Integrand size = 28, antiderivative size = 142 \[ \int \frac {(a+i a \tan (e+f x))^3}{(c+d \tan (e+f x))^2} \, dx=\frac {4 a^3 x}{(c-i d)^2}+\frac {i a^3 \log (\cos (e+f x))}{d^2 f}-\frac {a^3 (i c-d) (c-3 i d) \log (c \cos (e+f x)+d \sin (e+f x))}{(c-i d)^2 d^2 f}+\frac {(c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{(c-i d) d f (c+d \tan (e+f x))} \]
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Time = 0.43 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3634, 3670, 3556, 3612, 3611} \[ \int \frac {(a+i a \tan (e+f x))^3}{(c+d \tan (e+f x))^2} \, dx=-\frac {a^3 (-d+i c) (c-3 i d) \log (c \cos (e+f x)+d \sin (e+f x))}{d^2 f (c-i d)^2}+\frac {(c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{d f (c-i d) (c+d \tan (e+f x))}+\frac {4 a^3 x}{(c-i d)^2}+\frac {i a^3 \log (\cos (e+f x))}{d^2 f} \]
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Rule 3556
Rule 3611
Rule 3612
Rule 3634
Rule 3670
Rubi steps \begin{align*} \text {integral}& = \frac {(c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{(c-i d) d f (c+d \tan (e+f x))}-\frac {\int \frac {(a+i a \tan (e+f x)) \left (-a^2 (c+3 i d)+a^2 (i c+d) \tan (e+f x)\right )}{c+d \tan (e+f x)} \, dx}{d (i c+d)} \\ & = \frac {(c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{(c-i d) d f (c+d \tan (e+f x))}-\frac {\left (i a^3\right ) \int \tan (e+f x) \, dx}{d^2}-\frac {\int \frac {-a^3 (c+3 i d) d+a^3 \left (c^2-i c d+4 d^2\right ) \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{d^2 (i c+d)} \\ & = -\frac {4 a^3 x}{(i c+d)^2}+\frac {i a^3 \log (\cos (e+f x))}{d^2 f}+\frac {(c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{(c-i d) d f (c+d \tan (e+f x))}+\frac {\left (a^3 (c+i d) (c-3 i d)\right ) \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{(c-i d) d^2 (i c+d)} \\ & = -\frac {4 a^3 x}{(i c+d)^2}+\frac {i a^3 \log (\cos (e+f x))}{d^2 f}-\frac {a^3 (i c-d) (c-3 i d) \log (c \cos (e+f x)+d \sin (e+f x))}{(c-i d)^2 d^2 f}+\frac {(c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{(c-i d) d f (c+d \tan (e+f x))} \\ \end{align*}
Time = 1.88 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.33 \[ \int \frac {(a+i a \tan (e+f x))^3}{(c+d \tan (e+f x))^2} \, dx=\frac {i a^3 \left (-c d^2+2 i d^3-8 c d^2 \log (i+\tan (e+f x))+2 c^3 \log (c+d \tan (e+f x))-4 i c^2 d \log (c+d \tan (e+f x))+6 c d^2 \log (c+d \tan (e+f x))-d \left (2 c^2+2 i c d+3 d^2+8 d^2 \log (i+\tan (e+f x))-2 \left (c^2-2 i c d+3 d^2\right ) \log (c+d \tan (e+f x))\right ) \tan (e+f x)\right )}{2 d^2 (i c+d)^2 f (c+d \tan (e+f x))} \]
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Time = 0.41 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.23
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\frac {i c^{3}-3 i c \,d^{2}-3 c^{2} d +d^{3}}{d^{2} \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )}+\frac {\left (-i c^{4}-6 i c^{2} d^{2}+3 i d^{4}+8 c \,d^{3}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{2} d^{2}}+\frac {\frac {\left (4 i c^{2}-4 i d^{2}-8 c d \right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (8 i c d +4 c^{2}-4 d^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{2}}\right )}{f}\) | \(175\) |
default | \(\frac {a^{3} \left (-\frac {i c^{3}-3 i c \,d^{2}-3 c^{2} d +d^{3}}{d^{2} \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )}+\frac {\left (-i c^{4}-6 i c^{2} d^{2}+3 i d^{4}+8 c \,d^{3}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{2} d^{2}}+\frac {\frac {\left (4 i c^{2}-4 i d^{2}-8 c d \right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (8 i c d +4 c^{2}-4 d^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{2}}\right )}{f}\) | \(175\) |
norman | \(\frac {\frac {i \left (2 i a^{3} c d +a^{3} c^{2}-a^{3} d^{2}\right ) \tan \left (f x +e \right )}{c f \left (-i d +c \right ) d}+\frac {4 a^{3} c x}{-2 i c d +c^{2}-d^{2}}-\frac {4 d \,a^{3} x \tan \left (f x +e \right )}{2 i c d -c^{2}+d^{2}}}{c +d \tan \left (f x +e \right )}+\frac {2 i a^{3} \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{f \left (-2 i c d +c^{2}-d^{2}\right )}-\frac {i a^{3} \left (-2 i c d +c^{2}+3 d^{2}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{d^{2} f \left (-2 i c d +c^{2}-d^{2}\right )}\) | \(207\) |
parallelrisch | \(\frac {2 a^{3} c^{2} d^{3}+3 a^{3} c^{4} d +3 i \ln \left (c +d \tan \left (f x +e \right )\right ) \tan \left (f x +e \right ) a^{3} d^{5}-6 i \ln \left (c +d \tan \left (f x +e \right )\right ) a^{3} c^{3} d^{2}+3 i \ln \left (c +d \tan \left (f x +e \right )\right ) a^{3} c \,d^{4}-i \ln \left (c +d \tan \left (f x +e \right )\right ) a^{3} c^{5}+2 i a^{3} c^{3} d^{2}+3 i a^{3} c \,d^{4}-4 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) a^{3} c^{2} d^{3}-i a^{3} c^{5}+2 i \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) \tan \left (f x +e \right ) a^{3} c^{2} d^{3}+8 \ln \left (c +d \tan \left (f x +e \right )\right ) a^{3} c^{2} d^{3}-4 x \tan \left (f x +e \right ) a^{3} d^{5} f +4 x \,a^{3} c^{3} d^{2} f -4 x \,a^{3} c \,d^{4} f +8 \ln \left (c +d \tan \left (f x +e \right )\right ) \tan \left (f x +e \right ) a^{3} c \,d^{4}-a^{3} d^{5}-2 i \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) \tan \left (f x +e \right ) a^{3} d^{5}+2 i \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) a^{3} c^{3} d^{2}-2 i \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) a^{3} c \,d^{4}-4 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) \tan \left (f x +e \right ) a^{3} c \,d^{4}-i \ln \left (c +d \tan \left (f x +e \right )\right ) \tan \left (f x +e \right ) a^{3} c^{4} d -6 i \ln \left (c +d \tan \left (f x +e \right )\right ) \tan \left (f x +e \right ) a^{3} c^{2} d^{3}+8 i x \,a^{3} c^{2} d^{3} f +4 x \tan \left (f x +e \right ) a^{3} c^{2} d^{3} f +8 i x \tan \left (f x +e \right ) a^{3} c \,d^{4} f}{f \left (c^{2}+d^{2}\right )^{2} d^{2} \left (c +d \tan \left (f x +e \right )\right )}\) | \(521\) |
risch | \(-\frac {8 a^{3} x}{2 i c d -c^{2}+d^{2}}-\frac {4 a^{3} c x}{d \left (i c^{2}-i d^{2}+2 c d \right )}-\frac {4 a^{3} c e}{d f \left (i c^{2}-i d^{2}+2 c d \right )}+\frac {i a^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{d^{2} f}-\frac {6 i a^{3} e}{f \left (i c^{2}-i d^{2}+2 c d \right )}-\frac {4 i a^{3} c}{f \left (-i d +c \right )^{2} \left (-i {\mathrm e}^{2 i \left (f x +e \right )} d +{\mathrm e}^{2 i \left (f x +e \right )} c +i d +c \right )}-\frac {2 i a^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i d +c}{i d -c}\right ) c}{d f \left (i c^{2}-i d^{2}+2 c d \right )}+\frac {2 a^{3} x}{d^{2}}+\frac {2 a^{3} e}{d^{2} f}-\frac {6 i a^{3} x}{i c^{2}-i d^{2}+2 c d}-\frac {2 a^{3} c^{2}}{f d \left (-i d +c \right )^{2} \left (-i {\mathrm e}^{2 i \left (f x +e \right )} d +{\mathrm e}^{2 i \left (f x +e \right )} c +i d +c \right )}+\frac {2 a^{3} d}{f \left (-i d +c \right )^{2} \left (-i {\mathrm e}^{2 i \left (f x +e \right )} d +{\mathrm e}^{2 i \left (f x +e \right )} c +i d +c \right )}-\frac {2 i a^{3} c^{2} x}{d^{2} \left (i c^{2}-i d^{2}+2 c d \right )}+\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i d +c}{i d -c}\right ) c^{2}}{d^{2} f \left (i c^{2}-i d^{2}+2 c d \right )}+\frac {3 a^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i d +c}{i d -c}\right )}{f \left (i c^{2}-i d^{2}+2 c d \right )}-\frac {2 i a^{3} c^{2} e}{d^{2} f \left (i c^{2}-i d^{2}+2 c d \right )}\) | \(583\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (126) = 252\).
Time = 0.27 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.10 \[ \int \frac {(a+i a \tan (e+f x))^3}{(c+d \tan (e+f x))^2} \, dx=\frac {2 i \, a^{3} c^{2} d - 4 \, a^{3} c d^{2} - 2 i \, a^{3} d^{3} - {\left (a^{3} c^{3} - i \, a^{3} c^{2} d + 5 \, a^{3} c d^{2} + 3 i \, a^{3} d^{3} + {\left (a^{3} c^{3} - 3 i \, a^{3} c^{2} d + a^{3} c d^{2} - 3 i \, a^{3} d^{3}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \log \left (\frac {{\left (i \, c + d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c - d}{i \, c + d}\right ) + {\left (a^{3} c^{3} - i \, a^{3} c^{2} d + a^{3} c d^{2} - i \, a^{3} d^{3} + {\left (a^{3} c^{3} - 3 i \, a^{3} c^{2} d - 3 \, a^{3} c d^{2} + i \, a^{3} d^{3}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}{{\left (-i \, c^{3} d^{2} - 3 \, c^{2} d^{3} + 3 i \, c d^{4} + d^{5}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-i \, c^{3} d^{2} - c^{2} d^{3} - i \, c d^{4} - d^{5}\right )} f} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (116) = 232\).
Time = 14.03 (sec) , antiderivative size = 374, normalized size of antiderivative = 2.63 \[ \int \frac {(a+i a \tan (e+f x))^3}{(c+d \tan (e+f x))^2} \, dx=- \frac {i a^{3} \left (c - 3 i d\right ) \left (c + i d\right ) \log {\left (e^{2 i f x} + \frac {a^{3} c^{2} + \frac {i a^{3} c d \left (c - 3 i d\right ) \left (c + i d\right )}{\left (c - i d\right )^{2}} - i a^{3} c d + \frac {a^{3} d^{2} \left (c - 3 i d\right ) \left (c + i d\right )}{\left (c - i d\right )^{2}} + 2 a^{3} d^{2}}{a^{3} c^{2} e^{2 i e} - 2 i a^{3} c d e^{2 i e} + a^{3} d^{2} e^{2 i e}} \right )}}{d^{2} f \left (c - i d\right )^{2}} + \frac {i a^{3} \log {\left (\frac {a^{3} c^{2} - 2 i a^{3} c d + a^{3} d^{2}}{a^{3} c^{2} e^{2 i e} - 2 i a^{3} c d e^{2 i e} + a^{3} d^{2} e^{2 i e}} + e^{2 i f x} \right )}}{d^{2} f} + \frac {- 2 a^{3} c^{2} - 4 i a^{3} c d + 2 a^{3} d^{2}}{c^{3} d f - i c^{2} d^{2} f + c d^{3} f - i d^{4} f + \left (c^{3} d f e^{2 i e} - 3 i c^{2} d^{2} f e^{2 i e} - 3 c d^{3} f e^{2 i e} + i d^{4} f e^{2 i e}\right ) e^{2 i f x}} \]
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Time = 0.31 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.73 \[ \int \frac {(a+i a \tan (e+f x))^3}{(c+d \tan (e+f x))^2} \, dx=\frac {\frac {4 \, {\left (a^{3} c^{2} + 2 i \, a^{3} c d - a^{3} d^{2}\right )} {\left (f x + e\right )}}{c^{4} + 2 \, c^{2} d^{2} + d^{4}} + \frac {{\left (-i \, a^{3} c^{4} - 6 i \, a^{3} c^{2} d^{2} + 8 \, a^{3} c d^{3} + 3 i \, a^{3} d^{4}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{c^{4} d^{2} + 2 \, c^{2} d^{4} + d^{6}} - \frac {2 \, {\left (-i \, a^{3} c^{2} + 2 \, a^{3} c d + i \, a^{3} d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{4} + 2 \, c^{2} d^{2} + d^{4}} + \frac {-i \, a^{3} c^{3} + 3 \, a^{3} c^{2} d + 3 i \, a^{3} c d^{2} - a^{3} d^{3}}{c^{3} d^{2} + c d^{4} + {\left (c^{2} d^{3} + d^{5}\right )} \tan \left (f x + e\right )}}{f} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (126) = 252\).
Time = 0.57 (sec) , antiderivative size = 375, normalized size of antiderivative = 2.64 \[ \int \frac {(a+i a \tan (e+f x))^3}{(c+d \tan (e+f x))^2} \, dx=\frac {\frac {8 \, a^{3} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}{-i \, c^{2} - 2 \, c d + i \, d^{2}} + \frac {i \, a^{3} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{d^{2}} + \frac {i \, a^{3} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}{d^{2}} + \frac {{\left (-i \, a^{3} c^{2} - 2 \, a^{3} c d - 3 i \, a^{3} d^{2}\right )} \log \left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - c\right )}{c^{2} d^{2} - 2 i \, c d^{3} - d^{4}} - \frac {-i \, a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, a^{3} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 i \, a^{3} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 4 i \, a^{3} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, a^{3} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 8 i \, a^{3} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, a^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i \, a^{3} c^{4} + 2 \, a^{3} c^{3} d + 3 i \, a^{3} c^{2} d^{2}}{{\left (c^{3} d^{2} - 2 i \, c^{2} d^{3} - c d^{4}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - c\right )}}}{f} \]
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Time = 8.33 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.94 \[ \int \frac {(a+i a \tan (e+f x))^3}{(c+d \tan (e+f x))^2} \, dx=-\frac {4\,a^3\,\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )}{f\,\left (c^2\,1{}\mathrm {i}+2\,c\,d-d^2\,1{}\mathrm {i}\right )}+\frac {a^3\,\left (-c^2\,1{}\mathrm {i}+2\,c\,d+d^2\,1{}\mathrm {i}\right )}{d^3\,f\,\left (\mathrm {tan}\left (e+f\,x\right )+\frac {c}{d}\right )\,\left (c-d\,1{}\mathrm {i}\right )}+\frac {a^3\,\ln \left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (c^2\,1{}\mathrm {i}+2\,c\,d+d^2\,3{}\mathrm {i}\right )}{d^2\,f\,{\left (d+c\,1{}\mathrm {i}\right )}^2} \]
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