Optimal. Leaf size=75 \[ -\frac {a}{f (d+i c) (c+d \tan (e+f x))}-\frac {i a \log (c \cos (e+f x)+d \sin (e+f x))}{f (c-i d)^2}+\frac {a x}{(c-i d)^2} \]
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Rubi [A] time = 0.15, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {3529, 3531, 3530} \[ -\frac {a}{f (d+i c) (c+d \tan (e+f x))}-\frac {i a \log (c \cos (e+f x)+d \sin (e+f x))}{f (c-i d)^2}+\frac {a x}{(c-i d)^2} \]
Antiderivative was successfully verified.
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Rule 3529
Rule 3530
Rule 3531
Rubi steps
\begin {align*} \int \frac {a+i a \tan (e+f x)}{(c+d \tan (e+f x))^2} \, dx &=-\frac {a}{(i c+d) f (c+d \tan (e+f x))}+\frac {\int \frac {a (c+i d)+a (i c-d) \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{c^2+d^2}\\ &=\frac {a x}{(c-i d)^2}-\frac {a}{(i c+d) f (c+d \tan (e+f x))}-\frac {(i a) \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{(c-i d)^2}\\ &=\frac {a x}{(c-i d)^2}-\frac {i a \log (c \cos (e+f x)+d \sin (e+f x))}{(c-i d)^2 f}-\frac {a}{(i c+d) f (c+d \tan (e+f x))}\\ \end {align*}
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Mathematica [B] time = 2.93, size = 302, normalized size = 4.03 \[ \frac {(\cos (e)-i \sin (e)) \cos (e+f x) (\cos (f x)-i \sin (f x)) (a+i a \tan (e+f x)) \left (4 \tan ^{-1}\left (\frac {\left (d^2-c^2\right ) \sin (2 e+f x)+2 c d \cos (2 e+f x)}{\left (c^2-d^2\right ) \cos (2 e+f x)+2 c d \sin (2 e+f x)}\right )+\frac {\left (c^2+d^2\right ) \cos (f x) \left (4 f x-i \log \left ((c \cos (e+f x)+d \sin (e+f x))^2\right )\right )+\left (c^2-d^2\right ) \cos (2 e+f x) \left (4 f x-i \log \left ((c \cos (e+f x)+d \sin (e+f x))^2\right )\right )-2 d \left (c \sin (2 e+f x) \left (-4 f x+i \log \left ((c \cos (e+f x)+d \sin (e+f x))^2\right )\right )+2 (d+i c) \sin (f x)\right )}{(c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))}\right )}{4 f (c-i d)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 125, normalized size = 1.67 \[ \frac {-2 i \, a d - {\left (a c + i \, a d + {\left (a c - i \, a d\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \log \left (\frac {{\left (i \, c + d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c - d}{i \, c + d}\right )}{{\left (-i \, c^{3} - 3 \, c^{2} d + 3 i \, c d^{2} + d^{3}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-i \, c^{3} - c^{2} d - i \, c d^{2} - d^{3}\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.51, size = 186, normalized size = 2.48 \[ \frac {2 \, {\left (\frac {a \log \left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - c\right )}{2 i \, c^{2} + 4 \, c d - 2 i \, d^{2}} + \frac {a \log \left (-i \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{-i \, c^{2} - 2 \, c d + i \, d^{2}} - \frac {a c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 i \, a d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - a c^{2}}{{\left (2 i \, c^{3} + 4 \, c^{2} d - 2 i \, c d^{2}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - c\right )}}\right )}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.27, size = 309, normalized size = 4.12 \[ -\frac {i a \ln \left (c +d \tan \left (f x +e \right )\right ) c^{2}}{f \left (c^{2}+d^{2}\right )^{2}}+\frac {i a \ln \left (c +d \tan \left (f x +e \right )\right ) d^{2}}{f \left (c^{2}+d^{2}\right )^{2}}+\frac {2 a \ln \left (c +d \tan \left (f x +e \right )\right ) c d}{f \left (c^{2}+d^{2}\right )^{2}}+\frac {i a c}{f \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )}-\frac {a d}{f \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )}-\frac {a \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c d}{f \left (c^{2}+d^{2}\right )^{2}}+\frac {i a \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c^{2}}{2 f \left (c^{2}+d^{2}\right )^{2}}-\frac {i a \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) d^{2}}{2 f \left (c^{2}+d^{2}\right )^{2}}+\frac {2 i a \arctan \left (\tan \left (f x +e \right )\right ) c d}{f \left (c^{2}+d^{2}\right )^{2}}+\frac {a \arctan \left (\tan \left (f x +e \right )\right ) c^{2}}{f \left (c^{2}+d^{2}\right )^{2}}-\frac {a \arctan \left (\tan \left (f x +e \right )\right ) d^{2}}{f \left (c^{2}+d^{2}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.76, size = 180, normalized size = 2.40 \[ \frac {\frac {2 \, {\left (a c^{2} + 2 i \, a c d - a d^{2}\right )} {\left (f x + e\right )}}{c^{4} + 2 \, c^{2} d^{2} + d^{4}} + \frac {2 \, {\left (-i \, a c^{2} + 2 \, a c d + i \, a d^{2}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{c^{4} + 2 \, c^{2} d^{2} + d^{4}} + \frac {{\left (i \, a c^{2} - 2 \, a c d - i \, a d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{4} + 2 \, c^{2} d^{2} + d^{4}} + \frac {2 \, {\left (i \, a c - a d\right )}}{c^{3} + c d^{2} + {\left (c^{2} d + d^{3}\right )} \tan \left (f x + e\right )}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.13, size = 135, normalized size = 1.80 \[ -\frac {2\,a\,\mathrm {atan}\left (\frac {\left (c^2+d^2\right )\,1{}\mathrm {i}}{{\left (d+c\,1{}\mathrm {i}\right )}^2}-\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (2\,c^4\,d^2+4\,c^2\,d^4+2\,d^6\right )}{{\left (d+c\,1{}\mathrm {i}\right )}^2\,\left (c^3\,d\,1{}\mathrm {i}-c^2\,d^2+c\,d^3\,1{}\mathrm {i}-d^4\right )}\right )}{f\,{\left (d+c\,1{}\mathrm {i}\right )}^2}+\frac {a\,1{}\mathrm {i}}{d\,f\,\left (\mathrm {tan}\left (e+f\,x\right )+\frac {c}{d}\right )\,\left (c-d\,1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.56, size = 144, normalized size = 1.92 \[ \frac {2 i a d}{i c^{3} f + c^{2} d f + i c d^{2} f + d^{3} f + \left (i c^{3} f e^{2 i e} + 3 c^{2} d f e^{2 i e} - 3 i c d^{2} f e^{2 i e} - d^{3} f e^{2 i e}\right ) e^{2 i f x}} - \frac {i a \log {\left (\frac {i c - d}{i c e^{2 i e} + d e^{2 i e}} + e^{2 i f x} \right )}}{f \left (c - i d\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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