Optimal. Leaf size=71 \[ -\frac {e^a \left (-b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},-b x^n\right )}{2 n x}+\frac {e^{-a} \left (b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},b x^n\right )}{2 n x} \]
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Rubi [A]
time = 0.05, antiderivative size = 71, 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 = {5468, 2250}
\begin {gather*} \frac {e^{-a} \left (b x^n\right )^{\frac {1}{n}} \text {Gamma}\left (-\frac {1}{n},b x^n\right )}{2 n x}-\frac {e^a \left (-b x^n\right )^{\frac {1}{n}} \text {Gamma}\left (-\frac {1}{n},-b x^n\right )}{2 n x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2250
Rule 5468
Rubi steps
\begin {align*} \int \frac {\sinh \left (a+b x^n\right )}{x^2} \, dx &=-\left (\frac {1}{2} \int \frac {e^{-a-b x^n}}{x^2} \, dx\right )+\frac {1}{2} \int \frac {e^{a+b x^n}}{x^2} \, dx\\ &=-\frac {e^a \left (-b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},-b x^n\right )}{2 n x}+\frac {e^{-a} \left (b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},b x^n\right )}{2 n x}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 68, normalized size = 0.96 \begin {gather*} \frac {\left (b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},b x^n\right ) (\cosh (a)-\sinh (a))-\left (-b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},-b x^n\right ) (\cosh (a)+\sinh (a))}{2 n x} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
4.
time = 0.30, size = 77, normalized size = 1.08
method | result | size |
meijerg | \(-\frac {\hypergeom \left (\left [-\frac {1}{2 n}\right ], \left [\frac {1}{2}, 1-\frac {1}{2 n}\right ], \frac {x^{2 n} b^{2}}{4}\right ) \sinh \left (a \right )}{x}+\frac {x^{-1+n} b \hypergeom \left (\left [\frac {1}{2}-\frac {1}{2 n}\right ], \left [\frac {3}{2}, \frac {3}{2}-\frac {1}{2 n}\right ], \frac {x^{2 n} b^{2}}{4}\right ) \cosh \left (a \right )}{-1+n}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.08, size = 65, normalized size = 0.92 \begin {gather*} \frac {\left (b x^{n}\right )^{\left (\frac {1}{n}\right )} e^{\left (-a\right )} \Gamma \left (-\frac {1}{n}, b x^{n}\right )}{2 \, n x} - \frac {\left (-b x^{n}\right )^{\left (\frac {1}{n}\right )} e^{a} \Gamma \left (-\frac {1}{n}, -b x^{n}\right )}{2 \, n 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 {\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 )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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