Optimal. Leaf size=20 \[ -\frac {\tanh ^{-1}\left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
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Rubi [A] time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3770} \[ -\frac {\tanh ^{-1}\left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 3770
Rubi steps
\begin {align*} \int \frac {\csc \left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \csc (a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\tanh ^{-1}\left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ \end {align*}
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Mathematica [B] time = 0.06, size = 54, normalized size = 2.70 \[ \frac {\log \left (\sin \left (\frac {a}{2}+\frac {1}{2} b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\log \left (\cos \left (\frac {a}{2}+\frac {1}{2} b \log \left (c x^n\right )\right )\right )}{b n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 45, normalized size = 2.25 \[ -\frac {\log \left (\frac {1}{2} \, \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + \frac {1}{2}\right ) - \log \left (-\frac {1}{2} \, \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + \frac {1}{2}\right )}{2 \, b n} \]
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 \[ \int \frac {\csc \left (b \log \left (c x^{n}\right ) + a\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 33, normalized size = 1.65 \[ -\frac {\ln \left (\csc \left (a +b \ln \left (c \,x^{n}\right )\right )+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{n b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 32, normalized size = 1.60 \[ -\frac {\log \left (\cot \left (b \log \left (c x^{n}\right ) + a\right ) + \csc \left (b \log \left (c x^{n}\right ) + a\right )\right )}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.00, size = 68, normalized size = 3.40 \[ \frac {\ln \left (\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}\,2{}\mathrm {i}-2{}\mathrm {i}}{x}\right )}{b\,n}-\frac {\ln \left (\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}\,2{}\mathrm {i}+2{}\mathrm {i}}{x}\right )}{b\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.29, size = 49, normalized size = 2.45 \[ - \begin {cases} - \log {\relax (x )} \csc {\relax (a )} & \text {for}\: b = 0 \\- \log {\relax (x )} \csc {\left (a + b \log {\relax (c )} \right )} & \text {for}\: n = 0 \\\frac {\log {\left (\cot {\left (a + b \log {\left (c x^{n} \right )} \right )} + \csc {\left (a + b \log {\left (c x^{n} \right )} \right )} \right )}}{b n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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