3.11.95 \(\int \frac {(a+i a \tan (e+f x))^3}{(c+d \tan (e+f x))^3} \, dx\) [1095]

Optimal. Leaf size=134 \[ \frac {4 a^3 x}{(c-i d)^3}-\frac {4 a^3 \log (c \cos (e+f x)+d \sin (e+f x))}{(i c+d)^3 f}-\frac {a (a+i a \tan (e+f x))^2}{2 (i c+d) f (c+d \tan (e+f x))^2}+\frac {2 a^3 (c+i d)}{(c-i d)^2 d f (c+d \tan (e+f x))} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.18, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3626, 3623, 3612, 3611} \begin {gather*} \frac {2 a^3 (c+i d)}{d f (c-i d)^2 (c+d \tan (e+f x))}-\frac {4 a^3 \log (c \cos (e+f x)+d \sin (e+f x))}{f (d+i c)^3}+\frac {4 a^3 x}{(c-i d)^3}-\frac {a (a+i a \tan (e+f x))^2}{2 f (d+i c) (c+d \tan (e+f x))^2} \end {gather*}

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

Rule 3611

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c/(b*f))
*Log[RemoveContent[a*Cos[e + f*x] + b*Sin[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] && EqQ[a*c + b*d, 0]

Rule 3612

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

Rule 3623

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

Rule 3626

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(595\) vs. \(2(134)=268\).
time = 2.80, size = 595, normalized size = 4.44 \begin {gather*} \frac {a^3 \left (2 c^3 f x \cos (3 e+2 f x)-6 c d^2 f x \cos (3 e+2 f x)-i c^3 \cos (3 e+2 f x) \log \left ((c \cos (e+f x)+d \sin (e+f x))^2\right )+3 i c d^2 \cos (3 e+2 f x) \log \left ((c \cos (e+f x)+d \sin (e+f x))^2\right )+\left (c^2+d^2\right ) \cos (e+2 f x) \left (3 d+2 c f x-i c \log \left ((c \cos (e+f x)+d \sin (e+f x))^2\right )\right )+\left (c^2+d^2\right ) \cos (e) \left (-i c-3 d+4 c f x-2 i c \log \left ((c \cos (e+f x)+d \sin (e+f x))^2\right )\right )+3 c^3 \sin (e)-i c^2 d \sin (e)+3 c d^2 \sin (e)-i d^3 \sin (e)+4 c^2 d f x \sin (e)+4 d^3 f x \sin (e)-2 i c^2 d \log \left ((c \cos (e+f x)+d \sin (e+f x))^2\right ) \sin (e)-2 i d^3 \log \left ((c \cos (e+f x)+d \sin (e+f x))^2\right ) \sin (e)-3 c^3 \sin (e+2 f x)-3 c d^2 \sin (e+2 f x)+2 c^2 d f x \sin (e+2 f x)+2 d^3 f x \sin (e+2 f x)-i c^2 d \log \left ((c \cos (e+f x)+d \sin (e+f x))^2\right ) \sin (e+2 f x)-i d^3 \log \left ((c \cos (e+f x)+d \sin (e+f x))^2\right ) \sin (e+2 f x)+6 c^2 d f x \sin (3 e+2 f x)-2 d^3 f x \sin (3 e+2 f x)-3 i c^2 d \log \left ((c \cos (e+f x)+d \sin (e+f x))^2\right ) \sin (3 e+2 f x)+i d^3 \log \left ((c \cos (e+f x)+d \sin (e+f x))^2\right ) \sin (3 e+2 f x)\right )}{2 (c-i d)^3 f (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(2*c^3*f*x*Cos[3*e + 2*f*x] - 6*c*d^2*f*x*Cos[3*e + 2*f*x] - I*c^3*Cos[3*e + 2*f*x]*Log[(c*Cos[e + f*x] +
d*Sin[e + f*x])^2] + (3*I)*c*d^2*Cos[3*e + 2*f*x]*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2] + (c^2 + d^2)*Cos[
e + 2*f*x]*(3*d + 2*c*f*x - I*c*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]) + (c^2 + d^2)*Cos[e]*((-I)*c - 3*d +
4*c*f*x - (2*I)*c*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]) + 3*c^3*Sin[e] - I*c^2*d*Sin[e] + 3*c*d^2*Sin[e]
- I*d^3*Sin[e] + 4*c^2*d*f*x*Sin[e] + 4*d^3*f*x*Sin[e] - (2*I)*c^2*d*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*
Sin[e] - (2*I)*d^3*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*Sin[e] - 3*c^3*Sin[e + 2*f*x] - 3*c*d^2*Sin[e + 2*
f*x] + 2*c^2*d*f*x*Sin[e + 2*f*x] + 2*d^3*f*x*Sin[e + 2*f*x] - I*c^2*d*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2
]*Sin[e + 2*f*x] - I*d^3*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*Sin[e + 2*f*x] + 6*c^2*d*f*x*Sin[3*e + 2*f*x
] - 2*d^3*f*x*Sin[3*e + 2*f*x] - (3*I)*c^2*d*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*Sin[3*e + 2*f*x] + I*d^3
*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*Sin[3*e + 2*f*x]))/(2*(c - I*d)^3*f*(c*Cos[e] + d*Sin[e])*(c*Cos[e +
f*x] + d*Sin[e + f*x])^2)

________________________________________________________________________________________

Maple [A]
time = 0.34, size = 239, normalized size = 1.78

method	result	size
derivativedivides	\(\frac {a^{3} \left (\frac {\frac {\left (4 i c^{3}-12 i c \,d^{2}-12 c^{2} d +4 d^{3}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (12 i c^{2} d -4 i d^{3}+4 c^{3}-12 c \,d^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{3}}-\frac {-i c^{4}-6 i c^{2} d^{2}+3 i d^{4}+8 c \,d^{3}}{\left (c^{2}+d^{2}\right )^{2} d^{2} \left (c +d \tan \left (f x +e \right )\right )}-\frac {i c^{3}-3 i c \,d^{2}-3 c^{2} d +d^{3}}{2 d^{2} \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )^{2}}-\frac {4 \left (i c^{3}-3 i c \,d^{2}-3 c^{2} d +d^{3}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{3}}\right )}{f}\)	\(239\)
default	\(\frac {a^{3} \left (\frac {\frac {\left (4 i c^{3}-12 i c \,d^{2}-12 c^{2} d +4 d^{3}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (12 i c^{2} d -4 i d^{3}+4 c^{3}-12 c \,d^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{3}}-\frac {-i c^{4}-6 i c^{2} d^{2}+3 i d^{4}+8 c \,d^{3}}{\left (c^{2}+d^{2}\right )^{2} d^{2} \left (c +d \tan \left (f x +e \right )\right )}-\frac {i c^{3}-3 i c \,d^{2}-3 c^{2} d +d^{3}}{2 d^{2} \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )^{2}}-\frac {4 \left (i c^{3}-3 i c \,d^{2}-3 c^{2} d +d^{3}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{3}}\right )}{f}\)	\(239\)
risch	\(-\frac {8 a^{3} x}{3 i c^{2} d -i d^{3}-c^{3}+3 c \,d^{2}}-\frac {8 i a^{3} x}{i c^{3}-3 i c \,d^{2}+3 c^{2} d -d^{3}}-\frac {8 i a^{3} e}{f \left (i c^{3}-3 i c \,d^{2}+3 c^{2} d -d^{3}\right )}-\frac {2 i a^{3} \left (4 c^{2} {\mathrm e}^{2 i \left (f x +e \right )}+4 d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+6 i c d +3 c^{2}-3 d^{2}\right )}{f \left (-i d +c \right )^{3} \left (-i {\mathrm e}^{2 i \left (f x +e \right )} d +i d +{\mathrm e}^{2 i \left (f x +e \right )} c +c \right )^{2}}+\frac {4 a^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i d +c}{i d -c}\right )}{f \left (i c^{3}-3 i c \,d^{2}+3 c^{2} d -d^{3}\right )}\)	\(263\)
norman	\(\frac {\frac {\left (i a^{3} c^{2}+3 i a^{3} d^{2}+2 a^{3} c d \right ) \tan \left (f x +e \right )}{d f \left (-2 i c d +c^{2}-d^{2}\right )}+\frac {4 a^{3} c^{2} x}{\left (-i d +c \right )^{3}}+\frac {i a^{3} c^{3}+5 i a^{3} c \,d^{2}+5 a^{3} c^{2} d +a^{3} d^{3}}{2 d^{2} f \left (-2 i c d +c^{2}-d^{2}\right )}+\frac {8 c d \,a^{3} x \tan \left (f x +e \right )}{\left (-i d +c \right )^{3}}-\frac {4 i a^{3} d^{2} x \left (\tan ^{2}\left (f x +e \right )\right )}{\left (i c +d \right )^{3}}}{\left (c +d \tan \left (f x +e \right )\right )^{2}}+\frac {2 i a^{3} \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{f \left (-3 i c^{2} d +i d^{3}+c^{3}-3 c \,d^{2}\right )}-\frac {4 i a^{3} \ln \left (c +d \tan \left (f x +e \right )\right )}{f \left (-3 i c^{2} d +i d^{3}+c^{3}-3 c \,d^{2}\right )}\)	\(286\)

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,method=_RETURNVERBOSE)

[Out]

1/f*a^3*(1/(c^2+d^2)^3*(1/2*(4*I*c^3-12*I*c*d^2-12*c^2*d+4*d^3)*ln(1+tan(f*x+e)^2)+(12*I*c^2*d-4*I*d^3+4*c^3-1
2*c*d^2)*arctan(tan(f*x+e)))-(-I*c^4-6*I*c^2*d^2+3*I*d^4+8*c*d^3)/(c^2+d^2)^2/d^2/(c+d*tan(f*x+e))-1/2*(I*c^3-
3*I*c*d^2-3*c^2*d+d^3)/d^2/(c^2+d^2)/(c+d*tan(f*x+e))^2-4*(I*c^3-3*I*c*d^2-3*c^2*d+d^3)/(c^2+d^2)^3*ln(c+d*tan
(f*x+e)))

________________________________________________________________________________________

Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 409 vs. \(2 (125) = 250\).
time = 0.50, size = 409, normalized size = 3.05 \begin {gather*} \frac {\frac {8 \, {\left (a^{3} c^{3} + 3 i \, a^{3} c^{2} d - 3 \, a^{3} c d^{2} - i \, a^{3} d^{3}\right )} {\left (f x + e\right )}}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} - \frac {8 \, {\left (i \, a^{3} c^{3} - 3 \, a^{3} c^{2} d - 3 i \, a^{3} c d^{2} + a^{3} d^{3}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} - \frac {4 \, {\left (-i \, a^{3} c^{3} + 3 \, a^{3} c^{2} d + 3 i \, a^{3} c d^{2} - a^{3} d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} + \frac {i \, a^{3} c^{5} + 3 \, a^{3} c^{4} d + 14 i \, a^{3} c^{3} d^{2} - 14 \, a^{3} c^{2} d^{3} - 3 i \, a^{3} c d^{4} - a^{3} d^{5} + 2 \, {\left (i \, a^{3} c^{4} d + 6 i \, a^{3} c^{2} d^{3} - 8 \, a^{3} c d^{4} - 3 i \, a^{3} d^{5}\right )} \tan \left (f x + e\right )}{c^{6} d^{2} + 2 \, c^{4} d^{4} + c^{2} d^{6} + {\left (c^{4} d^{4} + 2 \, c^{2} d^{6} + d^{8}\right )} \tan \left (f x + e\right )^{2} + 2 \, {\left (c^{5} d^{3} + 2 \, c^{3} d^{5} + c d^{7}\right )} \tan \left (f x + e\right )}}{2 \, f} \end {gather*}

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]

1/2*(8*(a^3*c^3 + 3*I*a^3*c^2*d - 3*a^3*c*d^2 - I*a^3*d^3)*(f*x + e)/(c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6) - 8*(
I*a^3*c^3 - 3*a^3*c^2*d - 3*I*a^3*c*d^2 + a^3*d^3)*log(d*tan(f*x + e) + c)/(c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6)
- 4*(-I*a^3*c^3 + 3*a^3*c^2*d + 3*I*a^3*c*d^2 - a^3*d^3)*log(tan(f*x + e)^2 + 1)/(c^6 + 3*c^4*d^2 + 3*c^2*d^4
+ d^6) + (I*a^3*c^5 + 3*a^3*c^4*d + 14*I*a^3*c^3*d^2 - 14*a^3*c^2*d^3 - 3*I*a^3*c*d^4 - a^3*d^5 + 2*(I*a^3*c^
4*d + 6*I*a^3*c^2*d^3 - 8*a^3*c*d^4 - 3*I*a^3*d^5)*tan(f*x + e))/(c^6*d^2 + 2*c^4*d^4 + c^2*d^6 + (c^4*d^4 + 2
*c^2*d^6 + d^8)*tan(f*x + e)^2 + 2*(c^5*d^3 + 2*c^3*d^5 + c*d^7)*tan(f*x + e)))/f

________________________________________________________________________________________

Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (125) = 250\).
time = 1.55, size = 311, normalized size = 2.32 \begin {gather*} \frac {2 \, {\left (3 \, a^{3} c^{2} + 6 i \, a^{3} c d - 3 \, a^{3} d^{2} + 4 \, {\left (a^{3} c^{2} + a^{3} d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + 2 \, {\left (a^{3} c^{2} + 2 i \, a^{3} c d - a^{3} d^{2} + {\left (a^{3} c^{2} - 2 i \, a^{3} c d - a^{3} d^{2}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, {\left (a^{3} c^{2} + a^{3} d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \log \left (\frac {{\left (i \, c + d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c - d}{i \, c + d}\right )\right )}}{{\left (i \, c^{5} + 5 \, c^{4} d - 10 i \, c^{3} d^{2} - 10 \, c^{2} d^{3} + 5 i \, c d^{4} + d^{5}\right )} f e^{\left (4 i \, f x + 4 i \, e\right )} - 2 \, {\left (-i \, c^{5} - 3 \, c^{4} d + 2 i \, c^{3} d^{2} - 2 \, c^{2} d^{3} + 3 i \, c d^{4} + d^{5}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (i \, c^{5} + c^{4} d + 2 i \, c^{3} d^{2} + 2 \, c^{2} d^{3} + i \, c d^{4} + d^{5}\right )} f} \end {gather*}

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]

2*(3*a^3*c^2 + 6*I*a^3*c*d - 3*a^3*d^2 + 4*(a^3*c^2 + a^3*d^2)*e^(2*I*f*x + 2*I*e) + 2*(a^3*c^2 + 2*I*a^3*c*d
- a^3*d^2 + (a^3*c^2 - 2*I*a^3*c*d - a^3*d^2)*e^(4*I*f*x + 4*I*e) + 2*(a^3*c^2 + a^3*d^2)*e^(2*I*f*x + 2*I*e))
*log(((I*c + d)*e^(2*I*f*x + 2*I*e) + I*c - d)/(I*c + d)))/((I*c^5 + 5*c^4*d - 10*I*c^3*d^2 - 10*c^2*d^3 + 5*I
*c*d^4 + d^5)*f*e^(4*I*f*x + 4*I*e) - 2*(-I*c^5 - 3*c^4*d + 2*I*c^3*d^2 - 2*c^2*d^3 + 3*I*c*d^4 + d^5)*f*e^(2*
I*f*x + 2*I*e) + (I*c^5 + c^4*d + 2*I*c^3*d^2 + 2*c^2*d^3 + I*c*d^4 + d^5)*f)

________________________________________________________________________________________

Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (110) = 220\).
time = 4.03, size = 377, normalized size = 2.81 \begin {gather*} - \frac {4 i a^{3} \log {\left (\frac {c + i d}{c e^{2 i e} - i d e^{2 i e}} + e^{2 i f x} \right )}}{f \left (c - i d\right )^{3}} + \frac {- 6 i a^{3} c^{2} + 12 a^{3} c d + 6 i a^{3} d^{2} + \left (- 8 i a^{3} c^{2} e^{2 i e} - 8 i a^{3} d^{2} e^{2 i e}\right ) e^{2 i f x}}{c^{5} f - i c^{4} d f + 2 c^{3} d^{2} f - 2 i c^{2} d^{3} f + c d^{4} f - i d^{5} f + \left (2 c^{5} f e^{2 i e} - 6 i c^{4} d f e^{2 i e} - 4 c^{3} d^{2} f e^{2 i e} - 4 i c^{2} d^{3} f e^{2 i e} - 6 c d^{4} f e^{2 i e} + 2 i d^{5} f e^{2 i e}\right ) e^{2 i f x} + \left (c^{5} f e^{4 i e} - 5 i c^{4} d f e^{4 i e} - 10 c^{3} d^{2} f e^{4 i e} + 10 i c^{2} d^{3} f e^{4 i e} + 5 c d^{4} f e^{4 i e} - i d^{5} f e^{4 i e}\right ) e^{4 i f x}} \end {gather*}

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]

-4*I*a**3*log((c + I*d)/(c*exp(2*I*e) - I*d*exp(2*I*e)) + exp(2*I*f*x))/(f*(c - I*d)**3) + (-6*I*a**3*c**2 + 1
2*a**3*c*d + 6*I*a**3*d**2 + (-8*I*a**3*c**2*exp(2*I*e) - 8*I*a**3*d**2*exp(2*I*e))*exp(2*I*f*x))/(c**5*f - I*
c**4*d*f + 2*c**3*d**2*f - 2*I*c**2*d**3*f + c*d**4*f - I*d**5*f + (2*c**5*f*exp(2*I*e) - 6*I*c**4*d*f*exp(2*I
*e) - 4*c**3*d**2*f*exp(2*I*e) - 4*I*c**2*d**3*f*exp(2*I*e) - 6*c*d**4*f*exp(2*I*e) + 2*I*d**5*f*exp(2*I*e))*e
xp(2*I*f*x) + (c**5*f*exp(4*I*e) - 5*I*c**4*d*f*exp(4*I*e) - 10*c**3*d**2*f*exp(4*I*e) + 10*I*c**2*d**3*f*exp(
4*I*e) + 5*c*d**4*f*exp(4*I*e) - I*d**5*f*exp(4*I*e))*exp(4*I*f*x))

________________________________________________________________________________________

Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 470 vs. \(2 (125) = 250\).
time = 0.80, size = 470, normalized size = 3.51 \begin {gather*} \frac {2 \, {\left (\frac {2 \, a^{3} \log \left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - c\right )}{i \, c^{3} + 3 \, c^{2} d - 3 i \, c d^{2} - d^{3}} - \frac {4 \, a^{3} \log \left (-i \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{i \, c^{3} + 3 \, c^{2} d - 3 i \, c d^{2} - d^{3}} + \frac {3 \, a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 3 i \, a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 13 \, a^{3} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 i \, a^{3} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - a^{3} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 7 \, a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 6 i \, a^{3} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 12 \, a^{3} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 6 i \, a^{3} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 i \, a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 13 \, a^{3} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 i \, a^{3} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a^{3} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, a^{3} c^{4}}{{\left (-i \, c^{5} - 3 \, c^{4} d + 3 i \, c^{3} d^{2} + c^{2} d^{3}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - c\right )}^{2}}\right )}}{f} \end {gather*}

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]

2*(2*a^3*log(c*tan(1/2*f*x + 1/2*e)^2 - 2*d*tan(1/2*f*x + 1/2*e) - c)/(I*c^3 + 3*c^2*d - 3*I*c*d^2 - d^3) - 4*
a^3*log(-I*tan(1/2*f*x + 1/2*e) + 1)/(I*c^3 + 3*c^2*d - 3*I*c*d^2 - d^3) + (3*a^3*c^4*tan(1/2*f*x + 1/2*e)^4 -
3*I*a^3*c^4*tan(1/2*f*x + 1/2*e)^3 - 13*a^3*c^3*d*tan(1/2*f*x + 1/2*e)^3 - 3*I*a^3*c^2*d^2*tan(1/2*f*x + 1/2*
e)^3 - a^3*c*d^3*tan(1/2*f*x + 1/2*e)^3 - 7*a^3*c^4*tan(1/2*f*x + 1/2*e)^2 + 6*I*a^3*c^3*d*tan(1/2*f*x + 1/2*e
)^2 + 12*a^3*c^2*d^2*tan(1/2*f*x + 1/2*e)^2 + 6*I*a^3*c*d^3*tan(1/2*f*x + 1/2*e)^2 + a^3*d^4*tan(1/2*f*x + 1/2
*e)^2 + 3*I*a^3*c^4*tan(1/2*f*x + 1/2*e) + 13*a^3*c^3*d*tan(1/2*f*x + 1/2*e) + 3*I*a^3*c^2*d^2*tan(1/2*f*x + 1
/2*e) + a^3*c*d^3*tan(1/2*f*x + 1/2*e) + 3*a^3*c^4)/((-I*c^5 - 3*c^4*d + 3*I*c^3*d^2 + c^2*d^3)*(c*tan(1/2*f*x
+ 1/2*e)^2 - 2*d*tan(1/2*f*x + 1/2*e) - c)^2))/f

________________________________________________________________________________________

Mupad [B]
time = 5.62, size = 314, normalized size = 2.34 \begin {gather*} -\frac {\frac {a^3\,\left (c^3\,1{}\mathrm {i}+5\,c^2\,d+c\,d^2\,5{}\mathrm {i}+d^3\right )}{2\,d^4\,\left (-c^2+c\,d\,2{}\mathrm {i}+d^2\right )}+\frac {a^3\,\mathrm {tan}\left (e+f\,x\right )\,\left (c^2-c\,d\,2{}\mathrm {i}+3\,d^2\right )\,1{}\mathrm {i}}{d^3\,\left (-c^2+c\,d\,2{}\mathrm {i}+d^2\right )}}{f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^2+\frac {c^2}{d^2}+\frac {2\,c\,\mathrm {tan}\left (e+f\,x\right )}{d}\right )}+\frac {a^3\,\mathrm {atan}\left (\frac {c^3-c^2\,d\,1{}\mathrm {i}+c\,d^2-d^3\,1{}\mathrm {i}}{{\left (c-d\,1{}\mathrm {i}\right )}^2\,\left (d+c\,1{}\mathrm {i}\right )}-\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (2\,c^8\,d^2+8\,c^6\,d^4+12\,c^4\,d^6+8\,c^2\,d^8+2\,d^{10}\right )\,1{}\mathrm {i}}{{\left (c-d\,1{}\mathrm {i}\right )}^2\,\left (d+c\,1{}\mathrm {i}\right )\,\left (-c^6\,d\,1{}\mathrm {i}+2\,c^5\,d^2-c^4\,d^3\,1{}\mathrm {i}+4\,c^3\,d^4+c^2\,d^5\,1{}\mathrm {i}+2\,c\,d^6+d^7\,1{}\mathrm {i}\right )}\right )\,8{}\mathrm {i}}{f\,{\left (c-d\,1{}\mathrm {i}\right )}^2\,\left (d+c\,1{}\mathrm {i}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^3*atan((c*d^2 - c^2*d*1i + c^3 - d^3*1i)/((c - d*1i)^2*(c*1i + d)) - (tan(e + f*x)*(2*d^10 + 8*c^2*d^8 + 12
*c^4*d^6 + 8*c^6*d^4 + 2*c^8*d^2)*1i)/((c - d*1i)^2*(c*1i + d)*(2*c*d^6 - c^6*d*1i + d^7*1i + c^2*d^5*1i + 4*c
^3*d^4 - c^4*d^3*1i + 2*c^5*d^2)))*8i)/(f*(c - d*1i)^2*(c*1i + d)) - ((a^3*(c*d^2*5i + 5*c^2*d + c^3*1i + d^3)
)/(2*d^4*(c*d*2i - c^2 + d^2)) + (a^3*tan(e + f*x)*(c^2 - c*d*2i + 3*d^2)*1i)/(d^3*(c*d*2i - c^2 + d^2)))/(f*(
tan(e + f*x)^2 + c^2/d^2 + (2*c*tan(e + f*x))/d))

________________________________________________________________________________________