Optimal. Leaf size=95 \[ \frac {e^{-a} \left (b x^2\right )^{\frac {1}{2} (-m-1)} (e x)^{m+1} \Gamma \left (\frac {m+1}{2},b x^2\right )}{4 e}-\frac {e^a \left (-b x^2\right )^{\frac {1}{2} (-m-1)} (e x)^{m+1} \Gamma \left (\frac {m+1}{2},-b x^2\right )}{4 e} \]
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Rubi [A] time = 0.07, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5328, 2218} \[ \frac {e^{-a} \left (b x^2\right )^{\frac {1}{2} (-m-1)} (e x)^{m+1} \text {Gamma}\left (\frac {m+1}{2},b x^2\right )}{4 e}-\frac {e^a \left (-b x^2\right )^{\frac {1}{2} (-m-1)} (e x)^{m+1} \text {Gamma}\left (\frac {m+1}{2},-b x^2\right )}{4 e} \]
Antiderivative was successfully verified.
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Rule 2218
Rule 5328
Rubi steps
\begin {align*} \int (e x)^m \sinh \left (a+b x^2\right ) \, dx &=-\left (\frac {1}{2} \int e^{-a-b x^2} (e x)^m \, dx\right )+\frac {1}{2} \int e^{a+b x^2} (e x)^m \, dx\\ &=-\frac {e^a (e x)^{1+m} \left (-b x^2\right )^{\frac {1}{2} (-1-m)} \Gamma \left (\frac {1+m}{2},-b x^2\right )}{4 e}+\frac {e^{-a} (e x)^{1+m} \left (b x^2\right )^{\frac {1}{2} (-1-m)} \Gamma \left (\frac {1+m}{2},b x^2\right )}{4 e}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 98, normalized size = 1.03 \[ -\frac {1}{4} x \left (-b^2 x^4\right )^{\frac {1}{2} (-m-1)} (e x)^m \left ((\sinh (a)+\cosh (a)) \left (b x^2\right )^{\frac {m+1}{2}} \Gamma \left (\frac {m+1}{2},-b x^2\right )-(\cosh (a)-\sinh (a)) \left (-b x^2\right )^{\frac {m+1}{2}} \Gamma \left (\frac {m+1}{2},b x^2\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 124, normalized size = 1.31 \[ \frac {e \cosh \left (\frac {1}{2} \, {\left (m - 1\right )} \log \left (\frac {b}{e^{2}}\right ) + a\right ) \Gamma \left (\frac {1}{2} \, m + \frac {1}{2}, b x^{2}\right ) + e \cosh \left (\frac {1}{2} \, {\left (m - 1\right )} \log \left (-\frac {b}{e^{2}}\right ) - a\right ) \Gamma \left (\frac {1}{2} \, m + \frac {1}{2}, -b x^{2}\right ) - e \Gamma \left (\frac {1}{2} \, m + \frac {1}{2}, b x^{2}\right ) \sinh \left (\frac {1}{2} \, {\left (m - 1\right )} \log \left (\frac {b}{e^{2}}\right ) + a\right ) - e \Gamma \left (\frac {1}{2} \, m + \frac {1}{2}, -b x^{2}\right ) \sinh \left (\frac {1}{2} \, {\left (m - 1\right )} \log \left (-\frac {b}{e^{2}}\right ) - a\right )}{4 \, b} \]
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 \[ \int \left (e x\right )^{m} \sinh \left (b x^{2} + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.10, size = 77, normalized size = 0.81 \[ \frac {\left (e x \right )^{m} x \hypergeom \left (\left [\frac {m}{4}+\frac {1}{4}\right ], \left [\frac {1}{2}, \frac {5}{4}+\frac {m}{4}\right ], \frac {x^{4} b^{2}}{4}\right ) \sinh \relax (a )}{1+m}+\frac {\left (e x \right )^{m} b \,x^{3} \hypergeom \left (\left [\frac {3}{4}+\frac {m}{4}\right ], \left [\frac {3}{2}, \frac {7}{4}+\frac {m}{4}\right ], \frac {x^{4} b^{2}}{4}\right ) \cosh \relax (a )}{3+m} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{m} \sinh \left (b x^{2} + a\right )\,{d x} \]
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 \[ \int \mathrm {sinh}\left (b\,x^2+a\right )\,{\left (e\,x\right )}^m \,d x \]
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 \[ \int \left (e x\right )^{m} \sinh {\left (a + b x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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