Integrand size = 19, antiderivative size = 82 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {2 a b x}{\left (a^2+b^2\right )^2}-\frac {\left (a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {a}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))} \]
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Time = 0.13 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3610, 3612, 3611} \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {a}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {\left (a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^2}+\frac {2 a b x}{\left (a^2+b^2\right )^2} \]
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Rule 3610
Rule 3611
Rule 3612
Rubi steps \begin{align*} \text {integral}& = \frac {a}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {b+a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^2+b^2} \\ & = \frac {2 a b x}{\left (a^2+b^2\right )^2}+\frac {a}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {\left (a^2-b^2\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\left (a^2+b^2\right )^2} \\ & = \frac {2 a b x}{\left (a^2+b^2\right )^2}-\frac {\left (a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {a}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.50 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.21 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {a \left ((a-i b)^2 \log (i-\tan (c+d x))+(a+i b)^2 \log (i+\tan (c+d x))+2 \left (a^2+b^2+\left (-a^2+b^2\right ) \log (a+b \tan (c+d x))\right )\right )+b \left ((a-i b)^2 \log (i-\tan (c+d x))+(a+i b)^2 \log (i+\tan (c+d x))+2 \left (-a^2+b^2\right ) \log (a+b \tan (c+d x))\right ) \tan (c+d x)}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]
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Time = 0.37 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.27
method | result | size |
derivativedivides | \(\frac {\frac {a}{\left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}-\frac {\left (a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {\frac {\left (a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+2 a b \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(104\) |
default | \(\frac {\frac {a}{\left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}-\frac {\left (a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {\frac {\left (a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+2 a b \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(104\) |
norman | \(\frac {-\frac {b \tan \left (d x +c \right )}{d \left (a^{2}+b^{2}\right )}+\frac {2 a^{2} b x}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {2 b^{2} a x \tan \left (d x +c \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}}{a +b \tan \left (d x +c \right )}+\frac {\left (a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(175\) |
parallelrisch | \(\frac {4 x \tan \left (d x +c \right ) a^{2} b^{2} d +\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{3} b -\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right ) a \,b^{3}-2 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{3} b +2 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a \,b^{3}+4 a^{3} b d x +a^{4} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )-\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2} b^{2}-2 a^{4} \ln \left (a +b \tan \left (d x +c \right )\right )+2 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{2} b^{2}-2 a^{3} b \tan \left (d x +c \right )-2 a \,b^{3} \tan \left (d x +c \right )}{2 \left (a +b \tan \left (d x +c \right )\right ) \left (a^{2}+b^{2}\right )^{2} a d}\) | \(239\) |
risch | \(\frac {i x}{2 i a b -a^{2}+b^{2}}+\frac {2 i a^{2} x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {2 i x \,b^{2}}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {2 i a^{2} c}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {2 i b^{2} c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 i a b}{\left (i b +a \right ) d \left (-i b +a \right )^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right )}-\frac {a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) b^{2}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(287\) |
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Time = 0.25 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.91 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {4 \, a^{2} b d x + 2 \, a b^{2} - {\left (a^{3} - a b^{2} + {\left (a^{2} b - b^{3}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + 2 \, {\left (2 \, a b^{2} d x - a^{2} b\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d \tan \left (d x + c\right ) + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d\right )}} \]
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Result contains complex when optimal does not.
Time = 0.73 (sec) , antiderivative size = 1476, normalized size of antiderivative = 18.00 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]
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Time = 0.48 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.70 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {4 \, {\left (d x + c\right )} a b}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (a^{2} - b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, a}{a^{3} + a b^{2} + {\left (a^{2} b + b^{3}\right )} \tan \left (d x + c\right )}}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (82) = 164\).
Time = 0.43 (sec) , antiderivative size = 173, normalized size of antiderivative = 2.11 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {4 \, {\left (d x + c\right )} a b}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (a^{2} b - b^{3}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b + 2 \, a^{2} b^{3} + b^{5}} + \frac {2 \, {\left (a^{2} b \tan \left (d x + c\right ) - b^{3} \tan \left (d x + c\right ) + 2 \, a^{3}\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}}}{2 \, d} \]
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Time = 4.58 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.62 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^2+a\,b\,2{}\mathrm {i}-b^2\right )}-\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {1}{a^2+b^2}-\frac {2\,b^2}{{\left (a^2+b^2\right )}^2}\right )}{d}+\frac {a}{d\,\left (a^2+b^2\right )\,\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^2\,1{}\mathrm {i}+2\,a\,b-b^2\,1{}\mathrm {i}\right )} \]
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