Optimal. Leaf size=43 \[ \frac {2 \cosh (a+b x) e^{n \cosh (a+b x)}}{b n}-\frac {2 e^{n \cosh (a+b x)}}{b n^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {12, 2176, 2194} \[ \frac {2 \cosh (a+b x) e^{n \cosh (a+b x)}}{b n}-\frac {2 e^{n \cosh (a+b x)}}{b n^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2176
Rule 2194
Rubi steps
\begin {align*} \int e^{n \cosh (a+b x)} \sinh (2 a+2 b x) \, dx &=\frac {i \operatorname {Subst}\left (\int -2 i e^{n x} x \, dx,x,\cosh (a+b x)\right )}{b}\\ &=\frac {2 \operatorname {Subst}\left (\int e^{n x} x \, dx,x,\cosh (a+b x)\right )}{b}\\ &=\frac {2 e^{n \cosh (a+b x)} \cosh (a+b x)}{b n}-\frac {2 \operatorname {Subst}\left (\int e^{n x} \, dx,x,\cosh (a+b x)\right )}{b n}\\ &=-\frac {2 e^{n \cosh (a+b x)}}{b n^2}+\frac {2 e^{n \cosh (a+b x)} \cosh (a+b x)}{b n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 28, normalized size = 0.65 \[ \frac {2 e^{n \cosh (a+b x)} (n \cosh (a+b x)-1)}{b n^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 73, normalized size = 1.70 \[ \frac {2 \, {\left ({\left (n \cosh \left (b x + a\right ) - 1\right )} \cosh \left (n \cosh \left (b x + a\right )\right ) + {\left (n \cosh \left (b x + a\right ) - 1\right )} \sinh \left (n \cosh \left (b x + a\right )\right )\right )}}{b n^{2} \cosh \left (b x + a\right )^{2} - b n^{2} \sinh \left (b x + a\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\left (n \cosh \left (b x + a\right )\right )} \sinh \left (2 \, b x + 2 \, a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.47, size = 59, normalized size = 1.37 \[ \frac {\left ({\mathrm e}^{2 b x +2 a} n +n -2 \,{\mathrm e}^{b x +a}\right ) {\mathrm e}^{-b x -a +\frac {n \,{\mathrm e}^{b x +a}}{2}+\frac {n \,{\mathrm e}^{-b x -a}}{2}}}{n^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.41, size = 103, normalized size = 2.40 \[ \frac {e^{\left (b x + \frac {1}{2} \, n e^{\left (b x + a\right )} + \frac {1}{2} \, n e^{\left (-b x - a\right )} + a\right )}}{b n} + \frac {e^{\left (-b x + \frac {1}{2} \, n e^{\left (b x + a\right )} + \frac {1}{2} \, n e^{\left (-b x - a\right )} - a\right )}}{b n} - \frac {2 \, e^{\left (\frac {1}{2} \, n e^{\left (b x + a\right )} + \frac {1}{2} \, n e^{\left (-b x - a\right )}\right )}}{b n^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.76, size = 107, normalized size = 2.49 \[ \frac {{\mathrm {e}}^{-a}\,{\mathrm {e}}^{\frac {n\,{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a}{2}}\,{\mathrm {e}}^{-b\,x}\,{\mathrm {e}}^{\frac {n\,{\mathrm {e}}^{-a}\,{\mathrm {e}}^{-b\,x}}{2}}}{b\,n}-\frac {2\,{\mathrm {e}}^{\frac {n\,{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a}{2}}\,{\mathrm {e}}^{\frac {n\,{\mathrm {e}}^{-a}\,{\mathrm {e}}^{-b\,x}}{2}}}{b\,n^2}+\frac {{\mathrm {e}}^{\frac {n\,{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a}{2}}\,{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,{\mathrm {e}}^{\frac {n\,{\mathrm {e}}^{-a}\,{\mathrm {e}}^{-b\,x}}{2}}}{b\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{n \cosh {\left (a + b x \right )}} \sinh {\left (2 a + 2 b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________