Optimal. Leaf size=55 \[ \frac {a (A-i B)}{5 c^5 f (\tan (e+f x)+i)^5}+\frac {a B}{4 c^5 f (\tan (e+f x)+i)^4} \]
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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} \[ \frac {a (A-i B)}{5 c^5 f (\tan (e+f x)+i)^5}+\frac {a B}{4 c^5 f (\tan (e+f x)+i)^4} \]
Antiderivative was successfully verified.
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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))^5} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {A+B x}{(c-i c x)^6} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \operatorname {Subst}\left (\int \left (\frac {-A+i B}{c^6 (i+x)^6}-\frac {B}{c^6 (i+x)^5}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {a (A-i B)}{5 c^5 f (i+\tan (e+f x))^5}+\frac {a B}{4 c^5 f (i+\tan (e+f x))^4}\\ \end {align*}
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Mathematica [B] time = 3.06, size = 124, normalized size = 2.25 \[ -\frac {i a (\cos (6 (e+f x))+i \sin (6 (e+f x))) (5 (6 A+i B) \cos (2 (e+f x))+4 (3 A+2 i B) \cos (4 (e+f x))-10 i A \sin (2 (e+f x))-8 i A \sin (4 (e+f x))+20 A+15 B \sin (2 (e+f x))+12 B \sin (4 (e+f x)))}{320 c^5 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.06, size = 94, normalized size = 1.71 \[ \frac {{\left (-2 i \, A - 2 \, B\right )} a e^{\left (10 i \, f x + 10 i \, e\right )} + {\left (-10 i \, A - 5 \, B\right )} a e^{\left (8 i \, f x + 8 i \, e\right )} - 20 i \, A a e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (-20 i \, A + 10 \, B\right )} a e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (-10 i \, A + 10 \, B\right )} a e^{\left (2 i \, f x + 2 i \, e\right )}}{320 \, c^{5} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.57, size = 277, normalized size = 5.04 \[ -\frac {2 \, {\left (5 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 20 i \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 5 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 60 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 10 i \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 100 i \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 25 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 126 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 24 i \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 100 i \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 25 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 60 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 10 i \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 20 i \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 5 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 5 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{5 \, c^{5} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 45, normalized size = 0.82 \[ \frac {a \left (-\frac {i B -A}{5 \left (\tan \left (f x +e \right )+i\right )^{5}}+\frac {B}{4 \left (\tan \left (f x +e \right )+i\right )^{4}}\right )}{f \,c^{5}} \]
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 = 8.60, size = 82, normalized size = 1.49 \[ \frac {\frac {a\,\left (4\,A+B\,1{}\mathrm {i}\right )}{20}+\frac {B\,a\,\mathrm {tan}\left (e+f\,x\right )}{4}}{c^5\,f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^5+{\mathrm {tan}\left (e+f\,x\right )}^4\,5{}\mathrm {i}-10\,{\mathrm {tan}\left (e+f\,x\right )}^3-{\mathrm {tan}\left (e+f\,x\right )}^2\,10{}\mathrm {i}+5\,\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.90, size = 348, normalized size = 6.33 \[ \begin {cases} - \frac {10485760 i A a c^{20} f^{4} e^{6 i e} e^{6 i f x} + \left (5242880 i A a c^{20} f^{4} e^{2 i e} - 5242880 B a c^{20} f^{4} e^{2 i e}\right ) e^{2 i f x} + \left (10485760 i A a c^{20} f^{4} e^{4 i e} - 5242880 B a c^{20} f^{4} e^{4 i e}\right ) e^{4 i f x} + \left (5242880 i A a c^{20} f^{4} e^{8 i e} + 2621440 B a c^{20} f^{4} e^{8 i e}\right ) e^{8 i f x} + \left (1048576 i A a c^{20} f^{4} e^{10 i e} + 1048576 B a c^{20} f^{4} e^{10 i e}\right ) e^{10 i f x}}{167772160 c^{25} f^{5}} & \text {for}\: 167772160 c^{25} f^{5} \neq 0 \\\frac {x \left (A a e^{10 i e} + 4 A a e^{8 i e} + 6 A a e^{6 i e} + 4 A a e^{4 i e} + A a e^{2 i e} - i B a e^{10 i e} - 2 i B a e^{8 i e} + 2 i B a e^{4 i e} + i B a e^{2 i e}\right )}{16 c^{5}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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