Optimal. Leaf size=59 \[ \frac {c (A+i B)}{3 a^3 f (-\tan (e+f x)+i)^3}-\frac {B c}{2 a^3 f (-\tan (e+f x)+i)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {3588, 43} \[ \frac {c (A+i B)}{3 a^3 f (-\tan (e+f x)+i)^3}-\frac {B c}{2 a^3 f (-\tan (e+f x)+i)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 3588
Rubi steps
\begin {align*} \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))}{(a+i a \tan (e+f x))^3} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {A+B x}{(a+i a x)^4} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \operatorname {Subst}\left (\int \left (\frac {A+i B}{a^4 (-i+x)^4}+\frac {B}{a^4 (-i+x)^3}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(A+i B) c}{3 a^3 f (i-\tan (e+f x))^3}-\frac {B c}{2 a^3 f (i-\tan (e+f x))^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.39, size = 81, normalized size = 1.37 \[ \frac {c (\tan (e+f x)+i) \sec ^2(e+f x) (-2 (A-2 i B) \sin (2 (e+f x))+2 (B+2 i A) \cos (2 (e+f x))+3 i A)}{24 a^3 f (\tan (e+f x)-i)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.58, size = 58, normalized size = 0.98 \[ \frac {{\left ({\left (3 i \, A + 3 \, B\right )} c e^{\left (4 i \, f x + 4 i \, e\right )} + 3 i \, A c e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (i \, A - B\right )} c\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{24 \, a^{3} f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 2.64, size = 149, normalized size = 2.53 \[ -\frac {2 \, {\left (3 \, A c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 6 i \, A c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 3 \, B c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 10 \, A c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 i \, B c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 i \, A c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, B c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, A c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{3 \, a^{3} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - i\right )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.26, size = 43, normalized size = 0.73 \[ \frac {c \left (-\frac {B}{2 \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {i B +A}{3 \left (\tan \left (f x +e \right )-i\right )^{3}}\right )}{f \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 8.84, size = 62, normalized size = 1.05 \[ \frac {\frac {c\,\left (B+A\,2{}\mathrm {i}\right )}{6}+\frac {B\,c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}{2}}{a^3\,f\,\left (-{\mathrm {tan}\left (e+f\,x\right )}^3\,1{}\mathrm {i}-3\,{\mathrm {tan}\left (e+f\,x\right )}^2+\mathrm {tan}\left (e+f\,x\right )\,3{}\mathrm {i}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.57, size = 211, normalized size = 3.58 \[ \begin {cases} - \frac {\left (- 192 i A a^{6} c f^{2} e^{8 i e} e^{- 4 i f x} + \left (- 64 i A a^{6} c f^{2} e^{6 i e} + 64 B a^{6} c f^{2} e^{6 i e}\right ) e^{- 6 i f x} + \left (- 192 i A a^{6} c f^{2} e^{10 i e} - 192 B a^{6} c f^{2} e^{10 i e}\right ) e^{- 2 i f x}\right ) e^{- 12 i e}}{1536 a^{9} f^{3}} & \text {for}\: 1536 a^{9} f^{3} e^{12 i e} \neq 0 \\\frac {x \left (A c e^{4 i e} + 2 A c e^{2 i e} + A c - i B c e^{4 i e} + i B c\right ) e^{- 6 i e}}{4 a^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________