Optimal. Leaf size=143 \[ -\frac {e^{2 a} 2^{-\frac {m+2 n+1}{n}} (e x)^{m+1} \left (-b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-2 b x^n\right )}{e n}-\frac {e^{-2 a} 2^{-\frac {m+2 n+1}{n}} (e x)^{m+1} \left (b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},2 b x^n\right )}{e n}-\frac {(e x)^{m+1}}{2 e (m+1)} \]
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Rubi [A] time = 0.17, antiderivative size = 143, 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 = {5362, 5361, 2218} \[ -\frac {e^{2 a} 2^{-\frac {m+2 n+1}{n}} (e x)^{m+1} \left (-b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},-2 b x^n\right )}{e n}-\frac {e^{-2 a} 2^{-\frac {m+2 n+1}{n}} (e x)^{m+1} \left (b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},2 b x^n\right )}{e n}-\frac {(e x)^{m+1}}{2 e (m+1)} \]
Antiderivative was successfully verified.
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Rule 2218
Rule 5361
Rule 5362
Rubi steps
\begin {align*} \int (e x)^m \sinh ^2\left (a+b x^n\right ) \, dx &=\int \left (-\frac {1}{2} (e x)^m+\frac {1}{2} (e x)^m \cosh \left (2 a+2 b x^n\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^n\right ) \, dx\\ &=-\frac {(e x)^{1+m}}{2 e (1+m)}+\frac {1}{4} \int e^{-2 a-2 b x^n} (e x)^m \, dx+\frac {1}{4} \int e^{2 a+2 b x^n} (e x)^m \, dx\\ &=-\frac {(e x)^{1+m}}{2 e (1+m)}-\frac {2^{-\frac {1+m+2 n}{n}} e^{2 a} (e x)^{1+m} \left (-b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-2 b x^n\right )}{e n}-\frac {2^{-\frac {1+m+2 n}{n}} e^{-2 a} (e x)^{1+m} \left (b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},2 b x^n\right )}{e n}\\ \end {align*}
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Mathematica [A] time = 2.00, size = 117, normalized size = 0.82 \[ -\frac {x (e x)^m \left (e^{2 a} (m+1) 2^{-\frac {m+1}{n}} \left (-b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-2 b x^n\right )+e^{-2 a} (m+1) 2^{-\frac {m+1}{n}} \left (b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},2 b x^n\right )+2 n\right )}{4 (m+1) n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e x\right )^{m} \sinh \left (b x^{n} + a\right )^{2}, x\right ) \]
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 \[ \int \left (e x\right )^{m} \sinh \left (b x^{n} + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.26, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \left (\sinh ^{2}\left (a +b \,x^{n}\right )\right )\, dx \]
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 \[ \frac {1}{4} \, e^{m} \int e^{\left (2 \, b x^{n} + m \log \relax (x) + 2 \, a\right )}\,{d x} + \frac {1}{4} \, e^{m} \int e^{\left (-2 \, b x^{n} + m \log \relax (x) - 2 \, a\right )}\,{d x} - \frac {\left (e x\right )^{m + 1}}{2 \, e {\left (m + 1\right )}} \]
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 \[ \int {\mathrm {sinh}\left (a+b\,x^n\right )}^2\,{\left (e\,x\right )}^m \,d x \]
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 \[ \int \left (e x\right )^{m} \sinh ^{2}{\left (a + b x^{n} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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