Optimal. Leaf size=87 \[ -\frac {i a^3}{f (c-i c \tan (e+f x))^4}+\frac {4 i a^3}{3 c f (c-i c \tan (e+f x))^3}-\frac {i a^3}{2 f \left (c^2-i c^2 \tan (e+f x)\right )^2} \]
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Rubi [A]
time = 0.09, 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 = {3603, 3568, 45}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 3568
Rule 3603
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 ) \text {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 ) \text {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 = 0.97, size = 53, normalized size = 0.61 \begin {gather*} \frac {a^3 (7 \cos (e+f x)-i \sin (e+f x)) (-i \cos (7 (e+f x))+\sin (7 (e+f x)))}{48 c^4 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 53, normalized size = 0.61
method | result | size |
risch | \(-\frac {i a^{3} {\mathrm e}^{8 i \left (f x +e \right )}}{16 c^{4} f}-\frac {i a^{3} {\mathrm e}^{6 i \left (f x +e \right )}}{12 c^{4} f}\) | \(44\) |
derivativedivides | \(\frac {a^{3} \left (\frac {i}{2 \left (\tan \left (f x +e \right )+i\right )^{2}}-\frac {i}{\left (\tan \left (f x +e \right )+i\right )^{4}}+\frac {4}{3 \left (\tan \left (f x +e \right )+i\right )^{3}}\right )}{f \,c^{4}}\) | \(53\) |
default | \(\frac {a^{3} \left (\frac {i}{2 \left (\tan \left (f x +e \right )+i\right )^{2}}-\frac {i}{\left (\tan \left (f x +e \right )+i\right )^{4}}+\frac {4}{3 \left (\tan \left (f x +e \right )+i\right )^{3}}\right )}{f \,c^{4}}\) | \(53\) |
norman | \(\frac {\frac {a^{3} \tan \left (f x +e \right )}{c f}-\frac {i a^{3}}{6 c f}-\frac {14 a^{3} \left (\tan ^{3}\left (f x +e \right )\right )}{3 c f}+\frac {7 a^{3} \left (\tan ^{5}\left (f x +e \right )\right )}{3 c f}+\frac {i a^{3} \left (\tan ^{6}\left (f x +e \right )\right )}{2 c f}+\frac {17 i a^{3} \left (\tan ^{2}\left (f x +e \right )\right )}{6 c f}-\frac {9 i a^{3} \left (\tan ^{4}\left (f x +e \right )\right )}{2 c f}}{\left (1+\tan ^{2}\left (f x +e \right )\right )^{4} c^{3}}\) | \(144\) |
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.40, size = 39, normalized size = 0.45 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.27, size = 95, normalized size = 1.09 \begin {gather*} \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}\: 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.88, size = 140, normalized size = 1.61 \begin {gather*} -\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}} \end {gather*}
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 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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