Integrand size = 23, antiderivative size = 101 \[ \int \frac {a+b \tan (c+d x)}{(b+a \tan (c+d x))^2} \, dx=-\frac {a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^2}+\frac {b \left (3 a^2-b^2\right ) \log (b \cos (c+d x)+a \sin (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {a^2-b^2}{\left (a^2+b^2\right ) d (b+a \tan (c+d x))} \]
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Time = 0.16 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3610, 3612, 3611} \[ \int \frac {a+b \tan (c+d x)}{(b+a \tan (c+d x))^2} \, dx=-\frac {a^2-b^2}{d \left (a^2+b^2\right ) (a \tan (c+d x)+b)}+\frac {b \left (3 a^2-b^2\right ) \log (a \sin (c+d x)+b \cos (c+d x))}{d \left (a^2+b^2\right )^2}-\frac {a x \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^2} \]
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Rule 3610
Rule 3611
Rule 3612
Rubi steps \begin{align*} \text {integral}& = -\frac {a^2-b^2}{\left (a^2+b^2\right ) d (b+a \tan (c+d x))}+\frac {\int \frac {2 a b-\left (a^2-b^2\right ) \tan (c+d x)}{b+a \tan (c+d x)} \, dx}{a^2+b^2} \\ & = -\frac {a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^2}-\frac {a^2-b^2}{\left (a^2+b^2\right ) d (b+a \tan (c+d x))}+\frac {\left (b \left (3 a^2-b^2\right )\right ) \int \frac {a-b \tan (c+d x)}{b+a \tan (c+d x)} \, dx}{\left (a^2+b^2\right )^2} \\ & = -\frac {a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^2}+\frac {b \left (3 a^2-b^2\right ) \log (b \cos (c+d x)+a \sin (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {a^2-b^2}{\left (a^2+b^2\right ) d (b+a \tan (c+d x))} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.58 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.85 \[ \int \frac {a+b \tan (c+d x)}{(b+a \tan (c+d x))^2} \, dx=\frac {\frac {b (-((a+i b) \log (i-\tan (c+d x)))-(a-i b) \log (i+\tan (c+d x))+2 a \log (b+a \tan (c+d x)))}{a^2+b^2}+(a-b) (a+b) \left (\frac {i \log (i-\tan (c+d x))}{(a-i b)^2}-\frac {i \log (i+\tan (c+d x))}{(a+i b)^2}+\frac {2 a \left (2 b \log (b+a \tan (c+d x))-\frac {a^2+b^2}{b+a \tan (c+d x)}\right )}{\left (a^2+b^2\right )^2}\right )}{2 a d} \]
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Time = 0.08 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.24
method | result | size |
derivativedivides | \(\frac {-\frac {a^{2}-b^{2}}{\left (a^{2}+b^{2}\right ) \left (b +a \tan \left (d x +c \right )\right )}+\frac {b \left (3 a^{2}-b^{2}\right ) \ln \left (b +a \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {\frac {\left (-3 a^{2} b +b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-a^{3}+3 a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(125\) |
default | \(\frac {-\frac {a^{2}-b^{2}}{\left (a^{2}+b^{2}\right ) \left (b +a \tan \left (d x +c \right )\right )}+\frac {b \left (3 a^{2}-b^{2}\right ) \ln \left (b +a \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {\frac {\left (-3 a^{2} b +b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-a^{3}+3 a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(125\) |
norman | \(\frac {\frac {\left (a^{2}-b^{2}\right ) a \tan \left (d x +c \right )}{b d \left (a^{2}+b^{2}\right )}-\frac {a^{2} \left (a^{2}-3 b^{2}\right ) x \tan \left (d x +c \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {b a \left (a^{2}-3 b^{2}\right ) x}{a^{4}+2 a^{2} b^{2}+b^{4}}}{b +a \tan \left (d x +c \right )}+\frac {b \left (3 a^{2}-b^{2}\right ) \ln \left (b +a \tan \left (d x +c \right )\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b \left (3 a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(206\) |
parallelrisch | \(-\frac {2 x \tan \left (d x +c \right ) a^{4} b d -6 x \tan \left (d x +c \right ) a^{2} b^{3} d +3 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{3} b^{2}-\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right ) a \,b^{4}-6 \ln \left (b +a \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{3} b^{2}+2 \ln \left (b +a \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a \,b^{4}+2 x \,a^{3} b^{2} d -6 x a \,b^{4} d +3 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2} b^{3}-\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) b^{5}-6 \ln \left (b +a \tan \left (d x +c \right )\right ) a^{2} b^{3}+2 \ln \left (b +a \tan \left (d x +c \right )\right ) b^{5}-2 \tan \left (d x +c \right ) a^{5}+2 \tan \left (d x +c \right ) a \,b^{4}}{2 \left (b +a \tan \left (d x +c \right )\right ) \left (a^{2}+b^{2}\right )^{2} b d}\) | \(268\) |
risch | \(\frac {i x b}{2 i b a +a^{2}-b^{2}}-\frac {x a}{2 i b a +a^{2}-b^{2}}-\frac {6 i b \,a^{2} x}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {2 i b^{3} x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {6 i b \,a^{2} c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 i b^{3} c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 i a^{3}}{\left (-i b +a \right ) d \left (i b +a \right )^{2} \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b \,{\mathrm e}^{2 i \left (d x +c \right )}-a +i b \right )}+\frac {2 i a \,b^{2}}{\left (-i b +a \right ) d \left (i b +a \right )^{2} \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b \,{\mathrm e}^{2 i \left (d x +c \right )}-a +i b \right )}+\frac {3 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {i b -a}{i b +a}\right ) a^{2}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {i b -a}{i b +a}\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(373\) |
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Time = 0.26 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.89 \[ \int \frac {a+b \tan (c+d x)}{(b+a \tan (c+d x))^2} \, dx=-\frac {2 \, a^{4} - 2 \, a^{2} b^{2} + 2 \, {\left (a^{3} b - 3 \, a b^{3}\right )} d x - {\left (3 \, a^{2} b^{2} - b^{4} + {\left (3 \, a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {a^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + b^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (a^{3} b - a b^{3} - {\left (a^{4} - 3 \, a^{2} b^{2}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d \tan \left (d x + c\right ) + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d\right )}} \]
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Result contains complex when optimal does not.
Time = 0.75 (sec) , antiderivative size = 1348, normalized size of antiderivative = 13.35 \[ \int \frac {a+b \tan (c+d x)}{(b+a \tan (c+d x))^2} \, dx=\text {Too large to display} \]
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Time = 0.30 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.59 \[ \int \frac {a+b \tan (c+d x)}{(b+a \tan (c+d x))^2} \, dx=-\frac {\frac {2 \, {\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (3 \, a^{2} b - b^{3}\right )} \log \left (a \tan \left (d x + c\right ) + b\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (a^{2} - b^{2}\right )}}{a^{2} b + b^{3} + {\left (a^{3} + a b^{2}\right )} \tan \left (d x + c\right )}}{2 \, d} \]
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Time = 0.42 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.97 \[ \int \frac {a+b \tan (c+d x)}{(b+a \tan (c+d x))^2} \, dx=-\frac {\frac {2 \, {\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (3 \, a^{3} b - a b^{3}\right )} \log \left ({\left | a \tan \left (d x + c\right ) + b \right |}\right )}{a^{5} + 2 \, a^{3} b^{2} + a b^{4}} + \frac {2 \, {\left (3 \, a^{3} b \tan \left (d x + c\right ) - a b^{3} \tan \left (d x + c\right ) + a^{4} + 3 \, a^{2} b^{2} - 2 \, b^{4}\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (a \tan \left (d x + c\right ) + b\right )}}}{2 \, d} \]
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Time = 7.81 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.50 \[ \int \frac {a+b \tan (c+d x)}{(b+a \tan (c+d x))^2} \, dx=\frac {b\,\ln \left (b+a\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (3\,a^2-b^2\right )}{d\,{\left (a^2+b^2\right )}^2}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (a-b\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^2\,1{}\mathrm {i}+2\,a\,b+b^2\,1{}\mathrm {i}\right )}-\frac {a^2-b^2}{d\,\left (a^2+b^2\right )\,\left (b+a\,\mathrm {tan}\left (c+d\,x\right )\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (a+b\,1{}\mathrm {i}\right )}{2\,d\,\left (a^2\,1{}\mathrm {i}+2\,a\,b-b^2\,1{}\mathrm {i}\right )} \]
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