Integrand size = 41, antiderivative size = 91 \[ \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^2} \, dx=-\frac {i a^2 B x}{c^2}-\frac {a^2 B \log (\cos (e+f x))}{c^2 f}+\frac {a^2 (i A+B)}{c^2 f (i+\tan (e+f x))^2}-\frac {a^2 (A-3 i B)}{c^2 f (i+\tan (e+f x))} \]
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Time = 0.17 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {3669, 78} \[ \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^2} \, dx=-\frac {a^2 (A-3 i B)}{c^2 f (\tan (e+f x)+i)}+\frac {a^2 (B+i A)}{c^2 f (\tan (e+f x)+i)^2}-\frac {a^2 B \log (\cos (e+f x))}{c^2 f}-\frac {i a^2 B x}{c^2} \]
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Rule 78
Rule 3669
Rubi steps \begin{align*} \text {integral}& = \frac {(a c) \text {Subst}\left (\int \frac {(a+i a x) (A+B x)}{(c-i c x)^3} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {(a c) \text {Subst}\left (\int \left (-\frac {2 i a (A-i B)}{c^3 (i+x)^3}+\frac {a (A-3 i B)}{c^3 (i+x)^2}+\frac {a B}{c^3 (i+x)}\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {i a^2 B x}{c^2}-\frac {a^2 B \log (\cos (e+f x))}{c^2 f}+\frac {a^2 (i A+B)}{c^2 f (i+\tan (e+f x))^2}-\frac {a^2 (A-3 i B)}{c^2 f (i+\tan (e+f x))} \\ \end{align*}
Time = 4.53 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.62 \[ \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^2} \, dx=\frac {a^2 \left (B \log (i+\tan (e+f x))-\frac {2 B+(A-3 i B) \tan (e+f x)}{(i+\tan (e+f x))^2}\right )}{c^2 f} \]
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Time = 0.13 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.13
method | result | size |
risch | \(-\frac {{\mathrm e}^{4 i \left (f x +e \right )} a^{2} B}{4 c^{2} f}-\frac {i {\mathrm e}^{4 i \left (f x +e \right )} A \,a^{2}}{4 c^{2} f}+\frac {B \,a^{2} {\mathrm e}^{2 i \left (f x +e \right )}}{c^{2} f}+\frac {2 i B \,a^{2} e}{c^{2} f}-\frac {B \,a^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{c^{2} f}\) | \(103\) |
derivativedivides | \(\frac {i a^{2} A}{f \,c^{2} \left (i+\tan \left (f x +e \right )\right )^{2}}+\frac {a^{2} B}{f \,c^{2} \left (i+\tan \left (f x +e \right )\right )^{2}}+\frac {3 i a^{2} B}{f \,c^{2} \left (i+\tan \left (f x +e \right )\right )}-\frac {a^{2} A}{f \,c^{2} \left (i+\tan \left (f x +e \right )\right )}+\frac {a^{2} B \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2 f \,c^{2}}-\frac {i a^{2} B \arctan \left (\tan \left (f x +e \right )\right )}{f \,c^{2}}\) | \(138\) |
default | \(\frac {i a^{2} A}{f \,c^{2} \left (i+\tan \left (f x +e \right )\right )^{2}}+\frac {a^{2} B}{f \,c^{2} \left (i+\tan \left (f x +e \right )\right )^{2}}+\frac {3 i a^{2} B}{f \,c^{2} \left (i+\tan \left (f x +e \right )\right )}-\frac {a^{2} A}{f \,c^{2} \left (i+\tan \left (f x +e \right )\right )}+\frac {a^{2} B \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2 f \,c^{2}}-\frac {i a^{2} B \arctan \left (\tan \left (f x +e \right )\right )}{f \,c^{2}}\) | \(138\) |
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Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.68 \[ \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^2} \, dx=\frac {{\left (-i \, A - B\right )} a^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, B a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 4 \, B a^{2} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}{4 \, c^{2} f} \]
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Time = 0.29 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.76 \[ \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^2} \, dx=- \frac {B a^{2} \log {\left (e^{2 i f x} + e^{- 2 i e} \right )}}{c^{2} f} + \begin {cases} \frac {4 B a^{2} c^{2} f e^{2 i e} e^{2 i f x} + \left (- i A a^{2} c^{2} f e^{4 i e} - B a^{2} c^{2} f e^{4 i e}\right ) e^{4 i f x}}{4 c^{4} f^{2}} & \text {for}\: c^{4} f^{2} \neq 0 \\\frac {x \left (A a^{2} e^{4 i e} - i B a^{2} e^{4 i e} + 2 i B a^{2} e^{2 i e}\right )}{c^{2}} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^2} \, dx=\text {Exception raised: RuntimeError} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (81) = 162\).
Time = 0.54 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.10 \[ \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^2} \, dx=-\frac {\frac {6 \, B a^{2} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{c^{2}} - \frac {12 \, B a^{2} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}{c^{2}} + \frac {6 \, B a^{2} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}{c^{2}} + \frac {25 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 12 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 112 i \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 198 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 12 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 112 i \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 25 \, B a^{2}}{c^{2} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}^{4}}}{6 \, f} \]
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Time = 8.44 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.14 \[ \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^2} \, dx=-\frac {a^2\,\left (B\,2{}\mathrm {i}+A\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}+3\,B\,\mathrm {tan}\left (e+f\,x\right )+B\,\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}+2\,B\,\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\mathrm {tan}\left (e+f\,x\right )-B\,\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^2\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{c^2\,f\,{\left (-1+\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2} \]
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