Optimal. Leaf size=214 \[ -\frac {e^{3 a} 3^{-\frac {m}{2}-\frac {1}{2}} \left (-b x^2\right )^{\frac {1}{2} (-m-1)} (e x)^{m+1} \Gamma \left (\frac {m+1}{2},-3 b x^2\right )}{16 e}+\frac {3 e^a \left (-b x^2\right )^{\frac {1}{2} (-m-1)} (e x)^{m+1} \Gamma \left (\frac {m+1}{2},-b x^2\right )}{16 e}-\frac {3 e^{-a} \left (b x^2\right )^{\frac {1}{2} (-m-1)} (e x)^{m+1} \Gamma \left (\frac {m+1}{2},b x^2\right )}{16 e}+\frac {e^{-3 a} 3^{-\frac {m}{2}-\frac {1}{2}} \left (b x^2\right )^{\frac {1}{2} (-m-1)} (e x)^{m+1} \Gamma \left (\frac {m+1}{2},3 b x^2\right )}{16 e} \]
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Rubi [A] time = 0.20, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5340, 5328, 2218} \[ -\frac {e^{3 a} 3^{-\frac {m}{2}-\frac {1}{2}} \left (-b x^2\right )^{\frac {1}{2} (-m-1)} (e x)^{m+1} \text {Gamma}\left (\frac {m+1}{2},-3 b x^2\right )}{16 e}+\frac {3 e^a \left (-b x^2\right )^{\frac {1}{2} (-m-1)} (e x)^{m+1} \text {Gamma}\left (\frac {m+1}{2},-b x^2\right )}{16 e}-\frac {3 e^{-a} \left (b x^2\right )^{\frac {1}{2} (-m-1)} (e x)^{m+1} \text {Gamma}\left (\frac {m+1}{2},b x^2\right )}{16 e}+\frac {e^{-3 a} 3^{-\frac {m}{2}-\frac {1}{2}} \left (b x^2\right )^{\frac {1}{2} (-m-1)} (e x)^{m+1} \text {Gamma}\left (\frac {m+1}{2},3 b x^2\right )}{16 e} \]
Antiderivative was successfully verified.
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Rule 2218
Rule 5328
Rule 5340
Rubi steps
\begin {align*} \int (e x)^m \sinh ^3\left (a+b x^2\right ) \, dx &=\int \left (-\frac {3}{4} (e x)^m \sinh \left (a+b x^2\right )+\frac {1}{4} (e x)^m \sinh \left (3 a+3 b x^2\right )\right ) \, dx\\ &=\frac {1}{4} \int (e x)^m \sinh \left (3 a+3 b x^2\right ) \, dx-\frac {3}{4} \int (e x)^m \sinh \left (a+b x^2\right ) \, dx\\ &=-\left (\frac {1}{8} \int e^{-3 a-3 b x^2} (e x)^m \, dx\right )+\frac {1}{8} \int e^{3 a+3 b x^2} (e x)^m \, dx+\frac {3}{8} \int e^{-a-b x^2} (e x)^m \, dx-\frac {3}{8} \int e^{a+b x^2} (e x)^m \, dx\\ &=-\frac {3^{-\frac {1}{2}-\frac {m}{2}} e^{3 a} (e x)^{1+m} \left (-b x^2\right )^{\frac {1}{2} (-1-m)} \Gamma \left (\frac {1+m}{2},-3 b x^2\right )}{16 e}+\frac {3 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 )}{16 e}-\frac {3 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 )}{16 e}+\frac {3^{-\frac {1}{2}-\frac {m}{2}} e^{-3 a} (e x)^{1+m} \left (b x^2\right )^{\frac {1}{2} (-1-m)} \Gamma \left (\frac {1+m}{2},3 b x^2\right )}{16 e}\\ \end {align*}
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Mathematica [B] time = 12.55, size = 735, normalized size = 3.43 \[ \frac {1}{16} 3^{\frac {1}{2}-\frac {m}{2}} x \sinh (a) \cosh ^2(a) \left (-b^2 x^4\right )^{\frac {1}{2} (-m-1)} (e x)^m \left (\left (-b x^2\right )^{\frac {m+1}{2}} \left (3^{\frac {m+1}{2}} \Gamma \left (\frac {m+1}{2},b x^2\right )-\Gamma \left (\frac {m+1}{2},3 b x^2\right )\right )-\left (b x^2\right )^{\frac {m+1}{2}} \Gamma \left (\frac {m+1}{2},-3 b x^2\right )+3^{\frac {m+1}{2}} \left (b x^2\right )^{\frac {m+1}{2}} \Gamma \left (\frac {m+1}{2},-b x^2\right )\right )-\frac {1}{16} 3^{\frac {1}{2}-\frac {m}{2}} x \sinh ^2(a) \cosh (a) \left (-b^2 x^4\right )^{\frac {1}{2} (-m-1)} (e x)^m \left (-\left (\left (-b x^2\right )^{\frac {m+1}{2}} \left (3^{\frac {m+1}{2}} \Gamma \left (\frac {m+1}{2},b x^2\right )+\Gamma \left (\frac {m+1}{2},3 b x^2\right )\right )\right )+\left (b x^2\right )^{\frac {m+1}{2}} \Gamma \left (\frac {m+1}{2},-3 b x^2\right )+3^{\frac {m+1}{2}} \left (b x^2\right )^{\frac {m+1}{2}} \Gamma \left (\frac {m+1}{2},-b x^2\right )\right )+\cosh ^3(a) x^{-m} (e x)^m \left (\frac {1}{8} \left (\frac {1}{2} 3^{\frac {1}{2} (-m-1)} x^{m+1} \left (b x^2\right )^{\frac {1}{2} (-m-1)} \Gamma \left (\frac {m+1}{2},3 b x^2\right )-\frac {1}{2} 3^{\frac {1}{2} (-m-1)} x^{m+1} \left (-b x^2\right )^{\frac {1}{2} (-m-1)} \Gamma \left (\frac {m+1}{2},-3 b x^2\right )\right )-\frac {3}{8} \left (\frac {1}{2} x^{m+1} \left (b x^2\right )^{\frac {1}{2} (-m-1)} \Gamma \left (\frac {m+1}{2},b x^2\right )-\frac {1}{2} x^{m+1} \left (-b x^2\right )^{\frac {1}{2} (-m-1)} \Gamma \left (\frac {m+1}{2},-b x^2\right )\right )\right )+\sinh ^3(a) x^{-m} (e x)^m \left (\frac {3}{8} \left (-\frac {1}{2} x^{m+1} \left (-b x^2\right )^{\frac {1}{2} (-m-1)} \Gamma \left (\frac {m+1}{2},-b x^2\right )-\frac {1}{2} x^{m+1} \left (b x^2\right )^{\frac {1}{2} (-m-1)} \Gamma \left (\frac {m+1}{2},b x^2\right )\right )+\frac {1}{8} \left (-\frac {1}{2} 3^{\frac {1}{2} (-m-1)} x^{m+1} \left (-b x^2\right )^{\frac {1}{2} (-m-1)} \Gamma \left (\frac {m+1}{2},-3 b x^2\right )-\frac {1}{2} 3^{\frac {1}{2} (-m-1)} x^{m+1} \left (b x^2\right )^{\frac {1}{2} (-m-1)} \Gamma \left (\frac {m+1}{2},3 b x^2\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 252, normalized size = 1.18 \[ \frac {e \cosh \left (\frac {1}{2} \, {\left (m - 1\right )} \log \left (\frac {3 \, b}{e^{2}}\right ) + 3 \, a\right ) \Gamma \left (\frac {1}{2} \, m + \frac {1}{2}, 3 \, b x^{2}\right ) - 9 \, e \cosh \left (\frac {1}{2} \, {\left (m - 1\right )} \log \left (\frac {b}{e^{2}}\right ) + a\right ) \Gamma \left (\frac {1}{2} \, m + \frac {1}{2}, b x^{2}\right ) - 9 \, e \cosh \left (\frac {1}{2} \, {\left (m - 1\right )} \log \left (-\frac {b}{e^{2}}\right ) - a\right ) \Gamma \left (\frac {1}{2} \, m + \frac {1}{2}, -b x^{2}\right ) + e \cosh \left (\frac {1}{2} \, {\left (m - 1\right )} \log \left (-\frac {3 \, b}{e^{2}}\right ) - 3 \, a\right ) \Gamma \left (\frac {1}{2} \, m + \frac {1}{2}, -3 \, b x^{2}\right ) - e \Gamma \left (\frac {1}{2} \, m + \frac {1}{2}, 3 \, b x^{2}\right ) \sinh \left (\frac {1}{2} \, {\left (m - 1\right )} \log \left (\frac {3 \, b}{e^{2}}\right ) + 3 \, a\right ) + 9 \, e \Gamma \left (\frac {1}{2} \, m + \frac {1}{2}, b x^{2}\right ) \sinh \left (\frac {1}{2} \, {\left (m - 1\right )} \log \left (\frac {b}{e^{2}}\right ) + a\right ) + 9 \, e \Gamma \left (\frac {1}{2} \, m + \frac {1}{2}, -b x^{2}\right ) \sinh \left (\frac {1}{2} \, {\left (m - 1\right )} \log \left (-\frac {b}{e^{2}}\right ) - a\right ) - e \Gamma \left (\frac {1}{2} \, m + \frac {1}{2}, -3 \, b x^{2}\right ) \sinh \left (\frac {1}{2} \, {\left (m - 1\right )} \log \left (-\frac {3 \, b}{e^{2}}\right ) - 3 \, a\right )}{48 \, b} \]
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 \[ \int \left (e x\right )^{m} \sinh \left (b x^{2} + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \left (\sinh ^{3}\left (b \,x^{2}+a \right )\right )\, dx \]
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 \[ \int \left (e x\right )^{m} \sinh \left (b x^{2} + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {sinh}\left (b\,x^2+a\right )}^3\,{\left (e\,x\right )}^m \,d x \]
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 \[ \int \left (e x\right )^{m} \sinh ^{3}{\left (a + b x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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