Optimal. Leaf size=25 \[ -\frac {i a}{3 f (c-i c \tan (e+f x))^3} \]
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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 a}{3 f (c-i c \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 {a+i a \tan (e+f x)}{(c-i c \tan (e+f x))^3} \, dx &=(a c) \int \frac {\sec ^2(e+f x)}{(c-i c \tan (e+f x))^4} \, dx\\ &=\frac {(i a) \text {Subst}\left (\int \frac {1}{(c+x)^4} \, dx,x,-i c \tan (e+f x)\right )}{f}\\ &=-\frac {i a}{3 f (c-i c \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 = 0.36, size = 56, normalized size = 2.24 \begin {gather*} \frac {a (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 c^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 | \(-\frac {a}{3 f \,c^{3} \left (\tan \left (f x +e \right )+i\right )^{3}}\) | \(21\) |
default | \(-\frac {a}{3 f \,c^{3} \left (\tan \left (f x +e \right )+i\right )^{3}}\) | \(21\) |
risch | \(-\frac {i a \,{\mathrm e}^{6 i \left (f x +e \right )}}{24 c^{3} f}-\frac {i a \,{\mathrm e}^{4 i \left (f x +e \right )}}{8 c^{3} f}-\frac {i a \,{\mathrm e}^{2 i \left (f x +e \right )}}{8 c^{3} f}\) | \(59\) |
norman | \(\frac {\frac {a \tan \left (f x +e \right )}{c f}+\frac {i a \left (\tan ^{2}\left (f x +e \right )\right )}{c f}-\frac {i a}{3 c f}-\frac {a \left (\tan ^{3}\left (f x +e \right )\right )}{3 c f}}{c^{2} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}\) | \(77\) |
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 = 1.03, size = 48, normalized size = 1.92 \begin {gather*} \frac {-i \, a e^{\left (6 i \, f x + 6 i \, e\right )} - 3 i \, a e^{\left (4 i \, f x + 4 i \, e\right )} - 3 i \, a e^{\left (2 i \, f x + 2 i \, e\right )}}{24 \, c^{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. 129 vs. \(2 (20) = 40\).
time = 0.18, size = 129, normalized size = 5.16 \begin {gather*} \begin {cases} \frac {- 64 i a c^{6} f^{2} e^{6 i e} e^{6 i f x} - 192 i a c^{6} f^{2} e^{4 i e} e^{4 i f x} - 192 i a c^{6} f^{2} e^{2 i e} e^{2 i f x}}{1536 c^{9} f^{3}} & \text {for}\: c^{9} f^{3} \neq 0 \\\frac {x \left (a e^{6 i e} + 2 a e^{4 i e} + a e^{2 i e}\right )}{4 c^{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 = 0.68, size = 96, normalized size = 3.84 \begin {gather*} -\frac {2 \, {\left (3 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 6 i \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 10 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 6 i \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{3 \, c^{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 = 4.64, size = 20, normalized size = 0.80 \begin {gather*} -\frac {a}{3\,c^3\,f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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