Optimal. Leaf size=199 \[ \frac {\log \left (\cot \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2} b n}-\frac {\log \left (\cot \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2} b n}+\frac {2}{b n \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}-\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}+\frac {\tan ^{-1}\left (\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2} b n} \]
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Rubi [A] time = 0.13, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {3474, 3476, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac {\log \left (\cot \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2} b n}-\frac {\log \left (\cot \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2} b n}+\frac {2}{b n \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}-\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}+\frac {\tan ^{-1}\left (\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2} b n} \]
Antiderivative was successfully verified.
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Rule 204
Rule 297
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 3474
Rule 3476
Rubi steps
\begin {align*} \int \frac {1}{x \cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\cot ^{\frac {3}{2}}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {2}{b n \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}-\frac {\operatorname {Subst}\left (\int \sqrt {\cot (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {2}{b n \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\cot \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ &=\frac {2}{b n \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}+\frac {2 \operatorname {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}\\ &=\frac {2}{b n \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}-\frac {\operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}\\ &=\frac {2}{b n \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{2 b n}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{2 b n}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{2 \sqrt {2} b n}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{2 \sqrt {2} b n}\\ &=\frac {2}{b n \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}+\frac {\log \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\cot \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}-\frac {\log \left (1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\cot \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}\\ &=-\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}+\frac {\tan ^{-1}\left (1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}+\frac {2}{b n \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}+\frac {\log \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\cot \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}-\frac {\log \left (1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\cot \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}\\ \end {align*}
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Mathematica [C] time = 0.14, size = 46, normalized size = 0.23 \[ \frac {2 \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\cot ^2\left (a+b \log \left (c x^n\right )\right )\right )}{b n \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 161, normalized size = 0.81 \[ \frac {\arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right ) \sqrt {2}}{2 b n}+\frac {\arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right ) \sqrt {2}}{2 b n}+\frac {\sqrt {2}\, \ln \left (\frac {1+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )-\sqrt {2}\, \left (\sqrt {\cot }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{1+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )+\sqrt {2}\, \left (\sqrt {\cot }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}\right )}{4 b n}+\frac {2}{b n \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \cot \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.95, size = 79, normalized size = 0.40 \[ \frac {2}{b\,n\,\sqrt {\mathrm {cot}\left (a+b\,\ln \left (c\,x^n\right )\right )}}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {cot}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {cot}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \cot ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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