Integrand size = 17, antiderivative size = 43 \[ \int \frac {\tan ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\log \left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{b n}+\frac {\tan ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
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Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3554, 3556} \[ \int \frac {\tan ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\tan ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {\log \left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
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Rule 3554
Rule 3556
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \tan ^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {\tan ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}-\frac {\text {Subst}\left (\int \tan (a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {\log \left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{b n}+\frac {\tan ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.88 \[ \int \frac {\tan ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2 \log \left (\cos \left (a+b \log \left (c x^n\right )\right )\right )+\tan ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
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Time = 0.29 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.95
method | result | size |
parallelrisch | \(\frac {{\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}-\ln \left (1+{\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}\right )}{2 b n}\) | \(41\) |
derivativedivides | \(\frac {\frac {{\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{2}-\frac {\ln \left (1+{\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}\right )}{2}}{n b}\) | \(42\) |
default | \(\frac {\frac {{\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{2}-\frac {\ln \left (1+{\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}\right )}{2}}{n b}\) | \(42\) |
risch | \(i \ln \left (x \right )-\frac {2 i a}{n b}-\frac {2 i \ln \left (c \right )}{n}-\frac {2 i \ln \left (x^{n}\right )}{n}+\frac {\pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{n}-\frac {\pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{n}-\frac {\pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{n}+\frac {\pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{n}+\frac {2 \left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}}{b n {\left (\left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}+1\right )}^{2}}+\frac {\ln \left (\left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}+1\right )}{b n}\) | \(452\) |
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Time = 0.26 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.60 \[ \int \frac {\tan ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {{\left (\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + \frac {1}{2}\right ) + 2}{2 \, {\left (b n \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + b n\right )}} \]
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Time = 0.78 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.47 \[ \int \frac {\tan ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\begin {cases} \log {\left (x \right )} \tan ^{3}{\left (a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\left (x \right )} \tan ^{3}{\left (a + b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\- \frac {\log {\left (\tan ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} + 1 \right )}}{2 b n} + \frac {\tan ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{2 b n} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1242 vs. \(2 (41) = 82\).
Time = 0.24 (sec) , antiderivative size = 1242, normalized size of antiderivative = 28.88 \[ \int \frac {\tan ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\tan ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \]
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Time = 32.07 (sec) , antiderivative size = 105, normalized size of antiderivative = 2.44 \[ \int \frac {\tan ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\ln \left (x\right )\,1{}\mathrm {i}-\frac {2}{b\,n\,\left (2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}+1\right )}+\frac {2}{b\,n\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+1\right )}+\frac {\ln \left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+1\right )}{b\,n} \]
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