Optimal. Leaf size=49 \[ -\frac {1}{2} e^a x^m (-b x)^{-m} \Gamma (m,-b x)-\frac {1}{2} e^{-a} x^m (b x)^{-m} \Gamma (m,b x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3307, 2181} \[ -\frac {1}{2} e^a x^m (-b x)^{-m} \text {Gamma}(m,-b x)-\frac {1}{2} e^{-a} x^m (b x)^{-m} \text {Gamma}(m,b x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2181
Rule 3307
Rubi steps
\begin {align*} \int x^{-1+m} \cosh (a+b x) \, dx &=\frac {1}{2} \int e^{-i (i a+i b x)} x^{-1+m} \, dx+\frac {1}{2} \int e^{i (i a+i b x)} x^{-1+m} \, dx\\ &=-\frac {1}{2} e^a x^m (-b x)^{-m} \Gamma (m,-b x)-\frac {1}{2} e^{-a} x^m (b x)^{-m} \Gamma (m,b x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 49, normalized size = 1.00 \[ -\frac {1}{2} e^a x^m (-b x)^{-m} \Gamma (m,-b x)-\frac {1}{2} e^{-a} x^m (b x)^{-m} \Gamma (m,b x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 78, normalized size = 1.59 \[ -\frac {\cosh \left ({\left (m - 1\right )} \log \relax (b) + a\right ) \Gamma \left (m, b x\right ) - \cosh \left ({\left (m - 1\right )} \log \left (-b\right ) - a\right ) \Gamma \left (m, -b x\right ) + \Gamma \left (m, -b x\right ) \sinh \left ({\left (m - 1\right )} \log \left (-b\right ) - a\right ) - \Gamma \left (m, b x\right ) \sinh \left ({\left (m - 1\right )} \log \relax (b) + a\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m - 1} \cosh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.12, size = 67, normalized size = 1.37 \[ \frac {x^{m} \hypergeom \left (\left [\frac {m}{2}\right ], \left [\frac {1}{2}, 1+\frac {m}{2}\right ], \frac {x^{2} b^{2}}{4}\right ) \cosh \relax (a )}{m}+\frac {b \,x^{1+m} \hypergeom \left (\left [\frac {1}{2}+\frac {m}{2}\right ], \left [\frac {3}{2}, \frac {3}{2}+\frac {m}{2}\right ], \frac {x^{2} b^{2}}{4}\right ) \sinh \relax (a )}{1+m} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.77, size = 43, normalized size = 0.88 \[ -\frac {x^{m} e^{\left (-a\right )} \Gamma \left (m, b x\right )}{2 \, \left (b x\right )^{m}} - \frac {x^{m} e^{a} \Gamma \left (m, -b x\right )}{2 \, \left (-b x\right )^{m}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^{m-1}\,\mathrm {cosh}\left (a+b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________