Optimal. Leaf size=91 \[ \frac {1}{2 x}-\frac {2^{-2+\frac {1}{n}} e^{2 a} \left (-b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},-2 b x^n\right )}{n x}-\frac {2^{-2+\frac {1}{n}} e^{-2 a} \left (b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},2 b x^n\right )}{n x} \]
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Rubi [A]
time = 0.09, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5470, 5469,
2250} \begin {gather*} -\frac {e^{2 a} 2^{\frac {1}{n}-2} \left (-b x^n\right )^{\frac {1}{n}} \text {Gamma}\left (-\frac {1}{n},-2 b x^n\right )}{n x}-\frac {e^{-2 a} 2^{\frac {1}{n}-2} \left (b x^n\right )^{\frac {1}{n}} \text {Gamma}\left (-\frac {1}{n},2 b x^n\right )}{n x}+\frac {1}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2250
Rule 5469
Rule 5470
Rubi steps
\begin {align*} \int \frac {\sinh ^2\left (a+b x^n\right )}{x^2} \, dx &=\int \left (-\frac {1}{2 x^2}+\frac {\cosh \left (2 a+2 b x^n\right )}{2 x^2}\right ) \, dx\\ &=\frac {1}{2 x}+\frac {1}{2} \int \frac {\cosh \left (2 a+2 b x^n\right )}{x^2} \, dx\\ &=\frac {1}{2 x}+\frac {1}{4} \int \frac {e^{-2 a-2 b x^n}}{x^2} \, dx+\frac {1}{4} \int \frac {e^{2 a+2 b x^n}}{x^2} \, dx\\ &=\frac {1}{2 x}-\frac {2^{-2+\frac {1}{n}} e^{2 a} \left (-b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},-2 b x^n\right )}{n x}-\frac {2^{-2+\frac {1}{n}} e^{-2 a} \left (b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},2 b x^n\right )}{n x}\\ \end {align*}
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Mathematica [A]
time = 1.03, size = 79, normalized size = 0.87 \begin {gather*} -\frac {-2 n+2^{\frac {1}{n}} e^{2 a} \left (-b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},-2 b x^n\right )+2^{\frac {1}{n}} e^{-2 a} \left (b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},2 b x^n\right )}{4 n x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.56, size = 0, normalized size = 0.00 \[\int \frac {\sinh ^{2}\left (a +b \,x^{n}\right )}{x^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.09, size = 74, normalized size = 0.81 \begin {gather*} -\frac {\left (2 \, b x^{n}\right )^{\left (\frac {1}{n}\right )} e^{\left (-2 \, a\right )} \Gamma \left (-\frac {1}{n}, 2 \, b x^{n}\right )}{4 \, n x} - \frac {\left (-2 \, b x^{n}\right )^{\left (\frac {1}{n}\right )} e^{\left (2 \, a\right )} \Gamma \left (-\frac {1}{n}, -2 \, b x^{n}\right )}{4 \, n x} + \frac {1}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh ^{2}{\left (a + b x^{n} \right )}}{x^{2}}\, 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 = -0.01 \begin {gather*} \int \frac {{\mathrm {sinh}\left (a+b\,x^n\right )}^2}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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