (\cos ^2(e+f x)\right )-2 \sin \left (\frac{3 e}{2}\right ) \log \left (\cos ^2(e+f x)\right )\right ) (i \tan (e+f x) a+a)^3}{f (\cos (f x)+i \sin (f x))^3}+\frac{\cos (e+f x) \left (3 \cos (e) c^2-18 i d \cos (e) c+4 d \sin (e) c-15 d^2 \cos (e)-6 i d^2 \sin (e)\right ) \left (-\frac{1}{6} i \cos (3 e)-\frac{1}{6} \sin (3 e)\right ) (i \tan (e+f x) a+a)^3}{f \left (\cos \left (\frac{e}{2}\right )-\sin \left (\frac{e}{2}\right )\right ) \left (\cos \left (\frac{e}{2}\right )+\sin \left (\frac{e}{2}\right )\right ) (\cos (f x)+i \sin (f x))^3}+\frac{\sec (e+f x) \left (-\frac{1}{4} i \cos (3 e) d^2-\frac{1}{4} \sin (3 e) d^2\right ) (i \tan (e+f x) a+a)^3}{f (\cos (f x)+i \sin (f x))^3}+\frac{(c-i d)^2 \cos ^3(e+f x) (4 f x \cos (3 e)-4 i f x \sin (3 e)) (i \tan (e+f x) a+a)^3}{f (\cos (f x)+i \sin (f x))^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^2,x]

[Out]

(Cos[e + f*x]^3*(c^2*Cos[(3*e)/2] - (2*I)*c*d*Cos[(3*e)/2] - d^2*Cos[(3*e)/2] - I*c^2*Sin[(3*e)/2] - 2*c*d*Sin
[(3*e)/2] + I*d^2*Sin[(3*e)/2])*((-2*I)*Cos[(3*e)/2]*Log[Cos[e + f*x]^2] - 2*Log[Cos[e + f*x]^2]*Sin[(3*e)/2])
*(a + I*a*Tan[e + f*x])^3)/(f*(Cos[f*x] + I*Sin[f*x])^3) + (Cos[e + f*x]*(3*c^2*Cos[e] - (18*I)*c*d*Cos[e] - 1
5*d^2*Cos[e] + 4*c*d*Sin[e] - (6*I)*d^2*Sin[e])*((-I/6)*Cos[3*e] - Sin[3*e]/6)*(a + I*a*Tan[e + f*x])^3)/(f*(C
os[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2])*(Cos[f*x] + I*Sin[f*x])^3) + (Sec[e + f*x]*((-I/4)*d^2*Cos[3*e] - (d
^2*Sin[3*e])/4)*(a + I*a*Tan[e + f*x])^3)/(f*(Cos[f*x] + I*Sin[f*x])^3) + ((c - I*d)^2*Cos[e + f*x]^3*(4*f*x*C
os[3*e] - (4*I)*f*x*Sin[3*e])*(a + I*a*Tan[e + f*x])^3)/(f*(Cos[f*x] + I*Sin[f*x])^3) + ((Cos[3*e]/3 - (I/3)*S
in[3*e])*((-2*I)*c*d*Sin[f*x] - 3*d^2*Sin[f*x])*(a + I*a*Tan[e + f*x])^3)/(f*(Cos[e/2] - Sin[e/2])*(Cos[e/2] +
Sin[e/2])*(Cos[f*x] + I*Sin[f*x])^3) + (Cos[e + f*x]^2*(Cos[3*e]/3 - (I/3)*Sin[3*e])*(-9*c^2*Sin[f*x] + (26*I
)*c*d*Sin[f*x] + 15*d^2*Sin[f*x])*(a + I*a*Tan[e + f*x])^3)/(f*(Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2])*(Co
s[f*x] + I*Sin[f*x])^3) + (x*Cos[e + f*x]^3*(-2*c^2*Cos[e] + (4*I)*c*d*Cos[e] + 2*d^2*Cos[e] + 2*c^2*Cos[e]^3
- (4*I)*c*d*Cos[e]^3 - 2*d^2*Cos[e]^3 + (4*I)*c^2*Sin[e] + 8*c*d*Sin[e] - (4*I)*d^2*Sin[e] - (8*I)*c^2*Cos[e]^
2*Sin[e] - 16*c*d*Cos[e]^2*Sin[e] + (8*I)*d^2*Cos[e]^2*Sin[e] - 12*c^2*Cos[e]*Sin[e]^2 + (24*I)*c*d*Cos[e]*Sin
[e]^2 + 12*d^2*Cos[e]*Sin[e]^2 + (8*I)*c^2*Sin[e]^3 + 16*c*d*Sin[e]^3 - (8*I)*d^2*Sin[e]^3 + 2*c^2*Sin[e]*Tan[
e] - (4*I)*c*d*Sin[e]*Tan[e] - 2*d^2*Sin[e]*Tan[e] + 2*c^2*Sin[e]^3*Tan[e] - (4*I)*c*d*Sin[e]^3*Tan[e] - 2*d^2
*Sin[e]^3*Tan[e] + I*(c - I*d)^2*(4*Cos[3*e] - (4*I)*Sin[3*e])*Tan[e])*(a + I*a*Tan[e + f*x])^3)/(Cos[f*x] + I
*Sin[f*x])^3

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Maple [B]  time = 0.006, size = 290, normalized size = 1.9 \begin{align*}{\frac{-{\frac{i}{4}}{a}^{3}{d}^{2} \left ( \tan \left ( fx+e \right ) \right ) ^{4}}{f}}-{\frac{{\frac{2\,i}{3}}{a}^{3} \left ( \tan \left ( fx+e \right ) \right ) ^{3}cd}{f}}-{\frac{{\frac{i}{2}}{a}^{3} \left ( \tan \left ( fx+e \right ) \right ) ^{2}{c}^{2}}{f}}+{\frac{2\,i{a}^{3} \left ( \tan \left ( fx+e \right ) \right ) ^{2}{d}^{2}}{f}}-{\frac{{a}^{3}{d}^{2} \left ( \tan \left ( fx+e \right ) \right ) ^{3}}{f}}+{\frac{8\,i{a}^{3}cd\tan \left ( fx+e \right ) }{f}}-3\,{\frac{{a}^{3} \left ( \tan \left ( fx+e \right ) \right ) ^{2}cd}{f}}-3\,{\frac{{a}^{3}{c}^{2}\tan \left ( fx+e \right ) }{f}}+4\,{\frac{{a}^{3}\tan \left ( fx+e \right ){d}^{2}}{f}}+{\frac{2\,i{a}^{3}\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ){c}^{2}}{f}}-{\frac{2\,i{a}^{3}\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ){d}^{2}}{f}}+4\,{\frac{{a}^{3}\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) cd}{f}}-{\frac{8\,i{a}^{3}\arctan \left ( \tan \left ( fx+e \right ) \right ) cd}{f}}+4\,{\frac{{a}^{3}\arctan \left ( \tan \left ( fx+e \right ) \right ){c}^{2}}{f}}-4\,{\frac{{a}^{3}\arctan \left ( \tan \left ( fx+e \right ) \right ){d}^{2}}{f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.51523, size = 244, normalized size = 1.59 \begin{align*} -\frac{3 i \, a^{3} d^{2} \tan \left (f x + e\right )^{4} + 4 \,{\left (2 i \, a^{3} c d + 3 \, a^{3} d^{2}\right )} \tan \left (f x + e\right )^{3} -{\left (-6 i \, a^{3} c^{2} - 36 \, a^{3} c d + 24 i \, a^{3} d^{2}\right )} \tan \left (f x + e\right )^{2} - 48 \,{\left (a^{3} c^{2} - 2 i \, a^{3} c d - a^{3} d^{2}\right )}{\left (f x + e\right )} - 12 \,{\left (2 i \, a^{3} c^{2} + 4 \, a^{3} c d - 2 i \, a^{3} d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right ) + 12 \,{\left (3 \, a^{3} c^{2} - 8 i \, a^{3} c d - 4 \, a^{3} d^{2}\right )} \tan \left (f x + e\right )}{12 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(3*I*a^3*d^2*tan(f*x + e)^4 + 4*(2*I*a^3*c*d + 3*a^3*d^2)*tan(f*x + e)^3 - (-6*I*a^3*c^2 - 36*a^3*c*d +
24*I*a^3*d^2)*tan(f*x + e)^2 - 48*(a^3*c^2 - 2*I*a^3*c*d - a^3*d^2)*(f*x + e) - 12*(2*I*a^3*c^2 + 4*a^3*c*d -
2*I*a^3*d^2)*log(tan(f*x + e)^2 + 1) + 12*(3*a^3*c^2 - 8*I*a^3*c*d - 4*a^3*d^2)*tan(f*x + e))/f

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Fricas [B]  time = 1.64802, size = 941, normalized size = 6.15 \begin{align*} \frac{-18 i \, a^{3} c^{2} - 52 \, a^{3} c d + 30 i \, a^{3} d^{2} +{\left (-24 i \, a^{3} c^{2} - 96 \, a^{3} c d + 72 i \, a^{3} d^{2}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-66 i \, a^{3} c^{2} - 228 \, a^{3} c d + 138 i \, a^{3} d^{2}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (-60 i \, a^{3} c^{2} - 184 \, a^{3} c d + 108 i \, a^{3} d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )} +{\left (-12 i \, a^{3} c^{2} - 24 \, a^{3} c d + 12 i \, a^{3} d^{2} +{\left (-12 i \, a^{3} c^{2} - 24 \, a^{3} c d + 12 i \, a^{3} d^{2}\right )} e^{\left (8 i \, f x + 8 i \, e\right )} +{\left (-48 i \, a^{3} c^{2} - 96 \, a^{3} c d + 48 i \, a^{3} d^{2}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-72 i \, a^{3} c^{2} - 144 \, a^{3} c d + 72 i \, a^{3} d^{2}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (-48 i \, a^{3} c^{2} - 96 \, a^{3} c d + 48 i \, a^{3} d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}{3 \,{\left (f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(-18*I*a^3*c^2 - 52*a^3*c*d + 30*I*a^3*d^2 + (-24*I*a^3*c^2 - 96*a^3*c*d + 72*I*a^3*d^2)*e^(6*I*f*x + 6*I*
e) + (-66*I*a^3*c^2 - 228*a^3*c*d + 138*I*a^3*d^2)*e^(4*I*f*x + 4*I*e) + (-60*I*a^3*c^2 - 184*a^3*c*d + 108*I*
a^3*d^2)*e^(2*I*f*x + 2*I*e) + (-12*I*a^3*c^2 - 24*a^3*c*d + 12*I*a^3*d^2 + (-12*I*a^3*c^2 - 24*a^3*c*d + 12*I
*a^3*d^2)*e^(8*I*f*x + 8*I*e) + (-48*I*a^3*c^2 - 96*a^3*c*d + 48*I*a^3*d^2)*e^(6*I*f*x + 6*I*e) + (-72*I*a^3*c
^2 - 144*a^3*c*d + 72*I*a^3*d^2)*e^(4*I*f*x + 4*I*e) + (-48*I*a^3*c^2 - 96*a^3*c*d + 48*I*a^3*d^2)*e^(2*I*f*x
+ 2*I*e))*log(e^(2*I*f*x + 2*I*e) + 1))/(f*e^(8*I*f*x + 8*I*e) + 4*f*e^(6*I*f*x + 6*I*e) + 6*f*e^(4*I*f*x + 4*
I*e) + 4*f*e^(2*I*f*x + 2*I*e) + f)

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Sympy [B]  time = 26.6908, size = 287, normalized size = 1.88 \begin{align*} \frac{4 a^{3} \left (- i c^{2} - 2 c d + i d^{2}\right ) \log{\left (e^{2 i f x} + e^{- 2 i e} \right )}}{f} + \frac{- \frac{\left (8 i a^{3} c^{2} + 32 a^{3} c d - 24 i a^{3} d^{2}\right ) e^{- 2 i e} e^{6 i f x}}{f} - \frac{\left (18 i a^{3} c^{2} + 52 a^{3} c d - 30 i a^{3} d^{2}\right ) e^{- 8 i e}}{3 f} - \frac{\left (22 i a^{3} c^{2} + 76 a^{3} c d - 46 i a^{3} d^{2}\right ) e^{- 4 i e} e^{4 i f x}}{f} - \frac{\left (60 i a^{3} c^{2} + 184 a^{3} c d - 108 i a^{3} d^{2}\right ) e^{- 6 i e} e^{2 i f x}}{3 f}}{e^{8 i f x} + 4 e^{- 2 i e} e^{6 i f x} + 6 e^{- 4 i e} e^{4 i f x} + 4 e^{- 6 i e} e^{2 i f x} + e^{- 8 i e}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*a**3*(-I*c**2 - 2*c*d + I*d**2)*log(exp(2*I*f*x) + exp(-2*I*e))/f + (-(8*I*a**3*c**2 + 32*a**3*c*d - 24*I*a*
*3*d**2)*exp(-2*I*e)*exp(6*I*f*x)/f - (18*I*a**3*c**2 + 52*a**3*c*d - 30*I*a**3*d**2)*exp(-8*I*e)/(3*f) - (22*
I*a**3*c**2 + 76*a**3*c*d - 46*I*a**3*d**2)*exp(-4*I*e)*exp(4*I*f*x)/f - (60*I*a**3*c**2 + 184*a**3*c*d - 108*
I*a**3*d**2)*exp(-6*I*e)*exp(2*I*f*x)/(3*f))/(exp(8*I*f*x) + 4*exp(-2*I*e)*exp(6*I*f*x) + 6*exp(-4*I*e)*exp(4*
I*f*x) + 4*exp(-6*I*e)*exp(2*I*f*x) + exp(-8*I*e))

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Giac [B]  time = 1.71091, size = 905, normalized size = 5.92 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(-12*I*a^3*c^2*e^(8*I*f*x + 8*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) - 24*a^3*c*d*e^(8*I*f*x + 8*I*e)*log(e^(2*
I*f*x + 2*I*e) + 1) + 12*I*a^3*d^2*e^(8*I*f*x + 8*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) - 48*I*a^3*c^2*e^(6*I*f*x
+ 6*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) - 96*a^3*c*d*e^(6*I*f*x + 6*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) + 48*I*a^3
*d^2*e^(6*I*f*x + 6*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) - 72*I*a^3*c^2*e^(4*I*f*x + 4*I*e)*log(e^(2*I*f*x + 2*I*
e) + 1) - 144*a^3*c*d*e^(4*I*f*x + 4*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) + 72*I*a^3*d^2*e^(4*I*f*x + 4*I*e)*log(
e^(2*I*f*x + 2*I*e) + 1) - 48*I*a^3*c^2*e^(2*I*f*x + 2*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) - 96*a^3*c*d*e^(2*I*f
*x + 2*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) + 48*I*a^3*d^2*e^(2*I*f*x + 2*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) - 24*
I*a^3*c^2*e^(6*I*f*x + 6*I*e) - 96*a^3*c*d*e^(6*I*f*x + 6*I*e) + 72*I*a^3*d^2*e^(6*I*f*x + 6*I*e) - 66*I*a^3*c
^2*e^(4*I*f*x + 4*I*e) - 228*a^3*c*d*e^(4*I*f*x + 4*I*e) + 138*I*a^3*d^2*e^(4*I*f*x + 4*I*e) - 60*I*a^3*c^2*e^
(2*I*f*x + 2*I*e) - 184*a^3*c*d*e^(2*I*f*x + 2*I*e) + 108*I*a^3*d^2*e^(2*I*f*x + 2*I*e) - 12*I*a^3*c^2*log(e^(
2*I*f*x + 2*I*e) + 1) - 24*a^3*c*d*log(e^(2*I*f*x + 2*I*e) + 1) + 12*I*a^3*d^2*log(e^(2*I*f*x + 2*I*e) + 1) -
18*I*a^3*c^2 - 52*a^3*c*d + 30*I*a^3*d^2)/(f*e^(8*I*f*x + 8*I*e) + 4*f*e^(6*I*f*x + 6*I*e) + 6*f*e^(4*I*f*x +
4*I*e) + 4*f*e^(2*I*f*x + 2*I*e) + f)