Optimal. Leaf size=135 \[ -\frac {e^{2 a} 2^{-\frac {m}{2}-\frac {7}{2}} \left (-b x^2\right )^{\frac {1}{2} (-m-1)} (e x)^{m+1} \Gamma \left (\frac {m+1}{2},-2 b x^2\right )}{e}-\frac {e^{-2 a} 2^{-\frac {m}{2}-\frac {7}{2}} \left (b x^2\right )^{\frac {1}{2} (-m-1)} (e x)^{m+1} \Gamma \left (\frac {m+1}{2},2 b x^2\right )}{e}-\frac {(e x)^{m+1}}{2 e (m+1)} \]
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Rubi [A] time = 0.15, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5340, 5329, 2218} \[ -\frac {e^{2 a} 2^{-\frac {m}{2}-\frac {7}{2}} \left (-b x^2\right )^{\frac {1}{2} (-m-1)} (e x)^{m+1} \text {Gamma}\left (\frac {m+1}{2},-2 b x^2\right )}{e}-\frac {e^{-2 a} 2^{-\frac {m}{2}-\frac {7}{2}} \left (b x^2\right )^{\frac {1}{2} (-m-1)} (e x)^{m+1} \text {Gamma}\left (\frac {m+1}{2},2 b x^2\right )}{e}-\frac {(e x)^{m+1}}{2 e (m+1)} \]
Antiderivative was successfully verified.
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Rule 2218
Rule 5329
Rule 5340
Rubi steps
\begin {align*} \int (e x)^m \sinh ^2\left (a+b x^2\right ) \, dx &=\int \left (-\frac {1}{2} (e x)^m+\frac {1}{2} (e x)^m \cosh \left (2 a+2 b x^2\right )\right ) \, dx\\ &=-\frac {(e x)^{1+m}}{2 e (1+m)}+\frac {1}{2} \int (e x)^m \cosh \left (2 a+2 b x^2\right ) \, dx\\ &=-\frac {(e x)^{1+m}}{2 e (1+m)}+\frac {1}{4} \int e^{-2 a-2 b x^2} (e x)^m \, dx+\frac {1}{4} \int e^{2 a+2 b x^2} (e x)^m \, dx\\ &=-\frac {(e x)^{1+m}}{2 e (1+m)}-\frac {2^{-\frac {7}{2}-\frac {m}{2}} e^{2 a} (e x)^{1+m} \left (-b x^2\right )^{\frac {1}{2} (-1-m)} \Gamma \left (\frac {1+m}{2},-2 b x^2\right )}{e}-\frac {2^{-\frac {7}{2}-\frac {m}{2}} e^{-2 a} (e x)^{1+m} \left (b x^2\right )^{\frac {1}{2} (-1-m)} \Gamma \left (\frac {1+m}{2},2 b x^2\right )}{e}\\ \end {align*}
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Mathematica [A] time = 0.58, size = 152, normalized size = 1.13 \[ -\frac {2^{\frac {1}{2} (-m-7)} x \left (-b^2 x^4\right )^{\frac {1}{2} (-m-1)} (e x)^m \left ((m+1) (\cosh (2 a)-\sinh (2 a)) \left (-b x^2\right )^{\frac {m+1}{2}} \Gamma \left (\frac {m+1}{2},2 b x^2\right )+(m+1) (\sinh (2 a)+\cosh (2 a)) \left (b x^2\right )^{\frac {m+1}{2}} \Gamma \left (\frac {m+1}{2},-2 b x^2\right )+2^{\frac {m+5}{2}} \left (-b^2 x^4\right )^{\frac {m+1}{2}}\right )}{m+1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 174, normalized size = 1.29 \[ -\frac {8 \, b x \cosh \left (m \log \left (e x\right )\right ) + {\left (e m + e\right )} \cosh \left (\frac {1}{2} \, {\left (m - 1\right )} \log \left (\frac {2 \, b}{e^{2}}\right ) + 2 \, a\right ) \Gamma \left (\frac {1}{2} \, m + \frac {1}{2}, 2 \, b x^{2}\right ) - {\left (e m + e\right )} \cosh \left (\frac {1}{2} \, {\left (m - 1\right )} \log \left (-\frac {2 \, b}{e^{2}}\right ) - 2 \, a\right ) \Gamma \left (\frac {1}{2} \, m + \frac {1}{2}, -2 \, b x^{2}\right ) + 8 \, b x \sinh \left (m \log \left (e x\right )\right ) - {\left (e m + e\right )} \Gamma \left (\frac {1}{2} \, m + \frac {1}{2}, 2 \, b x^{2}\right ) \sinh \left (\frac {1}{2} \, {\left (m - 1\right )} \log \left (\frac {2 \, b}{e^{2}}\right ) + 2 \, a\right ) + {\left (e m + e\right )} \Gamma \left (\frac {1}{2} \, m + \frac {1}{2}, -2 \, b x^{2}\right ) \sinh \left (\frac {1}{2} \, {\left (m - 1\right )} \log \left (-\frac {2 \, b}{e^{2}}\right ) - 2 \, a\right )}{16 \, {\left (b m + b\right )}} \]
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 \[ \int \left (e x\right )^{m} \sinh \left (b x^{2} + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \left (\sinh ^{2}\left (b \,x^{2}+a \right )\right )\, dx \]
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 \[ \frac {1}{4} \, e^{m} \int e^{\left (2 \, b x^{2} + m \log \relax (x) + 2 \, a\right )}\,{d x} + \frac {1}{4} \, e^{m} \int e^{\left (-2 \, b x^{2} + m \log \relax (x) - 2 \, a\right )}\,{d x} - \frac {\left (e x\right )^{m + 1}}{2 \, e {\left (m + 1\right )}} \]
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 \[ \int {\mathrm {sinh}\left (b\,x^2+a\right )}^2\,{\left (e\,x\right )}^m \,d x \]
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 \[ \int \left (e x\right )^{m} \sinh ^{2}{\left (a + b x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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