Optimal. Leaf size=93 \[ -\frac {a^2 (3 B+i A)}{2 c^3 f (\tan (e+f x)+i)^2}-\frac {2 a^2 (A-i B)}{3 c^3 f (\tan (e+f x)+i)^3}-\frac {i a^2 B}{c^3 f (\tan (e+f x)+i)} \]
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Rubi [A] time = 0.15, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {3588, 77} \[ -\frac {a^2 (3 B+i A)}{2 c^3 f (\tan (e+f x)+i)^2}-\frac {2 a^2 (A-i B)}{3 c^3 f (\tan (e+f x)+i)^3}-\frac {i a^2 B}{c^3 f (\tan (e+f x)+i)} \]
Antiderivative was successfully verified.
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Rule 77
Rule 3588
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^3} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {(a+i a x) (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 {2 a (A-i B)}{c^4 (i+x)^4}+\frac {a (i A+3 B)}{c^4 (i+x)^3}+\frac {i a B}{c^4 (i+x)^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {2 a^2 (A-i B)}{3 c^3 f (i+\tan (e+f x))^3}-\frac {a^2 (i A+3 B)}{2 c^3 f (i+\tan (e+f x))^2}-\frac {i a^2 B}{c^3 f (i+\tan (e+f x))}\\ \end {align*}
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Mathematica [A] time = 2.99, size = 81, normalized size = 0.87 \[ \frac {a^2 (\cos (5 e+7 f x)+i \sin (5 e+7 f x)) ((B-5 i A) \cos (e+f x)-(A+5 i B) \sin (e+f x))}{24 c^3 f (\cos (f x)+i \sin (f x))^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 49, normalized size = 0.53 \[ \frac {{\left (-2 i \, A - 2 \, B\right )} a^{2} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (-3 i \, A + 3 \, B\right )} a^{2} e^{\left (4 i \, f x + 4 i \, e\right )}}{24 \, c^{3} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.79, size = 165, normalized size = 1.77 \[ -\frac {2 \, {\left (3 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3 i \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 3 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 8 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 i \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 i \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{3 \, c^{3} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 69, normalized size = 0.74 \[ \frac {a^{2} \left (-\frac {i B}{\tan \left (f x +e \right )+i}-\frac {i A +3 B}{2 \left (\tan \left (f x +e \right )+i\right )^{2}}-\frac {-2 i B +2 A}{3 \left (\tan \left (f x +e \right )+i\right )^{3}}\right )}{f \,c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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mupad [B] time = 8.69, size = 87, normalized size = 0.94 \[ \frac {\frac {a^2\,\left (A-B\,1{}\mathrm {i}\right )}{6}+\frac {a^2\,\mathrm {tan}\left (e+f\,x\right )\,\left (-3\,B+A\,3{}\mathrm {i}\right )}{6}+B\,a^2\,{\mathrm {tan}\left (e+f\,x\right )}^2\,1{}\mathrm {i}}{c^3\,f\,\left (-{\mathrm {tan}\left (e+f\,x\right )}^3-{\mathrm {tan}\left (e+f\,x\right )}^2\,3{}\mathrm {i}+3\,\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.56, size = 168, normalized size = 1.81 \[ \begin {cases} \frac {\left (- 12 i A a^{2} c^{3} f e^{4 i e} + 12 B a^{2} c^{3} f e^{4 i e}\right ) e^{4 i f x} + \left (- 8 i A a^{2} c^{3} f e^{6 i e} - 8 B a^{2} c^{3} f e^{6 i e}\right ) e^{6 i f x}}{96 c^{6} f^{2}} & \text {for}\: 96 c^{6} f^{2} \neq 0 \\\frac {x \left (A a^{2} e^{6 i e} + A a^{2} e^{4 i e} - i B a^{2} e^{6 i e} + i B a^{2} e^{4 i e}\right )}{2 c^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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