Optimal. Leaf size=50 \[ \frac {i c^4 \left (a^2-i a^2 \tan (e+f x)\right )^4}{8 f \left (a^3+i a^3 \tan (e+f x)\right )^4} \]
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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 c^4 \left (a^2-i a^2 \tan (e+f x)\right )^4}{8 f \left (a^3+i a^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 {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^4} \, dx &=\left (a^4 c^4\right ) \int \frac {\sec ^8(e+f x)}{(a+i a \tan (e+f x))^8} \, dx\\ &=-\frac {\left (i c^4\right ) \text {Subst}\left (\int \frac {(a-x)^3}{(a+x)^5} \, dx,x,i a \tan (e+f x)\right )}{a^3 f}\\ &=\frac {i c^4 (1-i \tan (e+f x))^4}{8 f (a+i a \tan (e+f x))^4}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 34, normalized size = 0.68 \begin {gather*} \frac {c^4 (i \cos (8 (e+f x))+\sin (8 (e+f x)))}{8 a^4 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 66, normalized size = 1.32
method | result | size |
risch | \(\frac {i c^{4} {\mathrm e}^{-8 i \left (f x +e \right )}}{8 a^{4} f}\) | \(22\) |
derivativedivides | \(\frac {c^{4} \left (-\frac {1}{\tan \left (f x +e \right )-i}+\frac {2 i}{\left (\tan \left (f x +e \right )-i\right )^{4}}-\frac {3 i}{\left (\tan \left (f x +e \right )-i\right )^{2}}+\frac {4}{\left (\tan \left (f x +e \right )-i\right )^{3}}\right )}{f \,a^{4}}\) | \(66\) |
default | \(\frac {c^{4} \left (-\frac {1}{\tan \left (f x +e \right )-i}+\frac {2 i}{\left (\tan \left (f x +e \right )-i\right )^{4}}-\frac {3 i}{\left (\tan \left (f x +e \right )-i\right )^{2}}+\frac {4}{\left (\tan \left (f x +e \right )-i\right )^{3}}\right )}{f \,a^{4}}\) | \(66\) |
norman | \(\frac {\frac {c^{4} \tan \left (f x +e \right )}{a f}-\frac {7 c^{4} \left (\tan ^{3}\left (f x +e \right )\right )}{a f}+\frac {7 c^{4} \left (\tan ^{5}\left (f x +e \right )\right )}{a f}-\frac {c^{4} \left (\tan ^{7}\left (f x +e \right )\right )}{a f}-\frac {4 i c^{4} \left (\tan ^{2}\left (f x +e \right )\right )}{a f}-\frac {4 i c^{4} \left (\tan ^{6}\left (f x +e \right )\right )}{a f}+\frac {8 i c^{4} \left (\tan ^{4}\left (f x +e \right )\right )}{a f}}{\left (1+\tan ^{2}\left (f x +e \right )\right )^{4} a^{3}}\) | \(151\) |
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 = 0.76, size = 21, normalized size = 0.42 \begin {gather*} \frac {i \, c^{4} e^{\left (-8 i \, f x - 8 i \, e\right )}}{8 \, a^{4} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.26, size = 51, normalized size = 1.02 \begin {gather*} \begin {cases} \frac {i c^{4} e^{- 8 i e} e^{- 8 i f x}}{8 a^{4} f} & \text {for}\: a^{4} f e^{8 i e} \neq 0 \\\frac {c^{4} x e^{- 8 i e}}{a^{4}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.96, size = 88, normalized size = 1.76 \begin {gather*} -\frac {2 \, {\left (c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 7 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 7 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{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 = 76, normalized size = 1.52 \begin {gather*} -\frac {c^4\,\mathrm {tan}\left (e+f\,x\right )\,\left ({\mathrm {tan}\left (e+f\,x\right )}^2\,1{}\mathrm {i}-\mathrm {i}\right )}{a^4\,f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^4\,1{}\mathrm {i}+4\,{\mathrm {tan}\left (e+f\,x\right )}^3-{\mathrm {tan}\left (e+f\,x\right )}^2\,6{}\mathrm {i}-4\,\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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