Integrand size = 16, antiderivative size = 94 \[ \int \frac {x}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\frac {\left (a^2-b^2\right ) x^2}{2 \left (a^2+b^2\right )^2}+\frac {a b \log \left (a \cos \left (c+d x^2\right )+b \sin \left (c+d x^2\right )\right )}{\left (a^2+b^2\right )^2 d}-\frac {b}{2 \left (a^2+b^2\right ) d \left (a+b \tan \left (c+d x^2\right )\right )} \]
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Time = 0.15 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3832, 3564, 3612, 3611} \[ \int \frac {x}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=-\frac {b}{2 d \left (a^2+b^2\right ) \left (a+b \tan \left (c+d x^2\right )\right )}+\frac {a b \log \left (a \cos \left (c+d x^2\right )+b \sin \left (c+d x^2\right )\right )}{d \left (a^2+b^2\right )^2}+\frac {x^2 \left (a^2-b^2\right )}{2 \left (a^2+b^2\right )^2} \]
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Rule 3564
Rule 3611
Rule 3612
Rule 3832
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{(a+b \tan (c+d x))^2} \, dx,x,x^2\right ) \\ & = -\frac {b}{2 \left (a^2+b^2\right ) d \left (a+b \tan \left (c+d x^2\right )\right )}+\frac {\text {Subst}\left (\int \frac {a-b \tan (c+d x)}{a+b \tan (c+d x)} \, dx,x,x^2\right )}{2 \left (a^2+b^2\right )} \\ & = \frac {\left (a^2-b^2\right ) x^2}{2 \left (a^2+b^2\right )^2}-\frac {b}{2 \left (a^2+b^2\right ) d \left (a+b \tan \left (c+d x^2\right )\right )}+\frac {(a b) \text {Subst}\left (\int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx,x,x^2\right )}{\left (a^2+b^2\right )^2} \\ & = \frac {\left (a^2-b^2\right ) x^2}{2 \left (a^2+b^2\right )^2}+\frac {a b \log \left (a \cos \left (c+d x^2\right )+b \sin \left (c+d x^2\right )\right )}{\left (a^2+b^2\right )^2 d}-\frac {b}{2 \left (a^2+b^2\right ) d \left (a+b \tan \left (c+d x^2\right )\right )} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.99 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.21 \[ \int \frac {x}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\frac {-\frac {i \log \left (i-\tan \left (c+d x^2\right )\right )}{(a+i b)^2}+\frac {i \log \left (i+\tan \left (c+d x^2\right )\right )}{(a-i b)^2}+\frac {2 b \left (2 a \log \left (a+b \tan \left (c+d x^2\right )\right )-\frac {a^2+b^2}{a+b \tan \left (c+d x^2\right )}\right )}{\left (a^2+b^2\right )^2}}{4 d} \]
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Time = 0.15 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.13
method | result | size |
derivativedivides | \(\frac {-\frac {b}{\left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d \,x^{2}+c \right )\right )}+\frac {2 a b \ln \left (a +b \tan \left (d \,x^{2}+c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {-a b \ln \left (1+\tan ^{2}\left (d \,x^{2}+c \right )\right )+\left (a^{2}-b^{2}\right ) \arctan \left (\tan \left (d \,x^{2}+c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{2 d}\) | \(106\) |
default | \(\frac {-\frac {b}{\left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d \,x^{2}+c \right )\right )}+\frac {2 a b \ln \left (a +b \tan \left (d \,x^{2}+c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {-a b \ln \left (1+\tan ^{2}\left (d \,x^{2}+c \right )\right )+\left (a^{2}-b^{2}\right ) \arctan \left (\tan \left (d \,x^{2}+c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{2 d}\) | \(106\) |
norman | \(\frac {\frac {\left (a^{2}-b^{2}\right ) a \,x^{2}}{2 a^{4}+4 a^{2} b^{2}+2 b^{4}}+\frac {b \left (a^{2}-b^{2}\right ) x^{2} \tan \left (d \,x^{2}+c \right )}{2 a^{4}+4 a^{2} b^{2}+2 b^{4}}+\frac {b^{2} \tan \left (d \,x^{2}+c \right )}{2 a \left (a^{2}+b^{2}\right ) d}}{a +b \tan \left (d \,x^{2}+c \right )}+\frac {a b \ln \left (a +b \tan \left (d \,x^{2}+c \right )\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {a b \ln \left (1+\tan ^{2}\left (d \,x^{2}+c \right )\right )}{2 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(191\) |
risch | \(-\frac {x^{2}}{2 \left (2 i a b -a^{2}+b^{2}\right )}-\frac {2 i a b \,x^{2}}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {2 i a b c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {i b^{2}}{\left (-i a +b \right ) d \left (i a +b \right )^{2} \left ({\mathrm e}^{2 i \left (d \,x^{2}+c \right )} b +i a \,{\mathrm e}^{2 i \left (d \,x^{2}+c \right )}-b +i a \right )}+\frac {a b \ln \left ({\mathrm e}^{2 i \left (d \,x^{2}+c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(191\) |
parallelrisch | \(-\frac {-x^{2} \tan \left (d \,x^{2}+c \right ) a^{2} b^{2} d +x^{2} \tan \left (d \,x^{2}+c \right ) b^{4} d -x^{2} a^{3} b d +x^{2} a \,b^{3} d +\ln \left (1+\tan ^{2}\left (d \,x^{2}+c \right )\right ) \tan \left (d \,x^{2}+c \right ) a \,b^{3}-2 \ln \left (a +b \tan \left (d \,x^{2}+c \right )\right ) \tan \left (d \,x^{2}+c \right ) a \,b^{3}+\ln \left (1+\tan ^{2}\left (d \,x^{2}+c \right )\right ) a^{2} b^{2}-2 \ln \left (a +b \tan \left (d \,x^{2}+c \right )\right ) a^{2} b^{2}+a^{2} b^{2}+b^{4}}{2 \left (a +b \tan \left (d \,x^{2}+c \right )\right ) \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b d}\) | \(200\) |
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Time = 0.25 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.80 \[ \int \frac {x}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\frac {{\left (a^{3} - a b^{2}\right )} d x^{2} - b^{3} + {\left (a b^{2} \tan \left (d x^{2} + c\right ) + a^{2} b\right )} \log \left (\frac {b^{2} \tan \left (d x^{2} + c\right )^{2} + 2 \, a b \tan \left (d x^{2} + c\right ) + a^{2}}{\tan \left (d x^{2} + c\right )^{2} + 1}\right ) + {\left ({\left (a^{2} b - b^{3}\right )} d x^{2} + a b^{2}\right )} \tan \left (d x^{2} + c\right )}{2 \, {\left ({\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d \tan \left (d x^{2} + c\right ) + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d\right )}} \]
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Result contains complex when optimal does not.
Time = 0.78 (sec) , antiderivative size = 1584, normalized size of antiderivative = 16.85 \[ \int \frac {x}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 556 vs. \(2 (90) = 180\).
Time = 0.37 (sec) , antiderivative size = 556, normalized size of antiderivative = 5.91 \[ \int \frac {x}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\frac {{\left (a^{4} - b^{4}\right )} d x^{2} \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + {\left (a^{4} - b^{4}\right )} d x^{2} \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + {\left (a^{4} - b^{4}\right )} d x^{2} - 2 \, {\left (2 \, a b^{3} - {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} d x^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right ) + {\left (4 \, a^{2} b^{2} \sin \left (2 \, d x^{2} + 2 \, c\right ) + a^{3} b + a b^{3} + {\left (a^{3} b + a b^{3}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + {\left (a^{3} b + a b^{3}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right )\right )} \log \left (\frac {{\left (a^{2} + b^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + 4 \, a b \sin \left (2 \, d x^{2} + 2 \, c\right ) + {\left (a^{2} + b^{2}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + a^{2} + b^{2} + 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right )}{{\left (a^{2} + b^{2}\right )} \cos \left (2 \, c\right )^{2} + {\left (a^{2} + b^{2}\right )} \sin \left (2 \, c\right )^{2}}\right ) + 2 \, {\left (a^{2} b^{2} - b^{4} + 2 \, {\left (a^{3} b - a b^{3}\right )} d x^{2}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )}{2 \, {\left ({\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} d \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} d \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, {\left (a^{6} + a^{4} b^{2} - a^{2} b^{4} - b^{6}\right )} d \cos \left (2 \, d x^{2} + 2 \, c\right ) + 4 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} d \sin \left (2 \, d x^{2} + 2 \, c\right ) + {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} d\right )}} \]
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Time = 0.36 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.69 \[ \int \frac {x}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\frac {a b^{2} \log \left ({\left | b \tan \left (d x^{2} + c\right ) + a \right |}\right )}{a^{4} b d + 2 \, a^{2} b^{3} d + b^{5} d} - \frac {a b \log \left (\tan \left (d x^{2} + c\right )^{2} + 1\right )}{2 \, {\left (a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right )}} + \frac {{\left (d x^{2} + c\right )} {\left (a^{2} - b^{2}\right )}}{2 \, {\left (a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right )}} - \frac {a^{2} b + b^{3}}{2 \, {\left (a^{2} + b^{2}\right )}^{2} {\left (b \tan \left (d x^{2} + c\right ) + a\right )} d} \]
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Time = 3.86 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.84 \[ \int \frac {x}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\frac {\frac {x^2\,\mathrm {tan}\left (d\,x^2+c\right )\,\left (\frac {a^2\,b}{2}-\frac {b^3}{2}\right )}{{\left (a^2+b^2\right )}^2}-\frac {x^2\,\left (\frac {a\,b^2}{2}-\frac {a^3}{2}\right )}{{\left (a^2+b^2\right )}^2}+\frac {b^2\,\mathrm {tan}\left (d\,x^2+c\right )}{2\,a\,d\,\left (a^2+b^2\right )}}{a+b\,\mathrm {tan}\left (d\,x^2+c\right )}-\frac {a\,b\,\ln \left ({\mathrm {tan}\left (d\,x^2+c\right )}^2+1\right )}{2\,\left (d\,a^4+2\,d\,a^2\,b^2+d\,b^4\right )}+\frac {a\,b\,\ln \left (a+b\,\mathrm {tan}\left (d\,x^2+c\right )\right )}{d\,{\left (a^2+b^2\right )}^2} \]
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