Optimal. Leaf size=57 \[ -\frac {a (B+i A)}{4 c^4 f (\tan (e+f x)+i)^4}-\frac {i a B}{3 c^4 f (\tan (e+f x)+i)^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 57, 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} \[ -\frac {a (B+i A)}{4 c^4 f (\tan (e+f x)+i)^4}-\frac {i a B}{3 c^4 f (\tan (e+f x)+i)^3} \]
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))^4} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {A+B x}{(c-i c x)^5} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \operatorname {Subst}\left (\int \left (\frac {i A+B}{c^5 (i+x)^5}+\frac {i B}{c^5 (i+x)^4}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {a (i A+B)}{4 c^4 f (i+\tan (e+f x))^4}-\frac {i a B}{3 c^4 f (i+\tan (e+f x))^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.58, size = 97, normalized size = 1.70 \[ \frac {a (\cos (5 (e+f x))+i \sin (5 (e+f x))) (-(3 A+5 i B) (2 \sin (e+f x)+3 \sin (3 (e+f x)))+2 (B-15 i A) \cos (e+f x)+3 (3 B-5 i A) \cos (3 (e+f x)))}{192 c^4 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.66, size = 81, normalized size = 1.42 \[ \frac {{\left (-3 i \, A - 3 \, B\right )} a e^{\left (8 i \, f x + 8 i \, e\right )} + {\left (-12 i \, A - 4 \, B\right )} a e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (-18 i \, A + 6 \, B\right )} a e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (-12 i \, A + 12 \, B\right )} a e^{\left (2 i \, f x + 2 i \, e\right )}}{192 \, c^{4} f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 2.68, size = 213, normalized size = 3.74 \[ -\frac {2 \, {\left (3 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 9 i \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 3 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 21 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 4 i \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 24 i \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 8 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 21 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 4 i \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 9 i \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{3 \, c^{4} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.23, size = 44, normalized size = 0.77 \[ \frac {a \left (-\frac {i A +B}{4 \left (\tan \left (f x +e \right )+i\right )^{4}}-\frac {i B}{3 \left (\tan \left (f x +e \right )+i\right )^{3}}\right )}{f \,c^{4}} \]
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 = 8.67, size = 73, normalized size = 1.28 \[ -\frac {\frac {a\,\left (-B+A\,3{}\mathrm {i}\right )}{12}+\frac {B\,a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}{3}}{c^4\,f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^4+{\mathrm {tan}\left (e+f\,x\right )}^3\,4{}\mathrm {i}-6\,{\mathrm {tan}\left (e+f\,x\right )}^2-\mathrm {tan}\left (e+f\,x\right )\,4{}\mathrm {i}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.64, size = 306, normalized size = 5.37 \[ \begin {cases} \frac {\left (- 98304 i A a c^{12} f^{3} e^{2 i e} + 98304 B a c^{12} f^{3} e^{2 i e}\right ) e^{2 i f x} + \left (- 147456 i A a c^{12} f^{3} e^{4 i e} + 49152 B a c^{12} f^{3} e^{4 i e}\right ) e^{4 i f x} + \left (- 98304 i A a c^{12} f^{3} e^{6 i e} - 32768 B a c^{12} f^{3} e^{6 i e}\right ) e^{6 i f x} + \left (- 24576 i A a c^{12} f^{3} e^{8 i e} - 24576 B a c^{12} f^{3} e^{8 i e}\right ) e^{8 i f x}}{1572864 c^{16} f^{4}} & \text {for}\: 1572864 c^{16} f^{4} \neq 0 \\\frac {x \left (A a e^{8 i e} + 3 A a e^{6 i e} + 3 A a e^{4 i e} + A a e^{2 i e} - i B a e^{8 i e} - i B a e^{6 i e} + i B a e^{4 i e} + i B a e^{2 i e}\right )}{8 c^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________