Optimal. Leaf size=42 \[ \frac {\coth \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\coth ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]
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Rubi [A] time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {3767} \[ \frac {\coth \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\coth ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]
Antiderivative was successfully verified.
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Rule 3767
Rubi steps
\begin {align*} \int \frac {\text {csch}^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \text {csch}^4(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {i \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \coth \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ &=\frac {\coth \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\coth ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 56, normalized size = 1.33 \[ \frac {2 \coth \left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac {\coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}^2\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 272, normalized size = 6.48 \[ -\frac {8 \, {\left (\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + 2 \, \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )\right )}}{3 \, {\left (b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{5} + 5 \, b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{4} + b n \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{5} - 3 \, b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} + {\left (10 \, b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} - 3 \, b n\right )} \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} + 2 \, b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + {\left (10 \, b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} - 9 \, b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )\right )} \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + {\left (5 \, b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{4} - 9 \, b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + 4 \, b n\right )} \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 47, normalized size = 1.12 \[ -\frac {4 \, {\left (3 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} - 1\right )}}{3 \, {\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} - 1\right )}^{3} b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 36, normalized size = 0.86 \[ \frac {\left (\frac {2}{3}-\frac {\mathrm {csch}\left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{3}\right ) \coth \left (a +b \ln \left (c \,x^{n}\right )\right )}{n b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 92, normalized size = 2.19 \[ -\frac {4 \, {\left (3 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - 1\right )}}{3 \, {\left (b c^{6 \, b} n e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} - 3 \, b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 3 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - b n\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.45, size = 55, normalized size = 1.31 \[ \frac {4\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}\,\left ({\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}-3\right )}{3\,b\,n\,{\left ({\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}-1\right )}^3} \]
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 \[ \int \frac {\operatorname {csch}^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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