Integrand size = 17, antiderivative size = 26 \[ \int \frac {\tan \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=-\frac {\log \left (\cos \left (a d+b d \log \left (c x^n\right )\right )\right )}{b d n} \]
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Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {3556} \[ \int \frac {\tan \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=-\frac {\log \left (\cos \left (a d+b d \log \left (c x^n\right )\right )\right )}{b d n} \]
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Rule 3556
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \tan (d (a+b x)) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {\log \left (\cos \left (a d+b d \log \left (c x^n\right )\right )\right )}{b d n} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {\tan \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=-\frac {\log \left (\cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )}{b d n} \]
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Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15
method | result | size |
derivativedivides | \(\frac {\ln \left (1+{\tan \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}^{2}\right )}{2 n b d}\) | \(30\) |
default | \(\frac {\ln \left (1+{\tan \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}^{2}\right )}{2 n b d}\) | \(30\) |
parallelrisch | \(\frac {\ln \left (1+{\tan \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}^{2}\right )}{2 n b d}\) | \(30\) |
risch | \(-i \ln \left (x \right )+\frac {2 i a}{n b}+\frac {2 i \ln \left (c \right )}{n}+\frac {2 i \ln \left (x^{n}\right )}{n}-\frac {\pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{n}+\frac {\pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{n}+\frac {\pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{n}-\frac {\pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{n}-\frac {\ln \left (\left (x^{n}\right )^{2 i b d} c^{2 i b d} {\mathrm e}^{d \left (-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )+b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 i a \right )}+1\right )}{b d n}\) | \(238\) |
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Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.35 \[ \int \frac {\tan \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=-\frac {\log \left (\frac {1}{2} \, \cos \left (2 \, b d n \log \left (x\right ) + 2 \, b d \log \left (c\right ) + 2 \, a d\right ) + \frac {1}{2}\right )}{2 \, b d n} \]
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Time = 1.40 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {\tan \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\begin {cases} \log {\left (x \right )} \tan {\left (a d \right )} & \text {for}\: b = 0 \\0 & \text {for}\: d = 0 \\\log {\left (x \right )} \tan {\left (a d + b d \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\- \frac {\log {\left (\cos {\left (a d + b d \log {\left (c x^{n} \right )} \right )} \right )}}{b d n} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {\tan \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\log \left (\sec \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\right )}{b d n} \]
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Timed out. \[ \int \frac {\tan \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\text {Timed out} \]
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Time = 29.34 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46 \[ \int \frac {\tan \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\ln \left (x\right )\,1{}\mathrm {i}-\frac {\ln \left ({\mathrm {e}}^{a\,d\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,d\,2{}\mathrm {i}}+1\right )}{b\,d\,n} \]
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