Optimal. Leaf size=50 \[ -\frac {i a^4 \left (c^2+i c^2 \tan (e+f x)\right )^4}{8 f \left (c^3-i c^3 \tan (e+f x)\right )^4} \]
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Rubi [A]
time = 0.07, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {3603, 3568, 37}
\begin {gather*} -\frac {i a^4 \left (c^2+i c^2 \tan (e+f x)\right )^4}{8 f \left (c^3-i c^3 \tan (e+f x)\right )^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 3568
Rule 3603
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x))^4}{(c-i c \tan (e+f x))^4} \, dx &=\left (a^4 c^4\right ) \int \frac {\sec ^8(e+f x)}{(c-i c \tan (e+f x))^8} \, dx\\ &=\frac {\left (i a^4\right ) \text {Subst}\left (\int \frac {(c-x)^3}{(c+x)^5} \, dx,x,-i c \tan (e+f x)\right )}{c^3 f}\\ &=-\frac {i a^4 (c+i c \tan (e+f x))^4}{8 f \left (c^2-i c^2 \tan (e+f x)\right )^4}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 34, normalized size = 0.68 \begin {gather*} \frac {a^4 (-i \cos (8 (e+f x))+\sin (8 (e+f x)))}{8 c^4 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 66, normalized size = 1.32
method | result | size |
risch | \(-\frac {i a^{4} {\mathrm e}^{8 i \left (f x +e \right )}}{8 c^{4} f}\) | \(22\) |
derivativedivides | \(\frac {a^{4} \left (-\frac {1}{\tan \left (f x +e \right )+i}+\frac {4}{\left (\tan \left (f x +e \right )+i\right )^{3}}+\frac {3 i}{\left (\tan \left (f x +e \right )+i\right )^{2}}-\frac {2 i}{\left (\tan \left (f x +e \right )+i\right )^{4}}\right )}{f \,c^{4}}\) | \(66\) |
default | \(\frac {a^{4} \left (-\frac {1}{\tan \left (f x +e \right )+i}+\frac {4}{\left (\tan \left (f x +e \right )+i\right )^{3}}+\frac {3 i}{\left (\tan \left (f x +e \right )+i\right )^{2}}-\frac {2 i}{\left (\tan \left (f x +e \right )+i\right )^{4}}\right )}{f \,c^{4}}\) | \(66\) |
norman | \(\frac {\frac {a^{4} \tan \left (f x +e \right )}{c f}-\frac {7 a^{4} \left (\tan ^{3}\left (f x +e \right )\right )}{c f}+\frac {7 a^{4} \left (\tan ^{5}\left (f x +e \right )\right )}{c f}-\frac {a^{4} \left (\tan ^{7}\left (f x +e \right )\right )}{c f}-\frac {8 i a^{4} \left (\tan ^{4}\left (f x +e \right )\right )}{c f}+\frac {4 i a^{4} \left (\tan ^{2}\left (f x +e \right )\right )}{c f}+\frac {4 i a^{4} \left (\tan ^{6}\left (f x +e \right )\right )}{c f}}{\left (1+\tan ^{2}\left (f x +e \right )\right )^{4} c^{3}}\) | \(151\) |
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 = 1.18, size = 21, normalized size = 0.42 \begin {gather*} -\frac {i \, a^{4} e^{\left (8 i \, f x + 8 i \, e\right )}}{8 \, c^{4} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.26, size = 46, normalized size = 0.92 \begin {gather*} \begin {cases} - \frac {i a^{4} e^{8 i e} e^{8 i f x}}{8 c^{4} f} & \text {for}\: c^{4} f \neq 0 \\\frac {a^{4} x e^{8 i e}}{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 = 0.98, size = 88, normalized size = 1.76 \begin {gather*} -\frac {2 \, {\left (a^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 7 \, a^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 7 \, a^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - a^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{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 = 4.72, size = 69, normalized size = 1.38 \begin {gather*} -\frac {a^4\,\mathrm {tan}\left (e+f\,x\right )\,\left ({\mathrm {tan}\left (e+f\,x\right )}^2-1\right )}{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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