Optimal. Leaf size=43 \[ \frac {\log \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\coth ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
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Rubi [A] time = 0.04, antiderivative size = 43, 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 = {3473, 3475} \[ \frac {\log \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\coth ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3475
Rubi steps
\begin {align*} \int \frac {\coth ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \coth ^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\coth ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {\operatorname {Subst}\left (\int \coth (a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\coth ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {\log \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 52, normalized size = 1.21 \[ -\frac {-2 \log \left (\tanh \left (a+b \log \left (c x^n\right )\right )\right )-2 \log \left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )+\coth ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 572, normalized size = 13.30 \[ -\frac {b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{4} \log \relax (x) + 4 \, b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) \log \relax (x) \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} + b n \log \relax (x) \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{4} - 2 \, {\left (b n \log \relax (x) - 1\right )} \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + b n \log \relax (x) + 2 \, {\left (3 \, b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} \log \relax (x) - b n \log \relax (x) + 1\right )} \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} - {\left (\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{4} + 4 \, \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} + \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} - 1\right )} \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} - 2 \, \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + 4 \, {\left (\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} - \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )\right )} \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + 1\right )} \log \left (\frac {2 \, \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) - \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )}\right ) + 4 \, {\left (b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} \log \relax (x) - {\left (b n \log \relax (x) - 1\right )} \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )\right )} \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{4} + 4 \, b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} + b n \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{4} - 2 \, b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + 2 \, {\left (3 \, b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} - b n\right )} \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + b n + 4 \, {\left (b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} - b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )\right )} \sinh \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 [B] time = 0.30, size = 127, normalized size = 2.95 \[ \frac {\log \left (\sqrt {-2 \, x^{2 \, b n} {\left | c \right |}^{2 \, b} \cos \left (\pi b \mathrm {sgn}\relax (c) - \pi b\right ) e^{\left (2 \, a\right )} + x^{4 \, b n} {\left | c \right |}^{4 \, b} e^{\left (4 \, a\right )} + 1}\right )}{b n} - \frac {3 \, c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} - 2 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 3}{2 \, {\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} - 1\right )}^{2} b n} - \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 67, normalized size = 1.56 \[ -\frac {\coth ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )}{2 b n}-\frac {\ln \left (\coth \left (a +b \ln \left (c \,x^{n}\right )\right )-1\right )}{2 n b}-\frac {\ln \left (\coth \left (a +b \ln \left (c \,x^{n}\right )\right )+1\right )}{2 n b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 330, normalized size = 7.67 \[ -\frac {4 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - 3}{4 \, {\left (b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} - 2 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} - \frac {3 \, {\left (2 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - 1\right )}}{4 \, {\left (b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} - 2 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} + \frac {2 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - 3}{4 \, {\left (b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} - 2 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} - \frac {3}{4 \, {\left (b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} - 2 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} + \frac {\log \left (\frac {{\left (c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )} + 1\right )} e^{\left (-a\right )}}{c^{b}}\right )}{b n} + \frac {\log \left (\frac {{\left (c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )} - 1\right )} e^{\left (-a\right )}}{c^{b}}\right )}{b n} - \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.23, size = 95, normalized size = 2.21 \[ \frac {2}{b\,n-b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}}-\ln \relax (x)-\frac {2}{b\,n-2\,b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+b\,n\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}}+\frac {\ln \left ({\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}-1\right )}{b\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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