Optimal. Leaf size=58 \[ \frac {2 i c^2}{3 f (a+i a \tan (e+f x))^3}-\frac {i c^2}{2 a f (a+i a \tan (e+f x))^2} \]
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Rubi [A] time = 0.11, antiderivative size = 58, 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 = {3522, 3487, 43} \[ \frac {2 i c^2}{3 f (a+i a \tan (e+f x))^3}-\frac {i c^2}{2 a f (a+i a \tan (e+f x))^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3487
Rule 3522
Rubi steps
\begin {align*} \int \frac {(c-i c \tan (e+f x))^2}{(a+i a \tan (e+f x))^3} \, dx &=\left (a^2 c^2\right ) \int \frac {\sec ^4(e+f x)}{(a+i a \tan (e+f x))^5} \, dx\\ &=-\frac {\left (i c^2\right ) \operatorname {Subst}\left (\int \frac {a-x}{(a+x)^4} \, dx,x,i a \tan (e+f x)\right )}{a f}\\ &=-\frac {\left (i c^2\right ) \operatorname {Subst}\left (\int \left (\frac {2 a}{(a+x)^4}-\frac {1}{(a+x)^3}\right ) \, dx,x,i a \tan (e+f x)\right )}{a f}\\ &=\frac {2 i c^2}{3 f (a+i a \tan (e+f x))^3}-\frac {i c^2}{2 a f (a+i a \tan (e+f x))^2}\\ \end {align*}
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Mathematica [A] time = 1.78, size = 53, normalized size = 0.91 \[ \frac {c^2 (5 \cos (e+f x)+i \sin (e+f x)) (\sin (5 (e+f x))+i \cos (5 (e+f x)))}{24 a^3 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 37, normalized size = 0.64 \[ \frac {{\left (3 i \, c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 2 i \, c^{2}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{24 \, a^{3} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.33, size = 106, normalized size = 1.83 \[ -\frac {2 \, {\left (3 \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 3 i \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 8 \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3 i \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{3 \, a^{3} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - i\right )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 39, normalized size = 0.67 \[ \frac {c^{2} \left (-\frac {2}{3 \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {i}{2 \left (\tan \left (f x +e \right )-i\right )^{2}}\right )}{f \,a^{3}} \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.62, size = 56, normalized size = 0.97 \[ \frac {c^2\,\left (3\,\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )}{6\,a^3\,f\,\left (-{\mathrm {tan}\left (e+f\,x\right )}^3\,1{}\mathrm {i}-3\,{\mathrm {tan}\left (e+f\,x\right )}^2+\mathrm {tan}\left (e+f\,x\right )\,3{}\mathrm {i}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 109, normalized size = 1.88 \[ \begin {cases} \frac {\left (12 i a^{3} c^{2} f e^{6 i e} e^{- 4 i f x} + 8 i a^{3} c^{2} f e^{4 i e} e^{- 6 i f x}\right ) e^{- 10 i e}}{96 a^{6} f^{2}} & \text {for}\: 96 a^{6} f^{2} e^{10 i e} \neq 0 \\\frac {x \left (c^{2} e^{2 i e} + c^{2}\right ) e^{- 6 i e}}{2 a^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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