Optimal. Leaf size=134 \[ \frac {e^{3 a} 3^{-m-1} x^m (-b x)^{-m} \Gamma (m+1,-3 b x)}{8 b}-\frac {e^a x^m (-b x)^{-m} \Gamma (m+1,-b x)}{8 b}+\frac {e^{-a} x^m (b x)^{-m} \Gamma (m+1,b x)}{8 b}-\frac {e^{-3 a} 3^{-m-1} x^m (b x)^{-m} \Gamma (m+1,3 b x)}{8 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.18, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5448, 3307, 2181} \[ \frac {e^{3 a} 3^{-m-1} x^m (-b x)^{-m} \text {Gamma}(m+1,-3 b x)}{8 b}-\frac {e^a x^m (-b x)^{-m} \text {Gamma}(m+1,-b x)}{8 b}+\frac {e^{-a} x^m (b x)^{-m} \text {Gamma}(m+1,b x)}{8 b}-\frac {e^{-3 a} 3^{-m-1} x^m (b x)^{-m} \text {Gamma}(m+1,3 b x)}{8 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2181
Rule 3307
Rule 5448
Rubi steps
\begin {align*} \int x^m \cosh (a+b x) \sinh ^2(a+b x) \, dx &=\int \left (-\frac {1}{4} x^m \cosh (a+b x)+\frac {1}{4} x^m \cosh (3 a+3 b x)\right ) \, dx\\ &=-\left (\frac {1}{4} \int x^m \cosh (a+b x) \, dx\right )+\frac {1}{4} \int x^m \cosh (3 a+3 b x) \, dx\\ &=-\left (\frac {1}{8} \int e^{-i (i a+i b x)} x^m \, dx\right )-\frac {1}{8} \int e^{i (i a+i b x)} x^m \, dx+\frac {1}{8} \int e^{-i (3 i a+3 i b x)} x^m \, dx+\frac {1}{8} \int e^{i (3 i a+3 i b x)} x^m \, dx\\ &=\frac {3^{-1-m} e^{3 a} x^m (-b x)^{-m} \Gamma (1+m,-3 b x)}{8 b}-\frac {e^a x^m (-b x)^{-m} \Gamma (1+m,-b x)}{8 b}+\frac {e^{-a} x^m (b x)^{-m} \Gamma (1+m,b x)}{8 b}-\frac {3^{-1-m} e^{-3 a} x^m (b x)^{-m} \Gamma (1+m,3 b x)}{8 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.23, size = 114, normalized size = 0.85 \[ \frac {e^{-3 a} x^m \left (3^{-m} \left (-b^2 x^2\right )^{-m} \left (e^{6 a} (b x)^m \Gamma (m+1,-3 b x)-(-b x)^m \Gamma (m+1,3 b x)\right )-3 e^{4 a} (-b x)^{-m} \Gamma (m+1,-b x)+3 e^{2 a} (b x)^{-m} \Gamma (m+1,b x)\right )}{24 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.78, size = 162, normalized size = 1.21 \[ -\frac {\cosh \left (m \log \left (3 \, b\right ) + 3 \, a\right ) \Gamma \left (m + 1, 3 \, b x\right ) - 3 \, \cosh \left (m \log \relax (b) + a\right ) \Gamma \left (m + 1, b x\right ) + 3 \, \cosh \left (m \log \left (-b\right ) - a\right ) \Gamma \left (m + 1, -b x\right ) - \cosh \left (m \log \left (-3 \, b\right ) - 3 \, a\right ) \Gamma \left (m + 1, -3 \, b x\right ) - \Gamma \left (m + 1, 3 \, b x\right ) \sinh \left (m \log \left (3 \, b\right ) + 3 \, a\right ) - 3 \, \Gamma \left (m + 1, -b x\right ) \sinh \left (m \log \left (-b\right ) - a\right ) + \Gamma \left (m + 1, -3 \, b x\right ) \sinh \left (m \log \left (-3 \, b\right ) - 3 \, a\right ) + 3 \, \Gamma \left (m + 1, b x\right ) \sinh \left (m \log \relax (b) + a\right )}{24 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.48, size = 0, normalized size = 0.00 \[ \int x^{m} \cosh \left (b x +a \right ) \left (\sinh ^{2}\left (b x +a \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.44, size = 113, normalized size = 0.84 \[ -\frac {1}{8} \, \left (3 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (-3 \, a\right )} \Gamma \left (m + 1, 3 \, b x\right ) + \frac {1}{8} \, \left (b x\right )^{-m - 1} x^{m + 1} e^{\left (-a\right )} \Gamma \left (m + 1, b x\right ) + \frac {1}{8} \, \left (-b x\right )^{-m - 1} x^{m + 1} e^{a} \Gamma \left (m + 1, -b x\right ) - \frac {1}{8} \, \left (-3 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (3 \, a\right )} \Gamma \left (m + 1, -3 \, b x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^m\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________