Optimal. Leaf size=220 \[ -\frac {e^{3 a} 3^{-\frac {m+1}{n}} (e x)^{m+1} \left (-b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-3 b x^n\right )}{8 e n}+\frac {3 e^a (e x)^{m+1} \left (-b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-b x^n\right )}{8 e n}-\frac {3 e^{-a} (e x)^{m+1} \left (b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},b x^n\right )}{8 e n}+\frac {e^{-3 a} 3^{-\frac {m+1}{n}} (e x)^{m+1} \left (b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},3 b x^n\right )}{8 e n} \]
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Rubi [A] time = 0.23, antiderivative size = 220, 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 = {5362, 5360, 2218} \[ -\frac {e^{3 a} 3^{-\frac {m+1}{n}} (e x)^{m+1} \left (-b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},-3 b x^n\right )}{8 e n}+\frac {3 e^a (e x)^{m+1} \left (-b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},-b x^n\right )}{8 e n}-\frac {3 e^{-a} (e x)^{m+1} \left (b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},b x^n\right )}{8 e n}+\frac {e^{-3 a} 3^{-\frac {m+1}{n}} (e x)^{m+1} \left (b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},3 b x^n\right )}{8 e n} \]
Antiderivative was successfully verified.
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Rule 2218
Rule 5360
Rule 5362
Rubi steps
\begin {align*} \int (e x)^m \sinh ^3\left (a+b x^n\right ) \, dx &=\int \left (-\frac {3}{4} (e x)^m \sinh \left (a+b x^n\right )+\frac {1}{4} (e x)^m \sinh \left (3 a+3 b x^n\right )\right ) \, dx\\ &=\frac {1}{4} \int (e x)^m \sinh \left (3 a+3 b x^n\right ) \, dx-\frac {3}{4} \int (e x)^m \sinh \left (a+b x^n\right ) \, dx\\ &=-\left (\frac {1}{8} \int e^{-3 a-3 b x^n} (e x)^m \, dx\right )+\frac {1}{8} \int e^{3 a+3 b x^n} (e x)^m \, dx+\frac {3}{8} \int e^{-a-b x^n} (e x)^m \, dx-\frac {3}{8} \int e^{a+b x^n} (e x)^m \, dx\\ &=-\frac {3^{-\frac {1+m}{n}} e^{3 a} (e x)^{1+m} \left (-b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-3 b x^n\right )}{8 e n}+\frac {3 e^a (e x)^{1+m} \left (-b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-b x^n\right )}{8 e n}-\frac {3 e^{-a} (e x)^{1+m} \left (b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},b x^n\right )}{8 e n}+\frac {3^{-\frac {1+m}{n}} e^{-3 a} (e x)^{1+m} \left (b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},3 b x^n\right )}{8 e n}\\ \end {align*}
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Mathematica [A] time = 2.15, size = 185, normalized size = 0.84 \[ \frac {e^{-3 a} x 3^{-\frac {m+1}{n}} (e x)^m \left (-b^2 x^{2 n}\right )^{-\frac {m+1}{n}} \left (\left (-b x^n\right )^{\frac {m+1}{n}} \left (\Gamma \left (\frac {m+1}{n},3 b x^n\right )-e^{2 a} 3^{\frac {m+n+1}{n}} \Gamma \left (\frac {m+1}{n},b x^n\right )\right )-e^{6 a} \left (b x^n\right )^{\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-3 b x^n\right )+e^{4 a} 3^{\frac {m+n+1}{n}} \left (b x^n\right )^{\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-b x^n\right )\right )}{8 n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e x\right )^{m} \sinh \left (b x^{n} + a\right )^{3}, x\right ) \]
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 \[ \int \left (e x\right )^{m} \sinh \left (b x^{n} + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.32, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \left (\sinh ^{3}\left (a +b \,x^{n}\right )\right )\, dx \]
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 \[ \int \left (e x\right )^{m} \sinh \left (b x^{n} + a\right )^{3}\,{d x} \]
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 \[ \int {\mathrm {sinh}\left (a+b\,x^n\right )}^3\,{\left (e\,x\right )}^m \,d x \]
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 \[ \int \left (e x\right )^{m} \sinh ^{3}{\left (a + b x^{n} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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