Optimal. Leaf size=146 \[ -\frac {1}{8} e^{3 a} b 3^{m+1} \left (-\frac {b}{x}\right )^m (e x)^m \Gamma \left (-m-1,-\frac {3 b}{x}\right )+\frac {3}{8} e^a b \left (-\frac {b}{x}\right )^m (e x)^m \Gamma \left (-m-1,-\frac {b}{x}\right )+\frac {3}{8} e^{-a} b \left (\frac {b}{x}\right )^m (e x)^m \Gamma \left (-m-1,\frac {b}{x}\right )-\frac {1}{8} e^{-3 a} b 3^{m+1} \left (\frac {b}{x}\right )^m (e x)^m \Gamma \left (-m-1,\frac {3 b}{x}\right ) \]
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Rubi [A] time = 0.25, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5350, 3312, 3308, 2181} \[ -\frac {1}{8} e^{3 a} b 3^{m+1} \left (-\frac {b}{x}\right )^m (e x)^m \text {Gamma}\left (-m-1,-\frac {3 b}{x}\right )+\frac {3}{8} e^a b \left (-\frac {b}{x}\right )^m (e x)^m \text {Gamma}\left (-m-1,-\frac {b}{x}\right )+\frac {3}{8} e^{-a} b \left (\frac {b}{x}\right )^m (e x)^m \text {Gamma}\left (-m-1,\frac {b}{x}\right )-\frac {1}{8} e^{-3 a} b 3^{m+1} \left (\frac {b}{x}\right )^m (e x)^m \text {Gamma}\left (-m-1,\frac {3 b}{x}\right ) \]
Antiderivative was successfully verified.
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Rule 2181
Rule 3308
Rule 3312
Rule 5350
Rubi steps
\begin {align*} \int (e x)^m \sinh ^3\left (a+\frac {b}{x}\right ) \, dx &=-\left (\left (\left (\frac {1}{x}\right )^m (e x)^m\right ) \operatorname {Subst}\left (\int x^{-2-m} \sinh ^3(a+b x) \, dx,x,\frac {1}{x}\right )\right )\\ &=-\left (\left (i \left (\frac {1}{x}\right )^m (e x)^m\right ) \operatorname {Subst}\left (\int \left (\frac {3}{4} i x^{-2-m} \sinh (a+b x)-\frac {1}{4} i x^{-2-m} \sinh (3 a+3 b x)\right ) \, dx,x,\frac {1}{x}\right )\right )\\ &=-\left (\frac {1}{4} \left (\left (\frac {1}{x}\right )^m (e x)^m\right ) \operatorname {Subst}\left (\int x^{-2-m} \sinh (3 a+3 b x) \, dx,x,\frac {1}{x}\right )\right )+\frac {1}{4} \left (3 \left (\frac {1}{x}\right )^m (e x)^m\right ) \operatorname {Subst}\left (\int x^{-2-m} \sinh (a+b x) \, dx,x,\frac {1}{x}\right )\\ &=-\left (\frac {1}{8} \left (\left (\frac {1}{x}\right )^m (e x)^m\right ) \operatorname {Subst}\left (\int e^{-i (3 i a+3 i b x)} x^{-2-m} \, dx,x,\frac {1}{x}\right )\right )+\frac {1}{8} \left (\left (\frac {1}{x}\right )^m (e x)^m\right ) \operatorname {Subst}\left (\int e^{i (3 i a+3 i b x)} x^{-2-m} \, dx,x,\frac {1}{x}\right )+\frac {1}{8} \left (3 \left (\frac {1}{x}\right )^m (e x)^m\right ) \operatorname {Subst}\left (\int e^{-i (i a+i b x)} x^{-2-m} \, dx,x,\frac {1}{x}\right )-\frac {1}{8} \left (3 \left (\frac {1}{x}\right )^m (e x)^m\right ) \operatorname {Subst}\left (\int e^{i (i a+i b x)} x^{-2-m} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {1}{8} 3^{1+m} b e^{3 a} \left (-\frac {b}{x}\right )^m (e x)^m \Gamma \left (-1-m,-\frac {3 b}{x}\right )+\frac {3}{8} b e^a \left (-\frac {b}{x}\right )^m (e x)^m \Gamma \left (-1-m,-\frac {b}{x}\right )+\frac {3}{8} b e^{-a} \left (\frac {b}{x}\right )^m (e x)^m \Gamma \left (-1-m,\frac {b}{x}\right )-\frac {1}{8} 3^{1+m} b e^{-3 a} \left (\frac {b}{x}\right )^m (e x)^m \Gamma \left (-1-m,\frac {3 b}{x}\right )\\ \end {align*}
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Mathematica [F] time = 180.00, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]
Verification is Not applicable to the result.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e x\right )^{m} \sinh \left (\frac {a x + b}{x}\right )^{3}, x\right ) \]
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 \[ \int \left (e x\right )^{m} \sinh \left (a + \frac {b}{x}\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \left (\sinh ^{3}\left (a +\frac {b}{x}\right )\right )\, dx \]
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 \[ \int \left (e x\right )^{m} \sinh \left (a + \frac {b}{x}\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {sinh}\left (a+\frac {b}{x}\right )}^3\,{\left (e\,x\right )}^m \,d x \]
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 \[ \int \left (e x\right )^{m} \sinh ^{3}{\left (a + \frac {b}{x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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