Integrand size = 21, antiderivative size = 88 \[ \int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}-\frac {2 a b \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {a^2}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))} \]
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Time = 0.16 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3623, 3612, 3611} \[ \int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {a^2}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {2 a b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^2}-\frac {x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^2} \]
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Rule 3611
Rule 3612
Rule 3623
Rubi steps \begin{align*} \text {integral}& = -\frac {a^2}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {-a+b \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^2+b^2} \\ & = -\frac {\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}-\frac {a^2}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {(2 a b) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\left (a^2+b^2\right )^2} \\ & = -\frac {\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}-\frac {2 a b \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {a^2}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.55 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.08 \[ \int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {a \left (i (a-i b)^2 \log (i-\tan (c+d x))-i (a+i b)^2 \log (i+\tan (c+d x))-4 a b \log (a+b \tan (c+d x))\right )+\left (i (a-i b)^2 b \log (i-\tan (c+d x))-i (a+i b)^2 b \log (i+\tan (c+d x))+2 a \left (a^2+b^2-2 b^2 \log (a+b \tan (c+d x))\right )\right ) \tan (c+d x)}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]
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Time = 0.37 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.15
method | result | size |
derivativedivides | \(\frac {-\frac {a^{2}}{\left (a^{2}+b^{2}\right ) b \left (a +b \tan \left (d x +c \right )\right )}-\frac {2 a b \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {a b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )+\left (-a^{2}+b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(101\) |
default | \(\frac {-\frac {a^{2}}{\left (a^{2}+b^{2}\right ) b \left (a +b \tan \left (d x +c \right )\right )}-\frac {2 a b \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {a b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )+\left (-a^{2}+b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(101\) |
norman | \(\frac {-\frac {a^{2}}{b d \left (a^{2}+b^{2}\right )}-\frac {\left (a^{2}-b^{2}\right ) a x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {b \left (a^{2}-b^{2}\right ) x \tan \left (d x +c \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}}{a +b \tan \left (d x +c \right )}+\frac {a b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 a b \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(171\) |
parallelrisch | \(\frac {-x \tan \left (d x +c \right ) a^{2} b^{2} d +x \tan \left (d x +c \right ) b^{4} d +\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right ) a \,b^{3}-2 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a \,b^{3}-a^{3} b d x +a \,b^{3} d x +\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2} b^{2}-2 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{2} b^{2}-a^{4}-a^{2} b^{2}}{\left (a +b \tan \left (d x +c \right )\right ) \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b d}\) | \(176\) |
risch | \(\frac {x}{2 i a b -a^{2}+b^{2}}+\frac {4 i a b x}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {4 i a b c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 i a^{2}}{\left (i b +a \right ) d \left (-i b +a \right )^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right )}-\frac {2 a b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(179\) |
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Time = 0.25 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.74 \[ \int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {a^{2} b + {\left (a^{3} - a b^{2}\right )} d x + {\left (a b^{2} \tan \left (d x + c\right ) + a^{2} b\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (a^{3} - {\left (a^{2} b - b^{3}\right )} d x\right )} \tan \left (d x + c\right )}{{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d \tan \left (d x + c\right ) + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d} \]
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Result contains complex when optimal does not.
Time = 0.78 (sec) , antiderivative size = 1314, normalized size of antiderivative = 14.93 \[ \int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]
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Time = 0.40 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.56 \[ \int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {2 \, a b \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {a^{2}}{a^{3} b + a b^{3} + {\left (a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )}}{d} \]
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Time = 0.57 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.88 \[ \int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {2 \, a b^{2} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b + 2 \, a^{2} b^{3} + b^{5}} - \frac {a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, a b^{3} \tan \left (d x + c\right ) - a^{4} + a^{2} b^{2}}{{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}}}{d} \]
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Time = 5.45 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.43 \[ \int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (-a^2\,1{}\mathrm {i}+2\,a\,b+b^2\,1{}\mathrm {i}\right )}-\frac {a^2}{b\,d\,\left (a^2+b^2\right )\,\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}-\frac {2\,a\,b\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}{d\,{\left (a^2+b^2\right )}^2}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-a^2+a\,b\,2{}\mathrm {i}+b^2\right )} \]
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