Optimal. Leaf size=75 \[ -\frac {e^a \left (-b x^n\right )^{2/n} \Gamma \left (-\frac {2}{n},-b x^n\right )}{2 n x^2}+\frac {e^{-a} \left (b x^n\right )^{2/n} \Gamma \left (-\frac {2}{n},b x^n\right )}{2 n x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, 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 = {5468, 2250}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2250
Rule 5468
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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 72, normalized size = 0.96 \begin {gather*} \frac {\left (b x^n\right )^{2/n} \Gamma \left (-\frac {2}{n},b x^n\right ) (\cosh (a)-\sinh (a))-\left (-b x^n\right )^{2/n} \Gamma \left (-\frac {2}{n},-b x^n\right ) (\cosh (a)+\sinh (a))}{2 n x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
4.
time = 0.21, size = 77, normalized size = 1.03
method | result | size |
meijerg | \(-\frac {\hypergeom \left (\left [-\frac {1}{n}\right ], \left [\frac {1}{2}, 1-\frac {1}{n}\right ], \frac {x^{2 n} b^{2}}{4}\right ) \sinh \left (a \right )}{2 x^{2}}+\frac {x^{-2+n} b \hypergeom \left (\left [\frac {1}{2}-\frac {1}{n}\right ], \left [\frac {3}{2}, \frac {3}{2}-\frac {1}{n}\right ], \frac {x^{2 n} b^{2}}{4}\right ) \cosh \left (a \right )}{-2+n}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.09, size = 69, normalized size = 0.92 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh {\left (a + b x^{n} \right )}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {sinh}\left (a+b\,x^n\right )}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________