Integrand size = 17, antiderivative size = 45 \[ \int \frac {\tan ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\log (x)-\frac {\tan \left (a+b \log \left (c x^n\right )\right )}{b n}+\frac {\tan ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]
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Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3554, 8} \[ \int \frac {\tan ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\tan ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac {\tan \left (a+b \log \left (c x^n\right )\right )}{b n}+\log (x) \]
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Rule 8
Rule 3554
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \tan ^4(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {\tan ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac {\text {Subst}\left (\int \tan ^2(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {\tan \left (a+b \log \left (c x^n\right )\right )}{b n}+\frac {\tan ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\text {Subst}\left (\int 1 \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \log (x)-\frac {\tan \left (a+b \log \left (c x^n\right )\right )}{b n}+\frac {\tan ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.38 \[ \int \frac {\tan ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\arctan \left (\tan \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\tan \left (a+b \log \left (c x^n\right )\right )}{b n}+\frac {\tan ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]
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Time = 0.42 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98
method | result | size |
parallelrisch | \(-\frac {-3 \ln \left (x \right ) b n -{\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}+3 \tan \left (a +b \ln \left (c \,x^{n}\right )\right )}{3 b n}\) | \(44\) |
derivativedivides | \(\frac {\frac {{\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}}{3}-\tan \left (a +b \ln \left (c \,x^{n}\right )\right )+\arctan \left (\tan \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{n b}\) | \(49\) |
default | \(\frac {\frac {{\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}}{3}-\tan \left (a +b \ln \left (c \,x^{n}\right )\right )+\arctan \left (\tan \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{n b}\) | \(49\) |
risch | \(\ln \left (x \right )-\frac {4 i \left (3 c^{4 i b} \left (x^{n}\right )^{4 i b} {\mathrm e}^{-2 b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{2 b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-2 b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{4 i a}+3 \left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}+2\right )}{3 b n {\left (\left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}+1\right )}^{3}}\) | \(335\) |
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Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (43) = 86\).
Time = 0.25 (sec) , antiderivative size = 140, normalized size of antiderivative = 3.11 \[ \int \frac {\tan ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {3 \, b n \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )^{2} \log \left (x\right ) + 6 \, b n \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) \log \left (x\right ) + 3 \, b n \log \left (x\right ) - 2 \, {\left (2 \, \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1\right )} \sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )}{3 \, {\left (b n \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )^{2} + 2 \, b n \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + b n\right )}} \]
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Time = 1.65 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.44 \[ \int \frac {\tan ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\begin {cases} \log {\left (x \right )} \tan ^{4}{\left (a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\left (x \right )} \tan ^{4}{\left (a + b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\frac {\log {\left (c x^{n} \right )}}{n} + \frac {\tan ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{3 b n} - \frac {\tan {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b n} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2171 vs. \(2 (43) = 86\).
Time = 0.27 (sec) , antiderivative size = 2171, normalized size of antiderivative = 48.24 \[ \int \frac {\tan ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\tan ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \]
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Time = 37.03 (sec) , antiderivative size = 183, normalized size of antiderivative = 4.07 \[ \int \frac {\tan ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\ln \left (x\right )-\frac {\frac {4{}\mathrm {i}}{3\,b\,n}+\frac {{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}\,4{}\mathrm {i}}{3\,b\,n}}{3\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+3\,{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}+{\mathrm {e}}^{a\,6{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,6{}\mathrm {i}}+1}-\frac {4{}\mathrm {i}}{3\,b\,n\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+1\right )}-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}\,4{}\mathrm {i}}{3\,b\,n\,\left (2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}+1\right )} \]
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