Integrand size = 21, antiderivative size = 114 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {2 a b x}{\left (a^2+b^2\right )^2}+\frac {\left (a^2-b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {a^2 \left (a^2+3 b^2\right ) \log (a+b \tan (c+d x))}{b^2 \left (a^2+b^2\right )^2 d}+\frac {a^3}{b^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))} \]
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Time = 0.20 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3646, 3707, 3698, 31, 3556} \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {a^2 \tan (c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {a^2 \left (a^2+3 b^2\right ) \log (a+b \tan (c+d x))}{b^2 d \left (a^2+b^2\right )^2}+\frac {\left (a^2-b^2\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^2}-\frac {2 a b x}{\left (a^2+b^2\right )^2} \]
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Rule 31
Rule 3556
Rule 3646
Rule 3698
Rule 3707
Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 \tan (c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {a^2-a b \tan (c+d x)+\left (a^2+b^2\right ) \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b \left (a^2+b^2\right )} \\ & = -\frac {2 a b x}{\left (a^2+b^2\right )^2}-\frac {a^2 \tan (c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {\left (a^2-b^2\right ) \int \tan (c+d x) \, dx}{\left (a^2+b^2\right )^2}+\frac {\left (a^2 \left (a^2+3 b^2\right )\right ) \int \frac {1+\tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b \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 (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {a^2 \tan (c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\left (a^2 \left (a^2+3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^2 \left (a^2+b^2\right )^2 d} \\ & = -\frac {2 a b x}{\left (a^2+b^2\right )^2}+\frac {\left (a^2-b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {a^2 \left (a^2+3 b^2\right ) \log (a+b \tan (c+d x))}{b^2 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan (c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))} \\ \end{align*}
Result contains complex when optimal does not.
Time = 3.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.98 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {\log (i-\tan (c+d x))}{(a+i b)^2}+\frac {\log (i+\tan (c+d x))}{(a-i b)^2}-\frac {2 a^2 \left (\left (3+\frac {a^2}{b^2}\right ) \log (a+b \tan (c+d x))+\frac {a \left (a^2+b^2\right )}{b^2 (a+b \tan (c+d x))}\right )}{\left (a^2+b^2\right )^2}}{2 d} \]
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Time = 0.34 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (-a^{2}+b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-2 a b \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {a^{2} \left (a^{2}+3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2} b^{2}}+\frac {a^{3}}{b^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}}{d}\) | \(114\) |
default | \(\frac {\frac {\frac {\left (-a^{2}+b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-2 a b \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {a^{2} \left (a^{2}+3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2} b^{2}}+\frac {a^{3}}{b^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}}{d}\) | \(114\) |
norman | \(\frac {\frac {a^{3}}{d \left (a^{2}+b^{2}\right ) b^{2}}-\frac {2 a^{2} b x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {2 b^{2} a x \tan \left (d x +c \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}}{a +b \tan \left (d x +c \right )}+\frac {a^{2} \left (a^{2}+3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b^{2} d}-\frac {\left (a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(178\) |
parallelrisch | \(-\frac {4 b^{4} a \tan \left (d x +c \right ) x d +\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{2} b^{3}-\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right ) b^{5}-2 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{4} b -6 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{2} b^{3}+4 a^{2} b^{3} x d +\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{3} b^{2}-a \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) b^{4}-2 a^{5} \ln \left (a +b \tan \left (d x +c \right )\right )-6 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{3} b^{2}-2 a^{5}-2 a^{3} b^{2}}{2 \left (a +b \tan \left (d x +c \right )\right ) \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b^{2} d}\) | \(240\) |
risch | \(-\frac {i x}{2 i a b -a^{2}+b^{2}}-\frac {2 i a^{4} x}{b^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 i a^{4} c}{b^{2} d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {6 i a^{2} x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {6 i a^{2} c}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {2 i x}{b^{2}}+\frac {2 i c}{b^{2} d}-\frac {2 i a^{3}}{\left (i b +a \right ) d b \left (-i b +a \right )^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right )}+\frac {a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{b^{2} d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {3 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{2} d}\) | \(337\) |
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Time = 0.27 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.98 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {4 \, a^{2} b^{3} d x - 2 \, a^{3} b^{2} - {\left (a^{5} + 3 \, a^{3} b^{2} + {\left (a^{4} b + 3 \, a^{2} b^{3}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 2 \, {\left (2 \, a b^{4} d x + a^{4} b\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} d \tan \left (d x + c\right ) + {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} d\right )}} \]
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Result contains complex when optimal does not.
Time = 0.88 (sec) , antiderivative size = 1992, normalized size of antiderivative = 17.47 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]
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Time = 0.36 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.36 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {2 \, a^{3}}{a^{3} b^{2} + a b^{4} + {\left (a^{2} b^{3} + b^{5}\right )} \tan \left (d x + c\right )} - \frac {4 \, {\left (d x + c\right )} a b}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (a^{4} + 3 \, a^{2} b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}} - \frac {{\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}}}{2 \, d} \]
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Time = 0.66 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.59 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {4 \, {\left (d x + c\right )} a b}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (a^{4} + 3 \, a^{2} b^{2}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (a^{4} \tan \left (d x + c\right ) + 3 \, a^{2} b^{2} \tan \left (d x + c\right ) + 2 \, a^{3} b\right )}}{{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}}}{2 \, d} \]
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Time = 4.76 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.20 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {a^3}{b^2\,d\,\left (a^2+b^2\right )\,\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^2+a\,b\,2{}\mathrm {i}-b^2\right )}+\frac {a^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a^2+3\,b^2\right )}{b^2\,d\,{\left (a^2+b^2\right )}^2}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^2\,1{}\mathrm {i}+2\,a\,b-b^2\,1{}\mathrm {i}\right )} \]
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