Optimal. Leaf size=75 \[ \frac {x \left (c^2-2 i c d+d^2\right )}{2 a}+\frac {i (c+i d)^2}{2 f (a+i a \tan (e+f x))}+\frac {i d^2 \log (\cos (e+f x))}{a f} \]
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Rubi [A] time = 0.08, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3540, 3475} \[ \frac {x \left (c^2-2 i c d+d^2\right )}{2 a}+\frac {i (c+i d)^2}{2 f (a+i a \tan (e+f x))}+\frac {i d^2 \log (\cos (e+f x))}{a f} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3540
Rubi steps
\begin {align*} \int \frac {(c+d \tan (e+f x))^2}{a+i a \tan (e+f x)} \, dx &=\frac {i (c+i d)^2}{2 f (a+i a \tan (e+f x))}+\frac {\int \left (a \left (c^2-2 i c d+d^2\right )-2 i a d^2 \tan (e+f x)\right ) \, dx}{2 a^2}\\ &=\frac {\left (c^2-2 i c d+d^2\right ) x}{2 a}+\frac {i (c+i d)^2}{2 f (a+i a \tan (e+f x))}-\frac {\left (i d^2\right ) \int \tan (e+f x) \, dx}{a}\\ &=\frac {\left (c^2-2 i c d+d^2\right ) x}{2 a}+\frac {i d^2 \log (\cos (e+f x))}{a f}+\frac {i (c+i d)^2}{2 f (a+i a \tan (e+f x))}\\ \end {align*}
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Mathematica [B] time = 1.45, size = 155, normalized size = 2.07 \[ \frac {\tan (e+f x) \left (c^2 (2 f x-i)+2 c (d-2 i d f x)+2 i d^2 \log \left (\cos ^2(e+f x)\right )+d^2 (-2 f x+i)\right )-2 i c^2 f x+c^2-4 c d f x+2 i c d+4 d^2 \tan ^{-1}(\tan (f x)) (\tan (e+f x)-i)+2 d^2 \log \left (\cos ^2(e+f x)\right )+2 i d^2 f x-d^2}{4 a f (\tan (e+f x)-i)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 85, normalized size = 1.13 \[ \frac {{\left ({\left (2 \, c^{2} - 4 i \, c d + 6 \, d^{2}\right )} f x e^{\left (2 i \, f x + 2 i \, e\right )} + 4 i \, d^{2} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + i \, c^{2} - 2 \, c d - i \, d^{2}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{4 \, a f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.97, size = 131, normalized size = 1.75 \[ -\frac {\frac {{\left (i \, c^{2} + 2 \, c d + 3 i \, d^{2}\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{a} + \frac {{\left (-i \, c^{2} - 2 \, c d + i \, d^{2}\right )} \log \left (i \, \tan \left (f x + e\right ) - 1\right )}{a} + \frac {-i \, c^{2} \tan \left (f x + e\right ) - 2 \, c d \tan \left (f x + e\right ) - 3 i \, d^{2} \tan \left (f x + e\right ) - 3 \, c^{2} - 2 i \, c d - d^{2}}{a {\left (\tan \left (f x + e\right ) - i\right )}}}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.22, size = 196, normalized size = 2.61 \[ \frac {\ln \left (\tan \left (f x +e \right )+i\right ) c d}{2 f a}+\frac {i \ln \left (\tan \left (f x +e \right )+i\right ) c^{2}}{4 f a}-\frac {i \ln \left (\tan \left (f x +e \right )+i\right ) d^{2}}{4 f a}-\frac {\ln \left (\tan \left (f x +e \right )-i\right ) c d}{2 f a}-\frac {i \ln \left (\tan \left (f x +e \right )-i\right ) c^{2}}{4 f a}-\frac {3 i \ln \left (\tan \left (f x +e \right )-i\right ) d^{2}}{4 f a}+\frac {i c d}{f a \left (\tan \left (f x +e \right )-i\right )}+\frac {c^{2}}{2 f a \left (\tan \left (f x +e \right )-i\right )}-\frac {d^{2}}{2 f a \left (\tan \left (f x +e \right )-i\right )} \]
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 = 5.62, size = 112, normalized size = 1.49 \[ -\frac {\frac {c\,d}{a}-\frac {c^2\,1{}\mathrm {i}}{2\,a}+\frac {d^2\,1{}\mathrm {i}}{2\,a}}{f\,\left (1+\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,\left (c^2-c\,d\,2{}\mathrm {i}+3\,d^2\right )\,1{}\mathrm {i}}{4\,a\,f}+\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (c^2\,1{}\mathrm {i}+2\,c\,d-d^2\,1{}\mathrm {i}\right )}{4\,a\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.53, size = 173, normalized size = 2.31 \[ \begin {cases} - \frac {\left (- i c^{2} + 2 c d + i d^{2}\right ) e^{- 2 i e} e^{- 2 i f x}}{4 a f} & \text {for}\: 4 a f e^{2 i e} \neq 0 \\x \left (- \frac {c^{2} - 2 i c d + 3 d^{2}}{2 a} + \frac {\left (c^{2} e^{2 i e} + c^{2} - 2 i c d e^{2 i e} + 2 i c d + 3 d^{2} e^{2 i e} - d^{2}\right ) e^{- 2 i e}}{2 a}\right ) & \text {otherwise} \end {cases} + \frac {i d^{2} \log {\left (e^{2 i f x} + e^{- 2 i e} \right )}}{a f} - \frac {x \left (- c^{2} + 2 i c d - 3 d^{2}\right )}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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