Optimal. Leaf size=87 \[ \frac {i c^3}{f (a+i a \tan (e+f x))^4}-\frac {4 i c^3}{3 a f (a+i a \tan (e+f x))^3}+\frac {i c^3}{2 f \left (a^2+i a^2 \tan (e+f x)\right )^2} \]
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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 c^3}{2 f \left (a^2+i a^2 \tan (e+f x)\right )^2}-\frac {4 i c^3}{3 a f (a+i a \tan (e+f x))^3}+\frac {i c^3}{f (a+i a \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 {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^4} \, dx &=\left (a^3 c^3\right ) \int \frac {\sec ^6(e+f x)}{(a+i a \tan (e+f x))^7} \, dx\\ &=-\frac {\left (i c^3\right ) \text {Subst}\left (\int \frac {(a-x)^2}{(a+x)^5} \, dx,x,i a \tan (e+f x)\right )}{a^2 f}\\ &=-\frac {\left (i c^3\right ) \text {Subst}\left (\int \left (\frac {4 a^2}{(a+x)^5}-\frac {4 a}{(a+x)^4}+\frac {1}{(a+x)^3}\right ) \, dx,x,i a \tan (e+f x)\right )}{a^2 f}\\ &=\frac {i c^3}{f (a+i a \tan (e+f x))^4}-\frac {4 i c^3}{3 a f (a+i a \tan (e+f x))^3}+\frac {i c^3}{2 f \left (a^2+i a^2 \tan (e+f x)\right )^2}\\ \end {align*}
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Mathematica [A]
time = 1.16, size = 64, normalized size = 0.74 \begin {gather*} -\frac {c^3 \sec ^3(e+f x) (\cos (3 (e+f x))-i \sin (3 (e+f x))) (-7 i+\tan (e+f x))}{48 a^4 f (-i+\tan (e+f x))^4} \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 | \(\frac {i c^{3} {\mathrm e}^{-6 i \left (f x +e \right )}}{12 a^{4} f}+\frac {i c^{3} {\mathrm e}^{-8 i \left (f x +e \right )}}{16 a^{4} f}\) | \(44\) |
derivativedivides | \(\frac {c^{3} \left (-\frac {i}{2 \left (\tan \left (f x +e \right )-i\right )^{2}}+\frac {i}{\left (\tan \left (f x +e \right )-i\right )^{4}}+\frac {4}{3 \left (\tan \left (f x +e \right )-i\right )^{3}}\right )}{f \,a^{4}}\) | \(53\) |
default | \(\frac {c^{3} \left (-\frac {i}{2 \left (\tan \left (f x +e \right )-i\right )^{2}}+\frac {i}{\left (\tan \left (f x +e \right )-i\right )^{4}}+\frac {4}{3 \left (\tan \left (f x +e \right )-i\right )^{3}}\right )}{f \,a^{4}}\) | \(53\) |
norman | \(\frac {\frac {c^{3} \tan \left (f x +e \right )}{a f}+\frac {i c^{3}}{6 a f}-\frac {14 c^{3} \left (\tan ^{3}\left (f x +e \right )\right )}{3 a f}+\frac {7 c^{3} \left (\tan ^{5}\left (f x +e \right )\right )}{3 a f}-\frac {17 i c^{3} \left (\tan ^{2}\left (f x +e \right )\right )}{6 a f}-\frac {i c^{3} \left (\tan ^{6}\left (f x +e \right )\right )}{2 a f}+\frac {9 i c^{3} \left (\tan ^{4}\left (f x +e \right )\right )}{2 a f}}{\left (1+\tan ^{2}\left (f x +e \right )\right )^{4} a^{3}}\) | \(144\) |
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 = 1.16, size = 39, normalized size = 0.45 \begin {gather*} \frac {{\left (4 i \, c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 3 i \, c^{3}\right )} e^{\left (-8 i \, f x - 8 i \, e\right )}}{48 \, a^{4} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.28, size = 107, normalized size = 1.23 \begin {gather*} \begin {cases} \frac {\left (16 i a^{4} c^{3} f e^{8 i e} e^{- 6 i f x} + 12 i a^{4} c^{3} f e^{6 i e} e^{- 8 i f x}\right ) e^{- 14 i e}}{192 a^{8} f^{2}} & \text {for}\: a^{8} f^{2} e^{14 i e} \neq 0 \\\frac {x \left (c^{3} e^{2 i e} + c^{3}\right ) e^{- 8 i e}}{2 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 = 0.88, size = 140, normalized size = 1.61 \begin {gather*} -\frac {2 \, {\left (3 \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 3 i \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 17 \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 10 i \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 17 \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 i \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{3 \, 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 = 4.85, size = 77, normalized size = 0.89 \begin {gather*} \frac {c^3\,\left (3\,{\mathrm {tan}\left (e+f\,x\right )}^2+\mathrm {tan}\left (e+f\,x\right )\,2{}\mathrm {i}-1\right )}{6\,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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