Optimal. Leaf size=25 \[ \frac {i c}{3 f (a+i a \tan (e+f x))^3} \]
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Rubi [A]
time = 0.05, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3603, 3568, 32}
\begin {gather*} \frac {i c}{3 f (a+i a \tan (e+f x))^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 3568
Rule 3603
Rubi steps
\begin {align*} \int \frac {c-i c \tan (e+f x)}{(a+i a \tan (e+f x))^3} \, dx &=(a c) \int \frac {\sec ^2(e+f x)}{(a+i a \tan (e+f x))^4} \, dx\\ &=-\frac {(i c) \text {Subst}\left (\int \frac {1}{(a+x)^4} \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=\frac {i c}{3 f (a+i a \tan (e+f x))^3}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(56\) vs. \(2(25)=50\).
time = 0.47, size = 56, normalized size = 2.24 \begin {gather*} \frac {c (3+4 \cos (2 (e+f x))+2 i \sin (2 (e+f x))) (i \cos (4 (e+f x))+\sin (4 (e+f x)))}{24 a^3 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 21, normalized size = 0.84
method | result | size |
derivativedivides | \(-\frac {c}{3 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )^{3}}\) | \(21\) |
default | \(-\frac {c}{3 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )^{3}}\) | \(21\) |
risch | \(\frac {i c \,{\mathrm e}^{-2 i \left (f x +e \right )}}{8 a^{3} f}+\frac {i c \,{\mathrm e}^{-4 i \left (f x +e \right )}}{8 a^{3} f}+\frac {i c \,{\mathrm e}^{-6 i \left (f x +e \right )}}{24 a^{3} f}\) | \(59\) |
norman | \(\frac {\frac {c \tan \left (f x +e \right )}{a f}-\frac {i c \left (\tan ^{2}\left (f x +e \right )\right )}{a f}+\frac {i c}{3 a f}-\frac {c \left (\tan ^{3}\left (f x +e \right )\right )}{3 a f}}{a^{2} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}\) | \(77\) |
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 [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 48 vs. \(2 (20) = 40\).
time = 1.22, size = 48, normalized size = 1.92 \begin {gather*} \frac {{\left (3 i \, c e^{\left (4 i \, f x + 4 i \, e\right )} + 3 i \, c e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{24 \, a^{3} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 141 vs. \(2 (19) = 38\).
time = 0.19, size = 141, normalized size = 5.64 \begin {gather*} \begin {cases} \frac {\left (192 i a^{6} c f^{2} e^{10 i e} e^{- 2 i f x} + 192 i a^{6} c f^{2} e^{8 i e} e^{- 4 i f x} + 64 i a^{6} c f^{2} e^{6 i e} e^{- 6 i f x}\right ) e^{- 12 i e}}{1536 a^{9} f^{3}} & \text {for}\: a^{9} f^{3} e^{12 i e} \neq 0 \\\frac {x \left (c e^{4 i e} + 2 c e^{2 i e} + c\right ) e^{- 6 i e}}{4 a^{3}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 96 vs. \(2 (20) = 40\).
time = 0.63, size = 96, normalized size = 3.84 \begin {gather*} -\frac {2 \, {\left (3 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 6 i \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 10 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 i \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{3 \, a^{3} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - i\right )}^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.79, size = 20, normalized size = 0.80 \begin {gather*} -\frac {c}{3\,a^3\,f\,{\left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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