Integrand size = 19, antiderivative size = 29 \[ \int \frac {\tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=-\log (x)+\frac {\tan \left (a d+b d \log \left (c x^n\right )\right )}{b d n} \]
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Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3554, 8} \[ \int \frac {\tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\tan \left (a d+b d \log \left (c x^n\right )\right )}{b d n}-\log (x) \]
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Rule 8
Rule 3554
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \tan ^2(d (a+b x)) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {\tan \left (a d+b d \log \left (c x^n\right )\right )}{b d n}-\frac {\text {Subst}\left (\int 1 \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\log (x)+\frac {\tan \left (a d+b d \log \left (c x^n\right )\right )}{b d n} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.76 \[ \int \frac {\tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=-\frac {\arctan \left (\tan \left (a d+b d \log \left (c x^n\right )\right )\right )}{b d n}+\frac {\tan \left (a d+b d \log \left (c x^n\right )\right )}{b d n} \]
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Time = 0.18 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21
method | result | size |
parallelrisch | \(\frac {-b d \ln \left (c \,x^{n}\right )+\tan \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{b d n}\) | \(35\) |
derivativedivides | \(\frac {\tan \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )-\arctan \left (\tan \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )}{n b d}\) | \(41\) |
default | \(\frac {\tan \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )-\arctan \left (\tan \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )}{n b d}\) | \(41\) |
risch | \(-\ln \left (x \right )+\frac {2 i}{d b n \left (\left (x^{n}\right )^{2 i b d} c^{2 i b d} {\mathrm e}^{d \left (-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )+b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 i a \right )}+1\right )}\) | \(127\) |
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Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (29) = 58\).
Time = 0.25 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.93 \[ \int \frac {\tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=-\frac {b d n \cos \left (2 \, b d n \log \left (x\right ) + 2 \, b d \log \left (c\right ) + 2 \, a d\right ) \log \left (x\right ) + b d n \log \left (x\right ) - \sin \left (2 \, b d n \log \left (x\right ) + 2 \, b d \log \left (c\right ) + 2 \, a d\right )}{b d n \cos \left (2 \, b d n \log \left (x\right ) + 2 \, b d \log \left (c\right ) + 2 \, a d\right ) + b d n} \]
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\[ \int \frac {\tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\int \frac {\tan ^{2}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 320 vs. \(2 (29) = 58\).
Time = 0.22 (sec) , antiderivative size = 320, normalized size of antiderivative = 11.03 \[ \int \frac {\tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=-\frac {{\left (b d \cos \left (2 \, b d \log \left (c\right )\right )^{2} + b d \sin \left (2 \, b d \log \left (c\right )\right )^{2}\right )} n \cos \left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )^{2} \log \left (x\right ) + {\left (b d \cos \left (2 \, b d \log \left (c\right )\right )^{2} + b d \sin \left (2 \, b d \log \left (c\right )\right )^{2}\right )} n \log \left (x\right ) \sin \left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )^{2} + b d n \log \left (x\right ) + 2 \, {\left (b d n \cos \left (2 \, b d \log \left (c\right )\right ) \log \left (x\right ) - \sin \left (2 \, b d \log \left (c\right )\right )\right )} \cos \left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right ) - 2 \, {\left (b d n \log \left (x\right ) \sin \left (2 \, b d \log \left (c\right )\right ) + \cos \left (2 \, b d \log \left (c\right )\right )\right )} \sin \left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )}{2 \, b d n \cos \left (2 \, b d \log \left (c\right )\right ) \cos \left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right ) - 2 \, b d n \sin \left (2 \, b d \log \left (c\right )\right ) \sin \left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right ) + {\left (b d \cos \left (2 \, b d \log \left (c\right )\right )^{2} + b d \sin \left (2 \, b d \log \left (c\right )\right )^{2}\right )} n \cos \left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )^{2} + {\left (b d \cos \left (2 \, b d \log \left (c\right )\right )^{2} + b d \sin \left (2 \, b d \log \left (c\right )\right )^{2}\right )} n \sin \left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )^{2} + b d n} \]
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Timed out. \[ \int \frac {\tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\text {Timed out} \]
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Time = 31.48 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {\tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=-\ln \left (x\right )+\frac {2{}\mathrm {i}}{b\,d\,n\,\left ({\mathrm {e}}^{a\,d\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,d\,2{}\mathrm {i}}+1\right )} \]
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