Optimal. Leaf size=19 \[ -\frac {\cot \left (a+b \log \left (c x^n\right )\right )}{b n} \]
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Rubi [A] time = 0.03, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3767, 8} \[ -\frac {\cot \left (a+b \log \left (c x^n\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rubi steps
\begin {align*} \int \frac {\csc ^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \csc ^2(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\operatorname {Subst}\left (\int 1 \, 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}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 19, normalized size = 1.00 \[ -\frac {\cot \left (a+b \log \left (c x^n\right )\right )}{b n} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.31, size = 34, normalized size = 1.79 \[ -\frac {\cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{b n \sin \left (b n \log \relax (x) + b \log \relax (c) + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc \left (b \log \left (c x^{n}\right ) + a\right )^{2}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 20, normalized size = 1.05 \[ -\frac {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.63, size = 168, normalized size = 8.84 \[ \frac {2 \, {\left (\cos \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) \sin \left (2 \, b \log \relax (c)\right ) + \cos \left (2 \, b \log \relax (c)\right ) \sin \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )\right )}}{2 \, b n \cos \left (2 \, b \log \relax (c)\right ) \cos \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) - {\left (b \cos \left (2 \, b \log \relax (c)\right )^{2} + b \sin \left (2 \, b \log \relax (c)\right )^{2}\right )} n \cos \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )^{2} - 2 \, b n \sin \left (2 \, b \log \relax (c)\right ) \sin \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) - {\left (b \cos \left (2 \, b \log \relax (c)\right )^{2} + b \sin \left (2 \, b \log \relax (c)\right )^{2}\right )} n \sin \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )^{2} - b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.90, size = 29, normalized size = 1.53 \[ -\frac {2{}\mathrm {i}}{b\,n\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}-1\right )} \]
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 \[ \int \frac {\csc ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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