Optimal. Leaf size=62 \[ -\frac {i a^2}{2 f (c-i c \tan (e+f x))^4}+\frac {i a^2 c^2}{3 f \left (c^2-i c^2 \tan (e+f x)\right )^3} \]
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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 a^2 c^2}{3 f \left (c^2-i c^2 \tan (e+f x)\right )^3}-\frac {i a^2}{2 f (c-i c \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 {(a+i a \tan (e+f x))^2}{(c-i c \tan (e+f x))^4} \, dx &=\left (a^2 c^2\right ) \int \frac {\sec ^4(e+f x)}{(c-i c \tan (e+f x))^6} \, dx\\ &=\frac {\left (i a^2\right ) \text {Subst}\left (\int \frac {c-x}{(c+x)^5} \, dx,x,-i c \tan (e+f x)\right )}{c f}\\ &=\frac {\left (i a^2\right ) \text {Subst}\left (\int \left (\frac {2 c}{(c+x)^5}-\frac {1}{(c+x)^4}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c f}\\ &=-\frac {i a^2}{2 f (c-i c \tan (e+f x))^4}+\frac {i a^2}{3 c f (c-i c \tan (e+f x))^3}\\ \end {align*}
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Mathematica [A]
time = 0.75, size = 75, normalized size = 1.21 \begin {gather*} \frac {a^2 (8+9 \cos (2 (e+f x))-3 i \sin (2 (e+f x))) (-i \cos (6 e+8 f x)+\sin (6 e+8 f x))}{96 c^4 f (\cos (f x)+i \sin (f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 39, normalized size = 0.63
method | result | size |
derivativedivides | \(\frac {a^{2} \left (\frac {1}{3 \left (\tan \left (f x +e \right )+i\right )^{3}}-\frac {i}{2 \left (\tan \left (f x +e \right )+i\right )^{4}}\right )}{f \,c^{4}}\) | \(39\) |
default | \(\frac {a^{2} \left (\frac {1}{3 \left (\tan \left (f x +e \right )+i\right )^{3}}-\frac {i}{2 \left (\tan \left (f x +e \right )+i\right )^{4}}\right )}{f \,c^{4}}\) | \(39\) |
risch | \(-\frac {i a^{2} {\mathrm e}^{8 i \left (f x +e \right )}}{32 c^{4} f}-\frac {i a^{2} {\mathrm e}^{6 i \left (f x +e \right )}}{12 c^{4} f}-\frac {i a^{2} {\mathrm e}^{4 i \left (f x +e \right )}}{16 c^{4} f}\) | \(65\) |
norman | \(\frac {\frac {a^{2} \tan \left (f x +e \right )}{c f}-\frac {i a^{2}}{6 c f}-\frac {8 a^{2} \left (\tan ^{3}\left (f x +e \right )\right )}{3 c f}+\frac {a^{2} \left (\tan ^{5}\left (f x +e \right )\right )}{3 c f}-\frac {3 i a^{2} \left (\tan ^{4}\left (f x +e \right )\right )}{2 c f}+\frac {7 i a^{2} \left (\tan ^{2}\left (f x +e \right )\right )}{3 c f}}{\left (1+\tan ^{2}\left (f x +e \right )\right )^{4} c^{3}}\) | \(124\) |
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 = 1.40, size = 54, normalized size = 0.87 \begin {gather*} \frac {-3 i \, a^{2} e^{\left (8 i \, f x + 8 i \, e\right )} - 8 i \, a^{2} e^{\left (6 i \, f x + 6 i \, e\right )} - 6 i \, a^{2} e^{\left (4 i \, f x + 4 i \, e\right )}}{96 \, 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. 139 vs. \(2 (49) = 98\).
time = 0.23, size = 139, normalized size = 2.24 \begin {gather*} \begin {cases} \frac {- 192 i a^{2} c^{8} f^{2} e^{8 i e} e^{8 i f x} - 512 i a^{2} c^{8} f^{2} e^{6 i e} e^{6 i f x} - 384 i a^{2} c^{8} f^{2} e^{4 i e} e^{4 i f x}}{6144 c^{12} f^{3}} & \text {for}\: c^{12} f^{3} \neq 0 \\\frac {x \left (a^{2} e^{8 i e} + 2 a^{2} e^{6 i e} + a^{2} e^{4 i e}\right )}{4 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. 140 vs. \(2 (52) = 104\).
time = 0.78, size = 140, normalized size = 2.26 \begin {gather*} -\frac {2 \, {\left (3 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 6 i \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 17 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 16 i \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 17 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 i \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{3 \, 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 = 4.71, size = 64, normalized size = 1.03 \begin {gather*} \frac {a^2\,\left (2\,\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )}{6\,c^4\,f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^4+{\mathrm {tan}\left (e+f\,x\right )}^3\,4{}\mathrm {i}-6\,{\mathrm {tan}\left (e+f\,x\right )}^2-\mathrm {tan}\left (e+f\,x\right )\,4{}\mathrm {i}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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