Integrand size = 17, antiderivative size = 43 \[ \int \frac {\csc ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\cot \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\cot ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]
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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 = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {3852} \[ \int \frac {\csc ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\cot ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac {\cot \left (a+b \log \left (c x^n\right )\right )}{b n} \]
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Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \csc ^4(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {\text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \\ & = -\frac {\cot \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\cot ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.30 \[ \int \frac {\csc ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2 \cot \left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac {\cot \left (a+b \log \left (c x^n\right )\right ) \csc ^2\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]
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Time = 4.48 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {\left (-\frac {2}{3}-\frac {{\csc \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{3}\right ) \cot \left (a +b \ln \left (c \,x^{n}\right )\right )}{n b}\) | \(36\) |
default | \(\frac {\left (-\frac {2}{3}-\frac {{\csc \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{3}\right ) \cot \left (a +b \ln \left (c \,x^{n}\right )\right )}{n b}\) | \(36\) |
parallelrisch | \(\frac {-{\cot \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{3}+{\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{3}+9 \tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )-9 \cot \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}{24 b n}\) | \(80\) |
risch | \(\frac {4 i \left (3 \left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}-1\right )}{3 b n {\left (\left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}-1\right )}^{3}}\) | \(225\) |
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Time = 0.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.65 \[ \int \frac {\csc ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 3 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{3 \, {\left (b n \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - b n\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} \]
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\[ \int \frac {\csc ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\csc ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 1332 vs. \(2 (41) = 82\).
Time = 0.24 (sec) , antiderivative size = 1332, normalized size of antiderivative = 30.98 \[ \int \frac {\csc ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Too large to display} \]
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\[ \int \frac {\csc ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\csc \left (b \log \left (c x^{n}\right ) + a\right )^{4}}{x} \,d x } \]
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Time = 38.37 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.14 \[ \int \frac {\csc ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {4\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}\,3{}\mathrm {i}-\mathrm {i}\right )}{3\,b\,n\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}-1\right )}^3} \]
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