Optimal. Leaf size=62 \[ \frac {i c^2}{2 f (a+i a \tan (e+f x))^4}-\frac {i a^2 c^2}{3 f \left (a^2+i a^2 \tan (e+f x)\right )^3} \]
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Rubi [A]
time = 0.08, antiderivative size = 62, 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^2}{2 f (a+i a \tan (e+f x))^4}-\frac {i a^2 c^2}{3 f \left (a^2+i a^2 \tan (e+f x)\right )^3} \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))^2}{(a+i a \tan (e+f x))^4} \, dx &=\left (a^2 c^2\right ) \int \frac {\sec ^4(e+f x)}{(a+i a \tan (e+f x))^6} \, dx\\ &=-\frac {\left (i c^2\right ) \text {Subst}\left (\int \frac {a-x}{(a+x)^5} \, dx,x,i a \tan (e+f x)\right )}{a f}\\ &=-\frac {\left (i c^2\right ) \text {Subst}\left (\int \left (\frac {2 a}{(a+x)^5}-\frac {1}{(a+x)^4}\right ) \, dx,x,i a \tan (e+f x)\right )}{a f}\\ &=\frac {i c^2}{2 f (a+i a \tan (e+f x))^4}-\frac {i c^2}{3 a f (a+i a \tan (e+f x))^3}\\ \end {align*}
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Mathematica [A]
time = 1.07, size = 58, normalized size = 0.94 \begin {gather*} \frac {c^2 (8+9 \cos (2 (e+f x))+3 i \sin (2 (e+f x))) (i \cos (6 (e+f x))+\sin (6 (e+f x)))}{96 a^4 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 39, normalized size = 0.63
method | result | size |
derivativedivides | \(\frac {c^{2} \left (\frac {i}{2 \left (\tan \left (f x +e \right )-i\right )^{4}}+\frac {1}{3 \left (\tan \left (f x +e \right )-i\right )^{3}}\right )}{f \,a^{4}}\) | \(39\) |
default | \(\frac {c^{2} \left (\frac {i}{2 \left (\tan \left (f x +e \right )-i\right )^{4}}+\frac {1}{3 \left (\tan \left (f x +e \right )-i\right )^{3}}\right )}{f \,a^{4}}\) | \(39\) |
risch | \(\frac {i c^{2} {\mathrm e}^{-4 i \left (f x +e \right )}}{16 a^{4} f}+\frac {i c^{2} {\mathrm e}^{-6 i \left (f x +e \right )}}{12 a^{4} f}+\frac {i c^{2} {\mathrm e}^{-8 i \left (f x +e \right )}}{32 a^{4} f}\) | \(65\) |
norman | \(\frac {\frac {c^{2} \tan \left (f x +e \right )}{a f}+\frac {i c^{2}}{6 a f}-\frac {8 c^{2} \left (\tan ^{3}\left (f x +e \right )\right )}{3 a f}+\frac {c^{2} \left (\tan ^{5}\left (f x +e \right )\right )}{3 a f}-\frac {7 i c^{2} \left (\tan ^{2}\left (f x +e \right )\right )}{3 a f}+\frac {3 i c^{2} \left (\tan ^{4}\left (f x +e \right )\right )}{2 a f}}{\left (1+\tan ^{2}\left (f x +e \right )\right )^{4} a^{3}}\) | \(124\) |
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 = 1.27, size = 54, normalized size = 0.87 \begin {gather*} \frac {{\left (6 i \, c^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 8 i \, c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 3 i \, c^{2}\right )} e^{\left (-8 i \, f x - 8 i \, e\right )}}{96 \, a^{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. 151 vs. \(2 (49) = 98\).
time = 0.26, size = 151, normalized size = 2.44 \begin {gather*} \begin {cases} \frac {\left (384 i a^{8} c^{2} f^{2} e^{14 i e} e^{- 4 i f x} + 512 i a^{8} c^{2} f^{2} e^{12 i e} e^{- 6 i f x} + 192 i a^{8} c^{2} f^{2} e^{10 i e} e^{- 8 i f x}\right ) e^{- 18 i e}}{6144 a^{12} f^{3}} & \text {for}\: a^{12} f^{3} e^{18 i e} \neq 0 \\\frac {x \left (c^{2} e^{4 i e} + 2 c^{2} e^{2 i e} + c^{2}\right ) e^{- 8 i e}}{4 a^{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. 140 vs. \(2 (52) = 104\).
time = 0.81, size = 140, normalized size = 2.26 \begin {gather*} -\frac {2 \, {\left (3 \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 6 i \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 17 \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 16 i \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 17 \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 6 i \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, c^{2} \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 = 4.72, size = 67, normalized size = 1.08 \begin {gather*} \frac {c^2\,\left (-1+\mathrm {tan}\left (e+f\,x\right )\,2{}\mathrm {i}\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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