Optimal. Leaf size=59 \[ \frac {e^a x^m (-b x)^{-m} \Gamma (m+1,-b x)}{2 b}-\frac {e^{-a} x^m (b x)^{-m} \Gamma (m+1,b x)}{2 b} \]
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Rubi [A] time = 0.07, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3307, 2181} \[ \frac {e^a x^m (-b x)^{-m} \text {Gamma}(m+1,-b x)}{2 b}-\frac {e^{-a} x^m (b x)^{-m} \text {Gamma}(m+1,b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 3307
Rubi steps
\begin {align*} \int x^m \cosh (a+b x) \, dx &=\frac {1}{2} \int e^{-i (i a+i b x)} x^m \, dx+\frac {1}{2} \int e^{i (i a+i b x)} x^m \, dx\\ &=\frac {e^a x^m (-b x)^{-m} \Gamma (1+m,-b x)}{2 b}-\frac {e^{-a} x^m (b x)^{-m} \Gamma (1+m,b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 54, normalized size = 0.92 \[ \frac {e^{-a} x^m \left (e^{2 a} (-b x)^{-m} \Gamma (m+1,-b x)-(b x)^{-m} \Gamma (m+1,b x)\right )}{2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 78, normalized size = 1.32 \[ -\frac {\cosh \left (m \log \relax (b) + a\right ) \Gamma \left (m + 1, b x\right ) - \cosh \left (m \log \left (-b\right ) - a\right ) \Gamma \left (m + 1, -b x\right ) + \Gamma \left (m + 1, -b x\right ) \sinh \left (m \log \left (-b\right ) - a\right ) - \Gamma \left (m + 1, b x\right ) \sinh \left (m \log \relax (b) + a\right )}{2 \, b} \]
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 \[ \int x^{m} \cosh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.11, size = 73, normalized size = 1.24 \[ \frac {x^{1+m} \hypergeom \left (\left [\frac {1}{2}+\frac {m}{2}\right ], \left [\frac {1}{2}, \frac {3}{2}+\frac {m}{2}\right ], \frac {x^{2} b^{2}}{4}\right ) \cosh \relax (a )}{1+m}+\frac {b \,x^{2+m} \hypergeom \left (\left [1+\frac {m}{2}\right ], \left [\frac {3}{2}, 2+\frac {m}{2}\right ], \frac {x^{2} b^{2}}{4}\right ) \sinh \relax (a )}{2+m} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 55, normalized size = 0.93 \[ -\frac {1}{2} \, \left (b x\right )^{-m - 1} x^{m + 1} e^{\left (-a\right )} \Gamma \left (m + 1, b x\right ) - \frac {1}{2} \, \left (-b x\right )^{-m - 1} x^{m + 1} e^{a} \Gamma \left (m + 1, -b x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^m\,\mathrm {cosh}\left (a+b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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