Optimal. Leaf size=55 \[ -\frac {a (A-i B)}{3 c^3 f (\tan (e+f x)+i)^3}-\frac {a B}{2 c^3 f (\tan (e+f x)+i)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 55, 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} \[ -\frac {a (A-i B)}{3 c^3 f (\tan (e+f x)+i)^3}-\frac {a B}{2 c^3 f (\tan (e+f x)+i)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 3588
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x)) (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^3} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {A+B x}{(c-i c x)^4} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \operatorname {Subst}\left (\int \left (\frac {A-i B}{c^4 (i+x)^4}+\frac {B}{c^4 (i+x)^3}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {a (A-i B)}{3 c^3 f (i+\tan (e+f x))^3}-\frac {a B}{2 c^3 f (i+\tan (e+f x))^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.41, size = 72, normalized size = 1.31 \[ \frac {a (\cos (4 (e+f x))+i \sin (4 (e+f x))) (-2 (A+2 i B) \sin (2 (e+f x))+2 (B-2 i A) \cos (2 (e+f x))-3 i A)}{24 c^3 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.84, size = 58, normalized size = 1.05 \[ \frac {{\left (-i \, A - B\right )} a e^{\left (6 i \, f x + 6 i \, e\right )} - 3 i \, A a e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (-3 i \, A + 3 \, B\right )} a e^{\left (2 i \, f x + 2 i \, e\right )}}{24 \, c^{3} f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 2.04, size = 149, normalized size = 2.71 \[ -\frac {2 \, {\left (3 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 6 i \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 3 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 10 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 i \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 6 i \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{3 \, c^{3} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.23, size = 43, normalized size = 0.78 \[ \frac {a \left (-\frac {B}{2 \left (\tan \left (f x +e \right )+i\right )^{2}}-\frac {-i B +A}{3 \left (\tan \left (f x +e \right )+i\right )^{3}}\right )}{f \,c^{3}} \]
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 = 8.63, size = 63, normalized size = 1.15 \[ \frac {\frac {a\,\left (2\,A+B\,1{}\mathrm {i}\right )}{6}+\frac {B\,a\,\mathrm {tan}\left (e+f\,x\right )}{2}}{c^3\,f\,\left (-{\mathrm {tan}\left (e+f\,x\right )}^3-{\mathrm {tan}\left (e+f\,x\right )}^2\,3{}\mathrm {i}+3\,\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.51, size = 202, normalized size = 3.67 \[ \begin {cases} - \frac {192 i A a c^{6} f^{2} e^{4 i e} e^{4 i f x} + \left (192 i A a c^{6} f^{2} e^{2 i e} - 192 B a c^{6} f^{2} e^{2 i e}\right ) e^{2 i f x} + \left (64 i A a c^{6} f^{2} e^{6 i e} + 64 B a c^{6} f^{2} e^{6 i e}\right ) e^{6 i f x}}{1536 c^{9} f^{3}} & \text {for}\: 1536 c^{9} f^{3} \neq 0 \\\frac {x \left (A a e^{6 i e} + 2 A a e^{4 i e} + A a e^{2 i e} - i B a e^{6 i e} + i B a e^{2 i e}\right )}{4 c^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________