3.11.78 \(\int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^3 \, dx\) [1078]

Optimal. Leaf size=141 \[ 2 a^2 (c-i d)^3 x+\frac {2 a^2 (i c+d)^3 \log (\cos (e+f x))}{f}+\frac {2 i a^2 (c-i d)^2 d \tan (e+f x)}{f}+\frac {a^2 (i c+d) (c+d \tan (e+f x))^2}{f}+\frac {2 i a^2 (c+d \tan (e+f x))^3}{3 f}-\frac {a^2 (c+d \tan (e+f x))^4}{4 d f} \]

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3624, 3609, 3606, 3556} \begin {gather*} -\frac {a^2 (c+d \tan (e+f x))^4}{4 d f}+\frac {2 i a^2 (c+d \tan (e+f x))^3}{3 f}+\frac {a^2 (d+i c) (c+d \tan (e+f x))^2}{f}+\frac {2 i a^2 d (c-i d)^2 \tan (e+f x)}{f}+\frac {2 a^2 (d+i c)^3 \log (\cos (e+f x))}{f}+2 a^2 x (c-i d)^3 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3556

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

Rule 3606

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 3609

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 3624

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])

Rubi steps

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

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(733\) vs. \(2(141)=282\).
time = 6.39, size = 733, normalized size = 5.20 \begin {gather*} \frac {\sec ^2(e+f x) \left ((c-i d)^3 \cos ^4(e+f x) \log \left (\cos ^2(e+f x)\right ) (-i \cos (2 e)-\sin (2 e))+2 (c-i d)^3 f x \cos ^4(e+f x) (\cos (2 e)-i \sin (2 e))-2 (c-i d)^3 \text {ArcTan}(\tan (3 e+f x)) \cos ^4(e+f x) (\cos (2 e)-i \sin (2 e))+\frac {1}{24} \sec (e) (\cos (2 e)-i \sin (2 e)) \left (6 \left (3 c^3 f x+3 c d^2 (2 i-3 f x)+d^3 (2+3 i f x)+c^2 d (-3-9 i f x)\right ) \cos (e)+3 (c-i d)^2 (-3 d+4 c f x-4 i d f x) \cos (e+2 f x)-9 c^2 d \cos (3 e+2 f x)+18 i c d^2 \cos (3 e+2 f x)+9 d^3 \cos (3 e+2 f x)+12 c^3 f x \cos (3 e+2 f x)-36 i c^2 d f x \cos (3 e+2 f x)-36 c d^2 f x \cos (3 e+2 f x)+12 i d^3 f x \cos (3 e+2 f x)+3 c^3 f x \cos (3 e+4 f x)-9 i c^2 d f x \cos (3 e+4 f x)-9 c d^2 f x \cos (3 e+4 f x)+3 i d^3 f x \cos (3 e+4 f x)+3 c^3 f x \cos (5 e+4 f x)-9 i c^2 d f x \cos (5 e+4 f x)-9 c d^2 f x \cos (5 e+4 f x)+3 i d^3 f x \cos (5 e+4 f x)+9 c^3 \sin (e)-54 i c^2 d \sin (e)-63 c d^2 \sin (e)+24 i d^3 \sin (e)-9 c^3 \sin (e+2 f x)+54 i c^2 d \sin (e+2 f x)+57 c d^2 \sin (e+2 f x)-20 i d^3 \sin (e+2 f x)+3 c^3 \sin (3 e+2 f x)-18 i c^2 d \sin (3 e+2 f x)-27 c d^2 \sin (3 e+2 f x)+12 i d^3 \sin (3 e+2 f x)-3 c^3 \sin (3 e+4 f x)+18 i c^2 d \sin (3 e+4 f x)+21 c d^2 \sin (3 e+4 f x)-8 i d^3 \sin (3 e+4 f x)\right )\right ) (a+i a \tan (e+f x))^2}{f (\cos (f x)+i \sin (f x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[e + f*x]^2*((c - I*d)^3*Cos[e + f*x]^4*Log[Cos[e + f*x]^2]*((-I)*Cos[2*e] - Sin[2*e]) + 2*(c - I*d)^3*f*x
*Cos[e + f*x]^4*(Cos[2*e] - I*Sin[2*e]) - 2*(c - I*d)^3*ArcTan[Tan[3*e + f*x]]*Cos[e + f*x]^4*(Cos[2*e] - I*Si
n[2*e]) + (Sec[e]*(Cos[2*e] - I*Sin[2*e])*(6*(3*c^3*f*x + 3*c*d^2*(2*I - 3*f*x) + d^3*(2 + (3*I)*f*x) + c^2*d*
(-3 - (9*I)*f*x))*Cos[e] + 3*(c - I*d)^2*(-3*d + 4*c*f*x - (4*I)*d*f*x)*Cos[e + 2*f*x] - 9*c^2*d*Cos[3*e + 2*f
*x] + (18*I)*c*d^2*Cos[3*e + 2*f*x] + 9*d^3*Cos[3*e + 2*f*x] + 12*c^3*f*x*Cos[3*e + 2*f*x] - (36*I)*c^2*d*f*x*
Cos[3*e + 2*f*x] - 36*c*d^2*f*x*Cos[3*e + 2*f*x] + (12*I)*d^3*f*x*Cos[3*e + 2*f*x] + 3*c^3*f*x*Cos[3*e + 4*f*x
] - (9*I)*c^2*d*f*x*Cos[3*e + 4*f*x] - 9*c*d^2*f*x*Cos[3*e + 4*f*x] + (3*I)*d^3*f*x*Cos[3*e + 4*f*x] + 3*c^3*f
*x*Cos[5*e + 4*f*x] - (9*I)*c^2*d*f*x*Cos[5*e + 4*f*x] - 9*c*d^2*f*x*Cos[5*e + 4*f*x] + (3*I)*d^3*f*x*Cos[5*e
+ 4*f*x] + 9*c^3*Sin[e] - (54*I)*c^2*d*Sin[e] - 63*c*d^2*Sin[e] + (24*I)*d^3*Sin[e] - 9*c^3*Sin[e + 2*f*x] + (
54*I)*c^2*d*Sin[e + 2*f*x] + 57*c*d^2*Sin[e + 2*f*x] - (20*I)*d^3*Sin[e + 2*f*x] + 3*c^3*Sin[3*e + 2*f*x] - (1
8*I)*c^2*d*Sin[3*e + 2*f*x] - 27*c*d^2*Sin[3*e + 2*f*x] + (12*I)*d^3*Sin[3*e + 2*f*x] - 3*c^3*Sin[3*e + 4*f*x]
+ (18*I)*c^2*d*Sin[3*e + 4*f*x] + 21*c*d^2*Sin[3*e + 4*f*x] - (8*I)*d^3*Sin[3*e + 4*f*x]))/24)*(a + I*a*Tan[e
+ f*x])^2)/(f*(Cos[f*x] + I*Sin[f*x])^2)

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Maple [A]
time = 0.16, size = 210, normalized size = 1.49

method	result	size
derivativedivides	\(\frac {a^{2} \left (\frac {2 i d^{3} \left (\tan ^{3}\left (f x +e \right )\right )}{3}-\frac {d^{3} \left (\tan ^{4}\left (f x +e \right )\right )}{4}+3 i c \,d^{2} \left (\tan ^{2}\left (f x +e \right )\right )-c \,d^{2} \left (\tan ^{3}\left (f x +e \right )\right )+6 i c^{2} d \tan \left (f x +e \right )-2 i d^{3} \tan \left (f x +e \right )-\frac {3 c^{2} d \left (\tan ^{2}\left (f x +e \right )\right )}{2}+d^{3} \left (\tan ^{2}\left (f x +e \right )\right )-c^{3} \tan \left (f x +e \right )+6 c \,d^{2} \tan \left (f x +e \right )+\frac {\left (2 i c^{3}-6 i c \,d^{2}+6 c^{2} d -2 d^{3}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (-6 i c^{2} d +2 i d^{3}+2 c^{3}-6 c \,d^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )\right )}{f}\)	\(210\)
default	\(\frac {a^{2} \left (\frac {2 i d^{3} \left (\tan ^{3}\left (f x +e \right )\right )}{3}-\frac {d^{3} \left (\tan ^{4}\left (f x +e \right )\right )}{4}+3 i c \,d^{2} \left (\tan ^{2}\left (f x +e \right )\right )-c \,d^{2} \left (\tan ^{3}\left (f x +e \right )\right )+6 i c^{2} d \tan \left (f x +e \right )-2 i d^{3} \tan \left (f x +e \right )-\frac {3 c^{2} d \left (\tan ^{2}\left (f x +e \right )\right )}{2}+d^{3} \left (\tan ^{2}\left (f x +e \right )\right )-c^{3} \tan \left (f x +e \right )+6 c \,d^{2} \tan \left (f x +e \right )+\frac {\left (2 i c^{3}-6 i c \,d^{2}+6 c^{2} d -2 d^{3}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (-6 i c^{2} d +2 i d^{3}+2 c^{3}-6 c \,d^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )\right )}{f}\)	\(210\)
norman	\(\left (-6 i a^{2} c^{2} d +2 i a^{2} d^{3}+2 a^{2} c^{3}-6 a^{2} c \,d^{2}\right ) x -\frac {\left (-2 i a^{2} d^{3}+3 a^{2} c \,d^{2}\right ) \left (\tan ^{3}\left (f x +e \right )\right )}{3 f}+\frac {\left (6 i a^{2} c \,d^{2}-3 a^{2} c^{2} d +2 a^{2} d^{3}\right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2 f}-\frac {\left (-6 i a^{2} c^{2} d +2 i a^{2} d^{3}+a^{2} c^{3}-6 a^{2} c \,d^{2}\right ) \tan \left (f x +e \right )}{f}-\frac {a^{2} d^{3} \left (\tan ^{4}\left (f x +e \right )\right )}{4 f}-\frac {\left (-i a^{2} c^{3}+3 i a^{2} c \,d^{2}-3 a^{2} c^{2} d +a^{2} d^{3}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{f}\)	\(232\)
risch	\(-\frac {2 i a^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) c^{3}}{f}+\frac {12 i a^{2} c^{2} d e}{f}-\frac {4 i a^{2} d^{3} e}{f}+\frac {6 i a^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) c \,d^{2}}{f}-\frac {4 a^{2} c^{3} e}{f}+\frac {12 a^{2} c \,d^{2} e}{f}+\frac {2 a^{2} \left (99 i c \,d^{2} {\mathrm e}^{4 i \left (f x +e \right )}-3 i c^{3} {\mathrm e}^{6 i \left (f x +e \right )}-27 c^{2} d \,{\mathrm e}^{6 i \left (f x +e \right )}+21 d^{3} {\mathrm e}^{6 i \left (f x +e \right )}-9 i c^{3} {\mathrm e}^{4 i \left (f x +e \right )}-9 i c^{3} {\mathrm e}^{2 i \left (f x +e \right )}-72 c^{2} d \,{\mathrm e}^{4 i \left (f x +e \right )}+36 d^{3} {\mathrm e}^{4 i \left (f x +e \right )}+75 i c \,d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-3 i c^{3}-63 c^{2} d \,{\mathrm e}^{2 i \left (f x +e \right )}+29 d^{3} {\mathrm e}^{2 i \left (f x +e \right )}+45 i c \,d^{2} {\mathrm e}^{6 i \left (f x +e \right )}+21 i c \,d^{2}-18 c^{2} d +8 d^{3}\right )}{3 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{4}}-\frac {6 a^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) c^{2} d}{f}+\frac {2 a^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) d^{3}}{f}\)	\(376\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/f*a^2*(2/3*I*d^3*tan(f*x+e)^3-1/4*d^3*tan(f*x+e)^4+3*I*c*d^2*tan(f*x+e)^2-c*d^2*tan(f*x+e)^3+6*I*c^2*d*tan(f
*x+e)-2*I*d^3*tan(f*x+e)-3/2*c^2*d*tan(f*x+e)^2+d^3*tan(f*x+e)^2-c^3*tan(f*x+e)+6*c*d^2*tan(f*x+e)+1/2*(-2*d^3
+6*c^2*d-6*I*c*d^2+2*I*c^3)*ln(1+tan(f*x+e)^2)+(-6*c*d^2+2*I*d^3+2*c^3-6*I*c^2*d)*arctan(tan(f*x+e)))

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Maxima [A]
time = 0.50, size = 224, normalized size = 1.59 \begin {gather*} -\frac {3 \, a^{2} d^{3} \tan \left (f x + e\right )^{4} + 4 \, {\left (3 \, a^{2} c d^{2} - 2 i \, a^{2} d^{3}\right )} \tan \left (f x + e\right )^{3} + 6 \, {\left (3 \, a^{2} c^{2} d - 6 i \, a^{2} c d^{2} - 2 \, a^{2} d^{3}\right )} \tan \left (f x + e\right )^{2} - 24 \, {\left (a^{2} c^{3} - 3 i \, a^{2} c^{2} d - 3 \, a^{2} c d^{2} + i \, a^{2} d^{3}\right )} {\left (f x + e\right )} - 12 \, {\left (i \, a^{2} c^{3} + 3 \, a^{2} c^{2} d - 3 i \, a^{2} c d^{2} - a^{2} d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right ) + 12 \, {\left (a^{2} c^{3} - 6 i \, a^{2} c^{2} d - 6 \, a^{2} c d^{2} + 2 i \, a^{2} d^{3}\right )} \tan \left (f x + e\right )}{12 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 469 vs. \(2 (130) = 260\).
time = 0.90, size = 469, normalized size = 3.33 \begin {gather*} -\frac {2 \, {\left (3 i \, a^{2} c^{3} + 18 \, a^{2} c^{2} d - 21 i \, a^{2} c d^{2} - 8 \, a^{2} d^{3} + 3 \, {\left (i \, a^{2} c^{3} + 9 \, a^{2} c^{2} d - 15 i \, a^{2} c d^{2} - 7 \, a^{2} d^{3}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + 9 \, {\left (i \, a^{2} c^{3} + 8 \, a^{2} c^{2} d - 11 i \, a^{2} c d^{2} - 4 \, a^{2} d^{3}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (9 i \, a^{2} c^{3} + 63 \, a^{2} c^{2} d - 75 i \, a^{2} c d^{2} - 29 \, a^{2} d^{3}\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + 3 \, {\left (i \, a^{2} c^{3} + 3 \, a^{2} c^{2} d - 3 i \, a^{2} c d^{2} - a^{2} d^{3} + {\left (i \, a^{2} c^{3} + 3 \, a^{2} c^{2} d - 3 i \, a^{2} c d^{2} - a^{2} d^{3}\right )} e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, {\left (i \, a^{2} c^{3} + 3 \, a^{2} c^{2} d - 3 i \, a^{2} c d^{2} - a^{2} d^{3}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, {\left (i \, a^{2} c^{3} + 3 \, a^{2} c^{2} d - 3 i \, a^{2} c d^{2} - a^{2} d^{3}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, {\left (i \, a^{2} c^{3} + 3 \, a^{2} c^{2} d - 3 i \, a^{2} c d^{2} - a^{2} 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 )\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3*I*a^2*c^3 + 18*a^2*c^2*d - 21*I*a^2*c*d^2 - 8*a^2*d^3 + 3*(I*a^2*c^3 + 9*a^2*c^2*d - 15*I*a^2*c*d^2 -
7*a^2*d^3)*e^(6*I*f*x + 6*I*e) + 9*(I*a^2*c^3 + 8*a^2*c^2*d - 11*I*a^2*c*d^2 - 4*a^2*d^3)*e^(4*I*f*x + 4*I*e)
+ (9*I*a^2*c^3 + 63*a^2*c^2*d - 75*I*a^2*c*d^2 - 29*a^2*d^3)*e^(2*I*f*x + 2*I*e) + 3*(I*a^2*c^3 + 3*a^2*c^2*d
- 3*I*a^2*c*d^2 - a^2*d^3 + (I*a^2*c^3 + 3*a^2*c^2*d - 3*I*a^2*c*d^2 - a^2*d^3)*e^(8*I*f*x + 8*I*e) + 4*(I*a^2
*c^3 + 3*a^2*c^2*d - 3*I*a^2*c*d^2 - a^2*d^3)*e^(6*I*f*x + 6*I*e) + 6*(I*a^2*c^3 + 3*a^2*c^2*d - 3*I*a^2*c*d^2
- a^2*d^3)*e^(4*I*f*x + 4*I*e) + 4*(I*a^2*c^3 + 3*a^2*c^2*d - 3*I*a^2*c*d^2 - a^2*d^3)*e^(2*I*f*x + 2*I*e))*l
og(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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (122) = 244\).
time = 0.73, size = 381, normalized size = 2.70 \begin {gather*} - \frac {2 i a^{2} \left (c - i d\right )^{3} \log {\left (e^{2 i f x} + e^{- 2 i e} \right )}}{f} + \frac {- 6 i a^{2} c^{3} - 36 a^{2} c^{2} d + 42 i a^{2} c d^{2} + 16 a^{2} d^{3} + \left (- 18 i a^{2} c^{3} e^{2 i e} - 126 a^{2} c^{2} d e^{2 i e} + 150 i a^{2} c d^{2} e^{2 i e} + 58 a^{2} d^{3} e^{2 i e}\right ) e^{2 i f x} + \left (- 18 i a^{2} c^{3} e^{4 i e} - 144 a^{2} c^{2} d e^{4 i e} + 198 i a^{2} c d^{2} e^{4 i e} + 72 a^{2} d^{3} e^{4 i e}\right ) e^{4 i f x} + \left (- 6 i a^{2} c^{3} e^{6 i e} - 54 a^{2} c^{2} d e^{6 i e} + 90 i a^{2} c d^{2} e^{6 i e} + 42 a^{2} d^{3} e^{6 i e}\right ) e^{6 i f x}}{3 f e^{8 i e} e^{8 i f x} + 12 f e^{6 i e} e^{6 i f x} + 18 f e^{4 i e} e^{4 i f x} + 12 f e^{2 i e} e^{2 i f x} + 3 f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*I*a**2*(c - I*d)**3*log(exp(2*I*f*x) + exp(-2*I*e))/f + (-6*I*a**2*c**3 - 36*a**2*c**2*d + 42*I*a**2*c*d**2
+ 16*a**2*d**3 + (-18*I*a**2*c**3*exp(2*I*e) - 126*a**2*c**2*d*exp(2*I*e) + 150*I*a**2*c*d**2*exp(2*I*e) + 58
*a**2*d**3*exp(2*I*e))*exp(2*I*f*x) + (-18*I*a**2*c**3*exp(4*I*e) - 144*a**2*c**2*d*exp(4*I*e) + 198*I*a**2*c*
d**2*exp(4*I*e) + 72*a**2*d**3*exp(4*I*e))*exp(4*I*f*x) + (-6*I*a**2*c**3*exp(6*I*e) - 54*a**2*c**2*d*exp(6*I*
e) + 90*I*a**2*c*d**2*exp(6*I*e) + 42*a**2*d**3*exp(6*I*e))*exp(6*I*f*x))/(3*f*exp(8*I*e)*exp(8*I*f*x) + 12*f*
exp(6*I*e)*exp(6*I*f*x) + 18*f*exp(4*I*e)*exp(4*I*f*x) + 12*f*exp(2*I*e)*exp(2*I*f*x) + 3*f)

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 904 vs. \(2 (130) = 260\).
time = 0.94, size = 904, normalized size = 6.41 \begin {gather*} -\frac {2 \, {\left (3 i \, a^{2} c^{3} e^{\left (8 i \, f x + 8 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 9 \, a^{2} c^{2} d e^{\left (8 i \, f x + 8 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 9 i \, a^{2} c d^{2} e^{\left (8 i \, f x + 8 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 3 \, a^{2} d^{3} e^{\left (8 i \, f x + 8 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 12 i \, a^{2} c^{3} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 36 \, a^{2} c^{2} d e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 36 i \, a^{2} c d^{2} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 12 \, a^{2} d^{3} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 18 i \, a^{2} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 54 \, a^{2} c^{2} d e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 54 i \, a^{2} c d^{2} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 18 \, a^{2} d^{3} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 12 i \, a^{2} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 36 \, a^{2} c^{2} d e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 36 i \, a^{2} c d^{2} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 12 \, a^{2} d^{3} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 3 i \, a^{2} c^{3} e^{\left (6 i \, f x + 6 i \, e\right )} + 27 \, a^{2} c^{2} d e^{\left (6 i \, f x + 6 i \, e\right )} - 45 i \, a^{2} c d^{2} e^{\left (6 i \, f x + 6 i \, e\right )} - 21 \, a^{2} d^{3} e^{\left (6 i \, f x + 6 i \, e\right )} + 9 i \, a^{2} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 72 \, a^{2} c^{2} d e^{\left (4 i \, f x + 4 i \, e\right )} - 99 i \, a^{2} c d^{2} e^{\left (4 i \, f x + 4 i \, e\right )} - 36 \, a^{2} d^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 9 i \, a^{2} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 63 \, a^{2} c^{2} d e^{\left (2 i \, f x + 2 i \, e\right )} - 75 i \, a^{2} c d^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 29 \, a^{2} d^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 3 i \, a^{2} c^{3} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 9 \, a^{2} c^{2} d \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 9 i \, a^{2} c d^{2} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 3 \, a^{2} d^{3} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 3 i \, a^{2} c^{3} + 18 \, a^{2} c^{2} d - 21 i \, a^{2} c d^{2} - 8 \, a^{2} d^{3}\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3*I*a^2*c^3*e^(8*I*f*x + 8*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) + 9*a^2*c^2*d*e^(8*I*f*x + 8*I*e)*log(e^(2*
I*f*x + 2*I*e) + 1) - 9*I*a^2*c*d^2*e^(8*I*f*x + 8*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) - 3*a^2*d^3*e^(8*I*f*x +
8*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) + 12*I*a^2*c^3*e^(6*I*f*x + 6*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) + 36*a^2*c
^2*d*e^(6*I*f*x + 6*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) - 36*I*a^2*c*d^2*e^(6*I*f*x + 6*I*e)*log(e^(2*I*f*x + 2*
I*e) + 1) - 12*a^2*d^3*e^(6*I*f*x + 6*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) + 18*I*a^2*c^3*e^(4*I*f*x + 4*I*e)*log
(e^(2*I*f*x + 2*I*e) + 1) + 54*a^2*c^2*d*e^(4*I*f*x + 4*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) - 54*I*a^2*c*d^2*e^(
4*I*f*x + 4*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) - 18*a^2*d^3*e^(4*I*f*x + 4*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) +
12*I*a^2*c^3*e^(2*I*f*x + 2*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) + 36*a^2*c^2*d*e^(2*I*f*x + 2*I*e)*log(e^(2*I*f*
x + 2*I*e) + 1) - 36*I*a^2*c*d^2*e^(2*I*f*x + 2*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) - 12*a^2*d^3*e^(2*I*f*x + 2*
I*e)*log(e^(2*I*f*x + 2*I*e) + 1) + 3*I*a^2*c^3*e^(6*I*f*x + 6*I*e) + 27*a^2*c^2*d*e^(6*I*f*x + 6*I*e) - 45*I*
a^2*c*d^2*e^(6*I*f*x + 6*I*e) - 21*a^2*d^3*e^(6*I*f*x + 6*I*e) + 9*I*a^2*c^3*e^(4*I*f*x + 4*I*e) + 72*a^2*c^2*
d*e^(4*I*f*x + 4*I*e) - 99*I*a^2*c*d^2*e^(4*I*f*x + 4*I*e) - 36*a^2*d^3*e^(4*I*f*x + 4*I*e) + 9*I*a^2*c^3*e^(2
*I*f*x + 2*I*e) + 63*a^2*c^2*d*e^(2*I*f*x + 2*I*e) - 75*I*a^2*c*d^2*e^(2*I*f*x + 2*I*e) - 29*a^2*d^3*e^(2*I*f*
x + 2*I*e) + 3*I*a^2*c^3*log(e^(2*I*f*x + 2*I*e) + 1) + 9*a^2*c^2*d*log(e^(2*I*f*x + 2*I*e) + 1) - 9*I*a^2*c*d
^2*log(e^(2*I*f*x + 2*I*e) + 1) - 3*a^2*d^3*log(e^(2*I*f*x + 2*I*e) + 1) + 3*I*a^2*c^3 + 18*a^2*c^2*d - 21*I*a
^2*c*d^2 - 8*a^2*d^3)/(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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Mupad [B]
time = 5.06, size = 223, normalized size = 1.58 \begin {gather*} \frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {a^2\,d^3}{2}+\frac {a^2\,d^2\,\left (d+c\,3{}\mathrm {i}\right )}{2}+\frac {a^2\,c\,d\,\left (d+c\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2}\right )}{f}+\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (a^2\,c^3\,2{}\mathrm {i}+6\,a^2\,c^2\,d-a^2\,c\,d^2\,6{}\mathrm {i}-2\,a^2\,d^3\right )}{f}-\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (a^2\,d^3\,1{}\mathrm {i}+a^2\,d^2\,\left (d+c\,3{}\mathrm {i}\right )\,1{}\mathrm {i}-a^2\,c^2\,\left (3\,d+c\,1{}\mathrm {i}\right )\,1{}\mathrm {i}-3\,a^2\,c\,d\,\left (d+c\,1{}\mathrm {i}\right )\right )}{f}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (\frac {a^2\,d^3\,1{}\mathrm {i}}{3}+\frac {a^2\,d^2\,\left (d+c\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{3}\right )}{f}-\frac {a^2\,d^3\,{\mathrm {tan}\left (e+f\,x\right )}^4}{4\,f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(tan(e + f*x)^2*((a^2*d^3)/2 + (a^2*d^2*(c*3i + d))/2 + (a^2*c*d*(c*1i + d)*3i)/2))/f + (log(tan(e + f*x) + 1i
)*(a^2*c^3*2i - 2*a^2*d^3 - a^2*c*d^2*6i + 6*a^2*c^2*d))/f - (tan(e + f*x)*(a^2*d^3*1i + a^2*d^2*(c*3i + d)*1i
- a^2*c^2*(c*1i + 3*d)*1i - 3*a^2*c*d*(c*1i + d)))/f + (tan(e + f*x)^3*((a^2*d^3*1i)/3 + (a^2*d^2*(c*3i + d)*
1i)/3))/f - (a^2*d^3*tan(e + f*x)^4)/(4*f)

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