Optimal. Leaf size=11 \[ \frac{\sinh (x)}{a \cosh (x)+b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.086682, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {3289, 2754, 8} \[ \frac{\sinh (x)}{a \cosh (x)+b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3289
Rule 2754
Rule 8
Rubi steps
\begin{align*} \int \frac{a+b \cosh (x)}{b^2+2 a b \cosh (x)+a^2 \cosh ^2(x)} \, dx &=\left (4 a^2\right ) \int \frac{a+b \cosh (x)}{\left (2 a b+2 a^2 \cosh (x)\right )^2} \, dx\\ &=\frac{\sinh (x)}{b+a \cosh (x)}+\frac{\int 0 \, dx}{a^2-b^2}\\ &=\frac{\sinh (x)}{b+a \cosh (x)}\\ \end{align*}
Mathematica [A] time = 0.0577431, size = 11, normalized size = 1. \[ \frac{\sinh (x)}{a \cosh (x)+b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.026, size = 29, normalized size = 2.6 \begin{align*} 2\,{\frac{\tanh \left ( x/2 \right ) }{a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}- \left ( \tanh \left ( x/2 \right ) \right ) ^{2}b+a+b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.53732, size = 159, normalized size = 14.45 \begin{align*} -\frac{2 \,{\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{a^{2} \cosh \left (x\right )^{2} + a^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + a^{2} + 2 \,{\left (a^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.13539, size = 35, normalized size = 3.18 \begin{align*} -\frac{2 \,{\left (b e^{x} + a\right )}}{{\left (a e^{\left (2 \, x\right )} + 2 \, b e^{x} + a\right )} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]