3.1077 \(\int (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^3 \, dx\)
Optimal. Leaf size=190 \[ \frac{a^3 (-11 d+i c) (c+d \tan (e+f x))^4}{20 d^2 f}-\frac{\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^4}{5 d f}+\frac{4 i a^3 (c+d \tan (e+f x))^3}{3 f}+\frac{2 a^3 (d+i c) (c+d \tan (e+f x))^2}{f}+\frac{4 i a^3 d (c-i d)^2 \tan (e+f x)}{f}+\frac{4 a^3 (d+i c)^3 \log (\cos (e+f x))}{f}+4 a^3 x (c-i d)^3 \]
[Out]
4*a^3*(c - I*d)^3*x + (4*a^3*(I*c + d)^3*Log[Cos[e + f*x]])/f + ((4*I)*a^3*(c - I*d)^2*d*Tan[e + f*x])/f + (2*
a^3*(I*c + d)*(c + d*Tan[e + f*x])^2)/f + (((4*I)/3)*a^3*(c + d*Tan[e + f*x])^3)/f + (a^3*(I*c - 11*d)*(c + d*
Tan[e + f*x])^4)/(20*d^2*f) - ((a^3 + I*a^3*Tan[e + f*x])*(c + d*Tan[e + f*x])^4)/(5*d*f)
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Rubi [A] time = 0.336672, antiderivative size = 190, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used =
{3556, 3592, 3528, 3525, 3475} \[ \frac{a^3 (-11 d+i c) (c+d \tan (e+f x))^4}{20 d^2 f}-\frac{\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^4}{5 d f}+\frac{4 i a^3 (c+d \tan (e+f x))^3}{3 f}+\frac{2 a^3 (d+i c) (c+d \tan (e+f x))^2}{f}+\frac{4 i a^3 d (c-i d)^2 \tan (e+f x)}{f}+\frac{4 a^3 (d+i c)^3 \log (\cos (e+f x))}{f}+4 a^3 x (c-i d)^3 \]
Antiderivative was successfully verified.
[In]
Int[(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^3,x]
[Out]
4*a^3*(c - I*d)^3*x + (4*a^3*(I*c + d)^3*Log[Cos[e + f*x]])/f + ((4*I)*a^3*(c - I*d)^2*d*Tan[e + f*x])/f + (2*
a^3*(I*c + d)*(c + d*Tan[e + f*x])^2)/f + (((4*I)/3)*a^3*(c + d*Tan[e + f*x])^3)/f + (a^3*(I*c - 11*d)*(c + d*
Tan[e + f*x])^4)/(20*d^2*f) - ((a^3 + I*a^3*Tan[e + f*x])*(c + d*Tan[e + f*x])^4)/(5*d*f)
Rule 3556
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(b^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(m + n - 1)), x] + Dist[a/(d*(m + n - 1
)), Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^n*Simp[b*c*(m - 2) + a*d*(m + 2*n) + (a*c*(m - 2) +
b*d*(3*m + 2*n - 4))*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] && IntegerQ[2*m] && GtQ[m, 1] && NeQ[m + n - 1, 0] && (IntegerQ[m] || Intege
rsQ[2*m, 2*n])
Rule 3592
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(B*d*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Int[(a + b*Tan[e
+ f*x])^m*Simp[A*c - B*d + (B*c + A*d)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b
*c - a*d, 0] && !LeQ[m, -1]
Rule 3528
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] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
Rule 3525
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*c - b
*d)*x, x] + (Dist[b*c + a*d, Int[Tan[e + f*x], x], x] + Simp[(b*d*Tan[e + f*x])/f, x]) /; FreeQ[{a, b, c, d, e
, f}, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]
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))^3 \, dx &=-\frac{\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^4}{5 d f}+\frac{a \int (a+i a \tan (e+f x)) (a (i c+9 d)+a (c+11 i d) \tan (e+f x)) (c+d \tan (e+f x))^3 \, dx}{5 d}\\ &=\frac{a^3 (i c-11 d) (c+d \tan (e+f x))^4}{20 d^2 f}-\frac{\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^4}{5 d f}+\frac{a \int (c+d \tan (e+f x))^3 \left (20 a^2 d+20 i a^2 d \tan (e+f x)\right ) \, dx}{5 d}\\ &=\frac{4 i a^3 (c+d \tan (e+f x))^3}{3 f}+\frac{a^3 (i c-11 d) (c+d \tan (e+f x))^4}{20 d^2 f}-\frac{\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^4}{5 d f}+\frac{a \int (c+d \tan (e+f x))^2 \left (20 a^2 (c-i d) d+20 a^2 d (i c+d) \tan (e+f x)\right ) \, dx}{5 d}\\ &=\frac{2 a^3 (i c+d) (c+d \tan (e+f x))^2}{f}+\frac{4 i a^3 (c+d \tan (e+f x))^3}{3 f}+\frac{a^3 (i c-11 d) (c+d \tan (e+f x))^4}{20 d^2 f}-\frac{\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^4}{5 d f}+\frac{a \int (c+d \tan (e+f x)) \left (20 a^2 (c-i d)^2 d+20 i a^2 (c-i d)^2 d \tan (e+f x)\right ) \, dx}{5 d}\\ &=4 a^3 (c-i d)^3 x+\frac{4 i a^3 (c-i d)^2 d \tan (e+f x)}{f}+\frac{2 a^3 (i c+d) (c+d \tan (e+f x))^2}{f}+\frac{4 i a^3 (c+d \tan (e+f x))^3}{3 f}+\frac{a^3 (i c-11 d) (c+d \tan (e+f x))^4}{20 d^2 f}-\frac{\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^4}{5 d f}-\left (4 a^3 (i c+d)^3\right ) \int \tan (e+f x) \, dx\\ &=4 a^3 (c-i d)^3 x+\frac{4 a^3 (i c+d)^3 \log (\cos (e+f x))}{f}+\frac{4 i a^3 (c-i d)^2 d \tan (e+f x)}{f}+\frac{2 a^3 (i c+d) (c+d \tan (e+f x))^2}{f}+\frac{4 i a^3 (c+d \tan (e+f x))^3}{3 f}+\frac{a^3 (i c-11 d) (c+d \tan (e+f x))^4}{20 d^2 f}-\frac{\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^4}{5 d f}\\ \end{align*}
Mathematica [B] time = 10.2547, size = 1564, normalized size = 8.23 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^3,x]
[Out]
(Cos[e + f*x]^3*((-I)*c^3*Cos[(3*e)/2] - 3*c^2*d*Cos[(3*e)/2] + (3*I)*c*d^2*Cos[(3*e)/2] + d^3*Cos[(3*e)/2] -
c^3*Sin[(3*e)/2] + (3*I)*c^2*d*Sin[(3*e)/2] + 3*c*d^2*Sin[(3*e)/2] - I*d^3*Sin[(3*e)/2])*(2*Cos[(3*e)/2]*Log[C
os[e + f*x]^2] - (2*I)*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) + (Sec[e]*Sec[e + f*x]^2*(Cos[3*e]/240 - (I/240)*Sin[3*e])*((-45*I)*c^3*Cos[f*x] - 405*c^2*d*Cos[f*x] + (58
5*I)*c*d^2*Cos[f*x] + 225*d^3*Cos[f*x] + 300*c^3*f*x*Cos[f*x] - (900*I)*c^2*d*f*x*Cos[f*x] - 900*c*d^2*f*x*Cos
[f*x] + (300*I)*d^3*f*x*Cos[f*x] - (45*I)*c^3*Cos[2*e + f*x] - 405*c^2*d*Cos[2*e + f*x] + (585*I)*c*d^2*Cos[2*
e + f*x] + 225*d^3*Cos[2*e + f*x] + 300*c^3*f*x*Cos[2*e + f*x] - (900*I)*c^2*d*f*x*Cos[2*e + f*x] - 900*c*d^2*
f*x*Cos[2*e + f*x] + (300*I)*d^3*f*x*Cos[2*e + f*x] - (15*I)*c^3*Cos[2*e + 3*f*x] - 135*c^2*d*Cos[2*e + 3*f*x]
+ (225*I)*c*d^2*Cos[2*e + 3*f*x] + 105*d^3*Cos[2*e + 3*f*x] + 150*c^3*f*x*Cos[2*e + 3*f*x] - (450*I)*c^2*d*f*
x*Cos[2*e + 3*f*x] - 450*c*d^2*f*x*Cos[2*e + 3*f*x] + (150*I)*d^3*f*x*Cos[2*e + 3*f*x] - (15*I)*c^3*Cos[4*e +
3*f*x] - 135*c^2*d*Cos[4*e + 3*f*x] + (225*I)*c*d^2*Cos[4*e + 3*f*x] + 105*d^3*Cos[4*e + 3*f*x] + 150*c^3*f*x*
Cos[4*e + 3*f*x] - (450*I)*c^2*d*f*x*Cos[4*e + 3*f*x] - 450*c*d^2*f*x*Cos[4*e + 3*f*x] + (150*I)*d^3*f*x*Cos[4
*e + 3*f*x] + 30*c^3*f*x*Cos[4*e + 5*f*x] - (90*I)*c^2*d*f*x*Cos[4*e + 5*f*x] - 90*c*d^2*f*x*Cos[4*e + 5*f*x]
+ (30*I)*d^3*f*x*Cos[4*e + 5*f*x] + 30*c^3*f*x*Cos[6*e + 5*f*x] - (90*I)*c^2*d*f*x*Cos[6*e + 5*f*x] - 90*c*d^2
*f*x*Cos[6*e + 5*f*x] + (30*I)*d^3*f*x*Cos[6*e + 5*f*x] - 270*c^3*Sin[f*x] + (1140*I)*c^2*d*Sin[f*x] + 1260*c*
d^2*Sin[f*x] - (470*I)*d^3*Sin[f*x] + 180*c^3*Sin[2*e + f*x] - (810*I)*c^2*d*Sin[2*e + f*x] - 990*c*d^2*Sin[2*
e + f*x] + (360*I)*d^3*Sin[2*e + f*x] - 180*c^3*Sin[2*e + 3*f*x] + (750*I)*c^2*d*Sin[2*e + 3*f*x] + 810*c*d^2*
Sin[2*e + 3*f*x] - (280*I)*d^3*Sin[2*e + 3*f*x] + 45*c^3*Sin[4*e + 3*f*x] - (225*I)*c^2*d*Sin[4*e + 3*f*x] - 3
15*c*d^2*Sin[4*e + 3*f*x] + (135*I)*d^3*Sin[4*e + 3*f*x] - 45*c^3*Sin[4*e + 5*f*x] + (195*I)*c^2*d*Sin[4*e + 5
*f*x] + 225*c*d^2*Sin[4*e + 5*f*x] - (83*I)*d^3*Sin[4*e + 5*f*x])*(a + I*a*Tan[e + f*x])^3)/(f*(Cos[f*x] + I*S
in[f*x])^3) + (x*Cos[e + f*x]^3*(-2*c^3*Cos[e] + (6*I)*c^2*d*Cos[e] + 6*c*d^2*Cos[e] - (2*I)*d^3*Cos[e] + 2*c^
3*Cos[e]^3 - (6*I)*c^2*d*Cos[e]^3 - 6*c*d^2*Cos[e]^3 + (2*I)*d^3*Cos[e]^3 + (4*I)*c^3*Sin[e] + 12*c^2*d*Sin[e]
- (12*I)*c*d^2*Sin[e] - 4*d^3*Sin[e] - (8*I)*c^3*Cos[e]^2*Sin[e] - 24*c^2*d*Cos[e]^2*Sin[e] + (24*I)*c*d^2*Co
s[e]^2*Sin[e] + 8*d^3*Cos[e]^2*Sin[e] - 12*c^3*Cos[e]*Sin[e]^2 + (36*I)*c^2*d*Cos[e]*Sin[e]^2 + 36*c*d^2*Cos[e
]*Sin[e]^2 - (12*I)*d^3*Cos[e]*Sin[e]^2 + (8*I)*c^3*Sin[e]^3 + 24*c^2*d*Sin[e]^3 - (24*I)*c*d^2*Sin[e]^3 - 8*d
^3*Sin[e]^3 + 2*c^3*Sin[e]*Tan[e] - (6*I)*c^2*d*Sin[e]*Tan[e] - 6*c*d^2*Sin[e]*Tan[e] + (2*I)*d^3*Sin[e]*Tan[e
] + 2*c^3*Sin[e]^3*Tan[e] - (6*I)*c^2*d*Sin[e]^3*Tan[e] - 6*c*d^2*Sin[e]^3*Tan[e] + (2*I)*d^3*Sin[e]^3*Tan[e]
+ ((-I)*c - d)^3*(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 = 443, normalized size = 2.3 \begin{align*}{\frac{-6\,i{a}^{3}\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) c{d}^{2}}{f}}-{\frac{{\frac{i}{2}}{a}^{3} \left ( \tan \left ( fx+e \right ) \right ) ^{2}{c}^{3}}{f}}-{\frac{{\frac{i}{5}}{a}^{3}{d}^{3} \left ( \tan \left ( fx+e \right ) \right ) ^{5}}{f}}+{\frac{2\,i{a}^{3}\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ){c}^{3}}{f}}-{\frac{3\,{a}^{3}{d}^{3} \left ( \tan \left ( fx+e \right ) \right ) ^{4}}{4\,f}}-{\frac{12\,i{a}^{3}\arctan \left ( \tan \left ( fx+e \right ) \right ){c}^{2}d}{f}}+{\frac{{\frac{4\,i}{3}}{a}^{3} \left ( \tan \left ( fx+e \right ) \right ) ^{3}{d}^{3}}{f}}-3\,{\frac{{a}^{3} \left ( \tan \left ( fx+e \right ) \right ) ^{3}c{d}^{2}}{f}}-{\frac{i{a}^{3} \left ( \tan \left ( fx+e \right ) \right ) ^{3}{c}^{2}d}{f}}-{\frac{{\frac{3\,i}{4}}{a}^{3} \left ( \tan \left ( fx+e \right ) \right ) ^{4}c{d}^{2}}{f}}-{\frac{9\,{a}^{3} \left ( \tan \left ( fx+e \right ) \right ) ^{2}{c}^{2}d}{2\,f}}+2\,{\frac{{a}^{3} \left ( \tan \left ( fx+e \right ) \right ) ^{2}{d}^{3}}{f}}-3\,{\frac{{a}^{3}{c}^{3}\tan \left ( fx+e \right ) }{f}}+12\,{\frac{{a}^{3}\tan \left ( fx+e \right ) c{d}^{2}}{f}}-{\frac{4\,i{a}^{3}{d}^{3}\tan \left ( fx+e \right ) }{f}}+{\frac{12\,i{a}^{3}{c}^{2}d\tan \left ( fx+e \right ) }{f}}+6\,{\frac{{a}^{3}\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ){c}^{2}d}{f}}-2\,{\frac{{a}^{3}\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ){d}^{3}}{f}}+{\frac{6\,i{a}^{3} \left ( \tan \left ( fx+e \right ) \right ) ^{2}c{d}^{2}}{f}}+{\frac{4\,i{a}^{3}\arctan \left ( \tan \left ( fx+e \right ) \right ){d}^{3}}{f}}+4\,{\frac{{a}^{3}\arctan \left ( \tan \left ( fx+e \right ) \right ){c}^{3}}{f}}-12\,{\frac{{a}^{3}\arctan \left ( \tan \left ( fx+e \right ) \right ) c{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))^3,x)
[Out]
-6*I/f*a^3*ln(1+tan(f*x+e)^2)*c*d^2-1/2*I/f*a^3*tan(f*x+e)^2*c^3-1/5*I/f*a^3*d^3*tan(f*x+e)^5+2*I/f*a^3*ln(1+t
an(f*x+e)^2)*c^3-3/4/f*a^3*d^3*tan(f*x+e)^4-12*I/f*a^3*arctan(tan(f*x+e))*c^2*d+4/3*I/f*a^3*tan(f*x+e)^3*d^3-3
/f*a^3*tan(f*x+e)^3*c*d^2-I/f*a^3*tan(f*x+e)^3*c^2*d-3/4*I/f*a^3*tan(f*x+e)^4*c*d^2-9/2/f*a^3*tan(f*x+e)^2*c^2
*d+2/f*a^3*tan(f*x+e)^2*d^3-3*a^3*c^3*tan(f*x+e)/f+12/f*a^3*tan(f*x+e)*c*d^2-4*I/f*a^3*d^3*tan(f*x+e)+12*I/f*a
^3*c^2*d*tan(f*x+e)+6/f*a^3*ln(1+tan(f*x+e)^2)*c^2*d-2/f*a^3*ln(1+tan(f*x+e)^2)*d^3+6*I/f*a^3*tan(f*x+e)^2*c*d
^2+4*I/f*a^3*arctan(tan(f*x+e))*d^3+4/f*a^3*arctan(tan(f*x+e))*c^3-12/f*a^3*arctan(tan(f*x+e))*c*d^2
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Maxima [A] time = 1.52325, size = 355, normalized size = 1.87 \begin{align*} -\frac{12 i \, a^{3} d^{3} \tan \left (f x + e\right )^{5} + 45 \,{\left (i \, a^{3} c d^{2} + a^{3} d^{3}\right )} \tan \left (f x + e\right )^{4} -{\left (-60 i \, a^{3} c^{2} d - 180 \, a^{3} c d^{2} + 80 i \, a^{3} d^{3}\right )} \tan \left (f x + e\right )^{3} -{\left (-30 i \, a^{3} c^{3} - 270 \, a^{3} c^{2} d + 360 i \, a^{3} c d^{2} + 120 \, a^{3} d^{3}\right )} \tan \left (f x + e\right )^{2} - 60 \,{\left (4 \, a^{3} c^{3} - 12 i \, a^{3} c^{2} d - 12 \, a^{3} c d^{2} + 4 i \, a^{3} d^{3}\right )}{\left (f x + e\right )} - 60 \,{\left (2 i \, a^{3} c^{3} + 6 \, a^{3} c^{2} d - 6 i \, a^{3} c d^{2} - 2 \, a^{3} d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right ) +{\left (180 \, a^{3} c^{3} - 720 i \, a^{3} c^{2} d - 720 \, a^{3} c d^{2} + 240 i \, a^{3} d^{3}\right )} \tan \left (f x + e\right )}{60 \, 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))^3,x, algorithm="maxima")
[Out]
-1/60*(12*I*a^3*d^3*tan(f*x + e)^5 + 45*(I*a^3*c*d^2 + a^3*d^3)*tan(f*x + e)^4 - (-60*I*a^3*c^2*d - 180*a^3*c*
d^2 + 80*I*a^3*d^3)*tan(f*x + e)^3 - (-30*I*a^3*c^3 - 270*a^3*c^2*d + 360*I*a^3*c*d^2 + 120*a^3*d^3)*tan(f*x +
e)^2 - 60*(4*a^3*c^3 - 12*I*a^3*c^2*d - 12*a^3*c*d^2 + 4*I*a^3*d^3)*(f*x + e) - 60*(2*I*a^3*c^3 + 6*a^3*c^2*d
- 6*I*a^3*c*d^2 - 2*a^3*d^3)*log(tan(f*x + e)^2 + 1) + (180*a^3*c^3 - 720*I*a^3*c^2*d - 720*a^3*c*d^2 + 240*I
*a^3*d^3)*tan(f*x + e))/f
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Fricas [B] time = 1.73062, size = 1480, normalized size = 7.79 \begin{align*} \frac{-90 i \, a^{3} c^{3} - 390 \, a^{3} c^{2} d + 450 i \, a^{3} c d^{2} + 166 \, a^{3} d^{3} +{\left (-120 i \, a^{3} c^{3} - 720 \, a^{3} c^{2} d + 1080 i \, a^{3} c d^{2} + 480 \, a^{3} d^{3}\right )} e^{\left (8 i \, f x + 8 i \, e\right )} +{\left (-450 i \, a^{3} c^{3} - 2430 \, a^{3} c^{2} d + 3150 i \, a^{3} c d^{2} + 1170 \, a^{3} d^{3}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-630 i \, a^{3} c^{3} - 3090 \, a^{3} c^{2} d + 3690 i \, a^{3} c d^{2} + 1390 \, a^{3} d^{3}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (-390 i \, a^{3} c^{3} - 1770 \, a^{3} c^{2} d + 2070 i \, a^{3} c d^{2} + 770 \, a^{3} d^{3}\right )} e^{\left (2 i \, f x + 2 i \, e\right )} +{\left (-60 i \, a^{3} c^{3} - 180 \, a^{3} c^{2} d + 180 i \, a^{3} c d^{2} + 60 \, a^{3} d^{3} +{\left (-60 i \, a^{3} c^{3} - 180 \, a^{3} c^{2} d + 180 i \, a^{3} c d^{2} + 60 \, a^{3} d^{3}\right )} e^{\left (10 i \, f x + 10 i \, e\right )} +{\left (-300 i \, a^{3} c^{3} - 900 \, a^{3} c^{2} d + 900 i \, a^{3} c d^{2} + 300 \, a^{3} d^{3}\right )} e^{\left (8 i \, f x + 8 i \, e\right )} +{\left (-600 i \, a^{3} c^{3} - 1800 \, a^{3} c^{2} d + 1800 i \, a^{3} c d^{2} + 600 \, a^{3} d^{3}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-600 i \, a^{3} c^{3} - 1800 \, a^{3} c^{2} d + 1800 i \, a^{3} c d^{2} + 600 \, a^{3} d^{3}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (-300 i \, a^{3} c^{3} - 900 \, a^{3} c^{2} d + 900 i \, a^{3} c d^{2} + 300 \, a^{3} d^{3}\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 )}{15 \,{\left (f e^{\left (10 i \, f x + 10 i \, e\right )} + 5 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 10 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 10 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, 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))^3,x, algorithm="fricas")
[Out]
1/15*(-90*I*a^3*c^3 - 390*a^3*c^2*d + 450*I*a^3*c*d^2 + 166*a^3*d^3 + (-120*I*a^3*c^3 - 720*a^3*c^2*d + 1080*I
*a^3*c*d^2 + 480*a^3*d^3)*e^(8*I*f*x + 8*I*e) + (-450*I*a^3*c^3 - 2430*a^3*c^2*d + 3150*I*a^3*c*d^2 + 1170*a^3
*d^3)*e^(6*I*f*x + 6*I*e) + (-630*I*a^3*c^3 - 3090*a^3*c^2*d + 3690*I*a^3*c*d^2 + 1390*a^3*d^3)*e^(4*I*f*x + 4
*I*e) + (-390*I*a^3*c^3 - 1770*a^3*c^2*d + 2070*I*a^3*c*d^2 + 770*a^3*d^3)*e^(2*I*f*x + 2*I*e) + (-60*I*a^3*c^
3 - 180*a^3*c^2*d + 180*I*a^3*c*d^2 + 60*a^3*d^3 + (-60*I*a^3*c^3 - 180*a^3*c^2*d + 180*I*a^3*c*d^2 + 60*a^3*d
^3)*e^(10*I*f*x + 10*I*e) + (-300*I*a^3*c^3 - 900*a^3*c^2*d + 900*I*a^3*c*d^2 + 300*a^3*d^3)*e^(8*I*f*x + 8*I*
e) + (-600*I*a^3*c^3 - 1800*a^3*c^2*d + 1800*I*a^3*c*d^2 + 600*a^3*d^3)*e^(6*I*f*x + 6*I*e) + (-600*I*a^3*c^3
- 1800*a^3*c^2*d + 1800*I*a^3*c*d^2 + 600*a^3*d^3)*e^(4*I*f*x + 4*I*e) + (-300*I*a^3*c^3 - 900*a^3*c^2*d + 900
*I*a^3*c*d^2 + 300*a^3*d^3)*e^(2*I*f*x + 2*I*e))*log(e^(2*I*f*x + 2*I*e) + 1))/(f*e^(10*I*f*x + 10*I*e) + 5*f*
e^(8*I*f*x + 8*I*e) + 10*f*e^(6*I*f*x + 6*I*e) + 10*f*e^(4*I*f*x + 4*I*e) + 5*f*e^(2*I*f*x + 2*I*e) + f)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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))**3,x)
[Out]
Timed out
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Giac [B] time = 2.30314, size = 1508, normalized size = 7.94 \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))^3,x, algorithm="giac")
[Out]
1/15*(-60*I*a^3*c^3*e^(10*I*f*x + 10*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) - 180*a^3*c^2*d*e^(10*I*f*x + 10*I*e)*l
og(e^(2*I*f*x + 2*I*e) + 1) + 180*I*a^3*c*d^2*e^(10*I*f*x + 10*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) + 60*a^3*d^3*
e^(10*I*f*x + 10*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) - 300*I*a^3*c^3*e^(8*I*f*x + 8*I*e)*log(e^(2*I*f*x + 2*I*e)
+ 1) - 900*a^3*c^2*d*e^(8*I*f*x + 8*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) + 900*I*a^3*c*d^2*e^(8*I*f*x + 8*I*e)*l
og(e^(2*I*f*x + 2*I*e) + 1) + 300*a^3*d^3*e^(8*I*f*x + 8*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) - 600*I*a^3*c^3*e^(
6*I*f*x + 6*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) - 1800*a^3*c^2*d*e^(6*I*f*x + 6*I*e)*log(e^(2*I*f*x + 2*I*e) + 1
) + 1800*I*a^3*c*d^2*e^(6*I*f*x + 6*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) + 600*a^3*d^3*e^(6*I*f*x + 6*I*e)*log(e^
(2*I*f*x + 2*I*e) + 1) - 600*I*a^3*c^3*e^(4*I*f*x + 4*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) - 1800*a^3*c^2*d*e^(4*
I*f*x + 4*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) + 1800*I*a^3*c*d^2*e^(4*I*f*x + 4*I*e)*log(e^(2*I*f*x + 2*I*e) + 1
) + 600*a^3*d^3*e^(4*I*f*x + 4*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) - 300*I*a^3*c^3*e^(2*I*f*x + 2*I*e)*log(e^(2*
I*f*x + 2*I*e) + 1) - 900*a^3*c^2*d*e^(2*I*f*x + 2*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) + 900*I*a^3*c*d^2*e^(2*I*
f*x + 2*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) + 300*a^3*d^3*e^(2*I*f*x + 2*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) - 120
*I*a^3*c^3*e^(8*I*f*x + 8*I*e) - 720*a^3*c^2*d*e^(8*I*f*x + 8*I*e) + 1080*I*a^3*c*d^2*e^(8*I*f*x + 8*I*e) + 48
0*a^3*d^3*e^(8*I*f*x + 8*I*e) - 450*I*a^3*c^3*e^(6*I*f*x + 6*I*e) - 2430*a^3*c^2*d*e^(6*I*f*x + 6*I*e) + 3150*
I*a^3*c*d^2*e^(6*I*f*x + 6*I*e) + 1170*a^3*d^3*e^(6*I*f*x + 6*I*e) - 630*I*a^3*c^3*e^(4*I*f*x + 4*I*e) - 3090*
a^3*c^2*d*e^(4*I*f*x + 4*I*e) + 3690*I*a^3*c*d^2*e^(4*I*f*x + 4*I*e) + 1390*a^3*d^3*e^(4*I*f*x + 4*I*e) - 390*
I*a^3*c^3*e^(2*I*f*x + 2*I*e) - 1770*a^3*c^2*d*e^(2*I*f*x + 2*I*e) + 2070*I*a^3*c*d^2*e^(2*I*f*x + 2*I*e) + 77
0*a^3*d^3*e^(2*I*f*x + 2*I*e) - 60*I*a^3*c^3*log(e^(2*I*f*x + 2*I*e) + 1) - 180*a^3*c^2*d*log(e^(2*I*f*x + 2*I
*e) + 1) + 180*I*a^3*c*d^2*log(e^(2*I*f*x + 2*I*e) + 1) + 60*a^3*d^3*log(e^(2*I*f*x + 2*I*e) + 1) - 90*I*a^3*c
^3 - 390*a^3*c^2*d + 450*I*a^3*c*d^2 + 166*a^3*d^3)/(f*e^(10*I*f*x + 10*I*e) + 5*f*e^(8*I*f*x + 8*I*e) + 10*f*
e^(6*I*f*x + 6*I*e) + 10*f*e^(4*I*f*x + 4*I*e) + 5*f*e^(2*I*f*x + 2*I*e) + f)