Optimal. Leaf size=55 \[ \frac {2 i c^2}{f (a+i a \tan (e+f x))}-\frac {i c^2 \log (\cos (e+f x))}{a f}-\frac {c^2 x}{a} \]
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Rubi [A] time = 0.11, antiderivative size = 55, 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} \[ \frac {2 i c^2}{f (a+i a \tan (e+f x))}-\frac {i c^2 \log (\cos (e+f x))}{a f}-\frac {c^2 x}{a} \]
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)} \, dx &=\left (a^2 c^2\right ) \int \frac {\sec ^4(e+f x)}{(a+i a \tan (e+f x))^3} \, dx\\ &=-\frac {\left (i c^2\right ) \operatorname {Subst}\left (\int \frac {a-x}{(a+x)^2} \, dx,x,i a \tan (e+f x)\right )}{a f}\\ &=-\frac {\left (i c^2\right ) \operatorname {Subst}\left (\int \left (\frac {1}{-a-x}+\frac {2 a}{(a+x)^2}\right ) \, dx,x,i a \tan (e+f x)\right )}{a f}\\ &=-\frac {c^2 x}{a}-\frac {i c^2 \log (\cos (e+f x))}{a f}+\frac {2 i c^2}{f (a+i a \tan (e+f x))}\\ \end {align*}
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Mathematica [A] time = 1.32, size = 74, normalized size = 1.35 \[ -\frac {c^2 \left (2 \tan ^{-1}(\tan (f x)) (\tan (e+f x)-i)+\log \left (\cos ^2(e+f x)\right )+i \tan (e+f x) \left (\log \left (\cos ^2(e+f x)\right )+2\right )-2\right )}{2 a f (\tan (e+f x)-i)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 65, normalized size = 1.18 \[ -\frac {{\left (2 \, c^{2} f x e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - i \, c^{2}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{a f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.87, size = 125, normalized size = 2.27 \[ -\frac {\frac {i \, c^{2} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a} - \frac {2 i \, c^{2} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - i\right )}{a} + \frac {i \, c^{2} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}{a} + \frac {3 i \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 10 \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 i \, c^{2}}{a {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - i\right )}^{2}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 46, normalized size = 0.84 \[ \frac {2 c^{2}}{f a \left (\tan \left (f x +e \right )-i\right )}+\frac {i c^{2} \ln \left (\tan \left (f x +e \right )-i\right )}{f a} \]
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 = 48, normalized size = 0.87 \[ \frac {c^2\,2{}\mathrm {i}}{a\,f\,\left (1+\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}+\frac {c^2\,\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{a\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 100, normalized size = 1.82 \[ \begin {cases} \frac {i c^{2} e^{- 2 i e} e^{- 2 i f x}}{a f} & \text {for}\: a f e^{2 i e} \neq 0 \\x \left (\frac {2 c^{2}}{a} + \frac {\left (- 2 c^{2} e^{2 i e} + 2 c^{2}\right ) e^{- 2 i e}}{a}\right ) & \text {otherwise} \end {cases} - \frac {2 c^{2} x}{a} - \frac {i c^{2} \log {\left (e^{2 i f x} + e^{- 2 i e} \right )}}{a f} \]
Verification of antiderivative is not currently implemented for this CAS.
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