Optimal. Leaf size=176 \[ \frac {\text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}-\frac {\text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}+\frac {\log \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\cot \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}-\frac {\log \left (1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\cot \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n} \]
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Rubi [A]
time = 0.08, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {3557, 335,
217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {\text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}-\frac {\text {ArcTan}\left (\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\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 {\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3557
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {\cot (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx,x,\cot \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ &=-\frac {2 \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}\\ &=-\frac {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 {\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 {\text {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 {\text {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 {\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 [A]
time = 0.17, size = 142, normalized size = 0.81 \begin {gather*} \frac {2 \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )-2 \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )+\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 )-\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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 122, normalized size = 0.69
method | result | size |
derivativedivides | \(-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )+\sqrt {2}\, \left (\sqrt {\cot }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{1+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )-\sqrt {2}\, \left (\sqrt {\cot }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )\right )}{4 n b}\) | \(122\) |
default | \(-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )+\sqrt {2}\, \left (\sqrt {\cot }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{1+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )-\sqrt {2}\, \left (\sqrt {\cot }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )\right )}{4 n b}\) | \(122\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {1}{x \sqrt {\cot {\left (a + b \log {\left (c x^{n} \right )} \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.94, size = 57, normalized size = 0.32 \begin {gather*} \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 )\,1{}\mathrm {i}}{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 )\,1{}\mathrm {i}}{b\,n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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