Optimal. Leaf size=25 \[ -\frac {i a}{4 f (c-i c \tan (e+f x))^4} \]
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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}{4 f (c-i c \tan (e+f x))^4} \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))^4} \, dx &=(a c) \int \frac {\sec ^2(e+f x)}{(c-i c \tan (e+f x))^5} \, dx\\ &=\frac {(i a) \text {Subst}\left (\int \frac {1}{(c+x)^5} \, dx,x,-i c \tan (e+f x)\right )}{f}\\ &=-\frac {i a}{4 f (c-i c \tan (e+f x))^4}\\ \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. \(74\) vs. \(2(25)=50\).
time = 0.45, size = 74, normalized size = 2.96 \begin {gather*} \frac {a (10 \cos (e+f x)+5 \cos (3 (e+f x))-i (2 \sin (e+f x)+3 \sin (3 (e+f x)))) (-i \cos (5 (e+f x))+\sin (5 (e+f x)))}{64 c^4 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 22, normalized size = 0.88
method | result | size |
derivativedivides | \(-\frac {i a}{4 f \,c^{4} \left (\tan \left (f x +e \right )+i\right )^{4}}\) | \(22\) |
default | \(-\frac {i a}{4 f \,c^{4} \left (\tan \left (f x +e \right )+i\right )^{4}}\) | \(22\) |
risch | \(-\frac {i a \,{\mathrm e}^{8 i \left (f x +e \right )}}{64 c^{4} f}-\frac {i a \,{\mathrm e}^{6 i \left (f x +e \right )}}{16 c^{4} f}-\frac {3 i a \,{\mathrm e}^{4 i \left (f x +e \right )}}{32 c^{4} f}-\frac {i a \,{\mathrm e}^{2 i \left (f x +e \right )}}{16 c^{4} f}\) | \(78\) |
norman | \(\frac {\frac {a \tan \left (f x +e \right )}{c f}-\frac {i a}{4 c f}-\frac {a \left (\tan ^{3}\left (f x +e \right )\right )}{c f}-\frac {i a \left (\tan ^{4}\left (f x +e \right )\right )}{4 c f}+\frac {3 i a \left (\tan ^{2}\left (f x +e \right )\right )}{2 c f}}{\left (1+\tan ^{2}\left (f x +e \right )\right )^{4} c^{3}}\) | \(95\) |
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. 61 vs. \(2 (20) = 40\).
time = 1.69, size = 61, normalized size = 2.44 \begin {gather*} \frac {-i \, a e^{\left (8 i \, f x + 8 i \, e\right )} - 4 i \, a e^{\left (6 i \, f x + 6 i \, e\right )} - 6 i \, a e^{\left (4 i \, f x + 4 i \, e\right )} - 4 i \, a e^{\left (2 i \, f x + 2 i \, e\right )}}{64 \, c^{4} 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. 167 vs. \(2 (20) = 40\).
time = 0.24, size = 167, normalized size = 6.68 \begin {gather*} \begin {cases} \frac {- 8192 i a c^{12} f^{3} e^{8 i e} e^{8 i f x} - 32768 i a c^{12} f^{3} e^{6 i e} e^{6 i f x} - 49152 i a c^{12} f^{3} e^{4 i e} e^{4 i f x} - 32768 i a c^{12} f^{3} e^{2 i e} e^{2 i f x}}{524288 c^{16} f^{4}} & \text {for}\: c^{16} f^{4} \neq 0 \\\frac {x \left (a e^{8 i e} + 3 a e^{6 i e} + 3 a e^{4 i e} + a e^{2 i e}\right )}{8 c^{4}} & \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. 125 vs. \(2 (20) = 40\).
time = 0.72, size = 125, normalized size = 5.00 \begin {gather*} -\frac {2 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 3 i \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 7 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 8 i \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 7 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3 i \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a \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 = 4.65, size = 21, normalized size = 0.84 \begin {gather*} -\frac {a\,1{}\mathrm {i}}{4\,c^4\,f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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