Optimal. Leaf size=43 \[ \frac {\cosh (2 a) \text {Chi}\left (2 b x^n\right )}{2 n}-\frac {\log (x)}{2}+\frac {\sinh (2 a) \text {Shi}\left (2 b x^n\right )}{2 n} \]
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Rubi [A]
time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5470, 5427,
5425, 5424} \begin {gather*} \frac {\cosh (2 a) \text {Chi}\left (2 b x^n\right )}{2 n}+\frac {\sinh (2 a) \text {Shi}\left (2 b x^n\right )}{2 n}-\frac {\log (x)}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 5424
Rule 5425
Rule 5427
Rule 5470
Rubi steps
\begin {align*} \int \frac {\sinh ^2\left (a+b x^n\right )}{x} \, dx &=\int \left (-\frac {1}{2 x}+\frac {\cosh \left (2 a+2 b x^n\right )}{2 x}\right ) \, dx\\ &=-\frac {\log (x)}{2}+\frac {1}{2} \int \frac {\cosh \left (2 a+2 b x^n\right )}{x} \, dx\\ &=-\frac {\log (x)}{2}+\frac {1}{2} \cosh (2 a) \int \frac {\cosh \left (2 b x^n\right )}{x} \, dx+\frac {1}{2} \sinh (2 a) \int \frac {\sinh \left (2 b x^n\right )}{x} \, dx\\ &=\frac {\cosh (2 a) \text {Chi}\left (2 b x^n\right )}{2 n}-\frac {\log (x)}{2}+\frac {\sinh (2 a) \text {Shi}\left (2 b x^n\right )}{2 n}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 39, normalized size = 0.91 \begin {gather*} -\frac {\log (x)}{2}+\frac {\cosh (2 a) \text {Chi}\left (2 b x^n\right )+\sinh (2 a) \text {Shi}\left (2 b x^n\right )}{2 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 14.08, size = 40, normalized size = 0.93
method | result | size |
risch | \(-\frac {\ln \left (x \right )}{2}-\frac {{\mathrm e}^{-2 a} \expIntegral \left (1, 2 b \,x^{n}\right )}{4 n}-\frac {{\mathrm e}^{2 a} \expIntegral \left (1, -2 b \,x^{n}\right )}{4 n}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 37, normalized size = 0.86 \begin {gather*} \frac {{\rm Ei}\left (2 \, b x^{n}\right ) e^{\left (2 \, a\right )}}{4 \, n} + \frac {{\rm Ei}\left (-2 \, b x^{n}\right ) e^{\left (-2 \, a\right )}}{4 \, n} - \frac {1}{2} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.54, size = 69, normalized size = 1.60 \begin {gather*} \frac {{\left (\cosh \left (2 \, a\right ) + \sinh \left (2 \, a\right )\right )} {\rm Ei}\left (2 \, b \cosh \left (n \log \left (x\right )\right ) + 2 \, b \sinh \left (n \log \left (x\right )\right )\right ) + {\left (\cosh \left (2 \, a\right ) - \sinh \left (2 \, a\right )\right )} {\rm Ei}\left (-2 \, b \cosh \left (n \log \left (x\right )\right ) - 2 \, b \sinh \left (n \log \left (x\right )\right )\right ) - 2 \, n \log \left (x\right )}{4 \, n} \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 {\sinh ^{2}{\left (a + b x^{n} \right )}}{x}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\mathrm {sinh}\left (a+b\,x^n\right )}^2}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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