Optimal. Leaf size=87 \[ -\frac {i a^3}{2 f \left (c^2-i c^2 \tan (e+f x)\right )^2}+\frac {4 i a^3}{3 c f (c-i c \tan (e+f x))^3}-\frac {i a^3}{f (c-i c \tan (e+f x))^4} \]
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Rubi [A] time = 0.12, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {3522, 3487, 43} \[ -\frac {i a^3}{2 f \left (c^2-i c^2 \tan (e+f x)\right )^2}+\frac {4 i a^3}{3 c f (c-i c \tan (e+f x))^3}-\frac {i a^3}{f (c-i c \tan (e+f x))^4} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3487
Rule 3522
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x))^3}{(c-i c \tan (e+f x))^4} \, dx &=\left (a^3 c^3\right ) \int \frac {\sec ^6(e+f x)}{(c-i c \tan (e+f x))^7} \, dx\\ &=\frac {\left (i a^3\right ) \operatorname {Subst}\left (\int \frac {(c-x)^2}{(c+x)^5} \, dx,x,-i c \tan (e+f x)\right )}{c^2 f}\\ &=\frac {\left (i a^3\right ) \operatorname {Subst}\left (\int \left (\frac {4 c^2}{(c+x)^5}-\frac {4 c}{(c+x)^4}+\frac {1}{(c+x)^3}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c^2 f}\\ &=-\frac {i a^3}{f (c-i c \tan (e+f x))^4}+\frac {4 i a^3}{3 c f (c-i c \tan (e+f x))^3}-\frac {i a^3}{2 f \left (c^2-i c^2 \tan (e+f x)\right )^2}\\ \end {align*}
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Mathematica [A] time = 2.28, size = 53, normalized size = 0.61 \[ \frac {a^3 (7 \cos (e+f x)-i \sin (e+f x)) (\sin (7 (e+f x))-i \cos (7 (e+f x)))}{48 c^4 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 37, normalized size = 0.43 \[ \frac {-3 i \, a^{3} e^{\left (8 i \, f x + 8 i \, e\right )} - 4 i \, a^{3} e^{\left (6 i \, f x + 6 i \, e\right )}}{48 \, c^{4} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.17, size = 140, normalized size = 1.61 \[ -\frac {2 \, {\left (3 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 3 i \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 17 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 10 i \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 17 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3 i \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{3 \, c^{4} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 53, normalized size = 0.61 \[ \frac {a^{3} \left (\frac {4}{3 \left (\tan \left (f x +e \right )+i\right )^{3}}-\frac {i}{\left (\tan \left (f x +e \right )+i\right )^{4}}+\frac {i}{2 \left (\tan \left (f x +e \right )+i\right )^{2}}\right )}{f \,c^{4}} \]
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 = 4.77, size = 75, normalized size = 0.86 \[ \frac {a^3\,\left ({\mathrm {tan}\left (e+f\,x\right )}^2\,3{}\mathrm {i}+2\,\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )}{6\,c^4\,f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^4+{\mathrm {tan}\left (e+f\,x\right )}^3\,4{}\mathrm {i}-6\,{\mathrm {tan}\left (e+f\,x\right )}^2-\mathrm {tan}\left (e+f\,x\right )\,4{}\mathrm {i}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 97, normalized size = 1.11 \[ \begin {cases} \frac {- 12 i a^{3} c^{4} f e^{8 i e} e^{8 i f x} - 16 i a^{3} c^{4} f e^{6 i e} e^{6 i f x}}{192 c^{8} f^{2}} & \text {for}\: 192 c^{8} f^{2} \neq 0 \\\frac {x \left (a^{3} e^{8 i e} + a^{3} e^{6 i e}\right )}{2 c^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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