Optimal. Leaf size=75 \[ \frac {e^{-a} \left (b x^n\right )^{2/n} \Gamma \left (-\frac {2}{n},b x^n\right )}{2 n x^2}-\frac {e^a \left (-b x^n\right )^{2/n} \Gamma \left (-\frac {2}{n},-b x^n\right )}{2 n x^2} \]
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Rubi [A] time = 0.06, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5360, 2218} \[ \frac {e^{-a} \left (b x^n\right )^{2/n} \text {Gamma}\left (-\frac {2}{n},b x^n\right )}{2 n x^2}-\frac {e^a \left (-b x^n\right )^{2/n} \text {Gamma}\left (-\frac {2}{n},-b x^n\right )}{2 n x^2} \]
Antiderivative was successfully verified.
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Rule 2218
Rule 5360
Rubi steps
\begin {align*} \int \frac {\sinh \left (a+b x^n\right )}{x^3} \, dx &=-\left (\frac {1}{2} \int \frac {e^{-a-b x^n}}{x^3} \, dx\right )+\frac {1}{2} \int \frac {e^{a+b x^n}}{x^3} \, dx\\ &=-\frac {e^a \left (-b x^n\right )^{2/n} \Gamma \left (-\frac {2}{n},-b x^n\right )}{2 n x^2}+\frac {e^{-a} \left (b x^n\right )^{2/n} \Gamma \left (-\frac {2}{n},b x^n\right )}{2 n x^2}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 72, normalized size = 0.96 \[ \frac {(\cosh (a)-\sinh (a)) \left (b x^n\right )^{2/n} \Gamma \left (-\frac {2}{n},b x^n\right )-(\sinh (a)+\cosh (a)) \left (-b x^n\right )^{2/n} \Gamma \left (-\frac {2}{n},-b x^n\right )}{2 n x^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sinh \left (b x^{n} + a\right )}{x^{3}}, x\right ) \]
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 \[ \int \frac {\sinh \left (b x^{n} + a\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.05, size = 77, normalized size = 1.03 \[ -\frac {\hypergeom \left (\left [-\frac {1}{n}\right ], \left [\frac {1}{2}, 1-\frac {1}{n}\right ], \frac {x^{2 n} b^{2}}{4}\right ) \sinh \relax (a )}{2 x^{2}}+\frac {x^{-2+n} b \hypergeom \left (\left [\frac {1}{2}-\frac {1}{n}\right ], \left [\frac {3}{2}, \frac {3}{2}-\frac {1}{n}\right ], \frac {x^{2 n} b^{2}}{4}\right ) \cosh \relax (a )}{-2+n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 69, normalized size = 0.92 \[ \frac {\left (b x^{n}\right )^{\frac {2}{n}} e^{\left (-a\right )} \Gamma \left (-\frac {2}{n}, b x^{n}\right )}{2 \, n x^{2}} - \frac {\left (-b x^{n}\right )^{\frac {2}{n}} e^{a} \Gamma \left (-\frac {2}{n}, -b x^{n}\right )}{2 \, n x^{2}} \]
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 \[ \int \frac {\mathrm {sinh}\left (a+b\,x^n\right )}{x^3} \,d x \]
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 \[ \int \frac {\sinh {\left (a + b x^{n} \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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