Optimal. Leaf size=11 \[ \frac {\sinh (x)}{a \cosh (x)+b} \]
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Rubi [A] time = 0.09, antiderivative size = 11, normalized size of antiderivative = 1.00, 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.
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Rule 8
Rule 2754
Rule 3289
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*}
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Mathematica [A] time = 0.06, size = 11, normalized size = 1.00 \[ \frac {\sinh (x)}{a \cosh (x)+b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 54, normalized size = 4.91 \[ -\frac {2 \, {\left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}}{a^{2} \cosh \relax (x)^{2} + a^{2} \sinh \relax (x)^{2} + 2 \, a b \cosh \relax (x) + a^{2} + 2 \, {\left (a^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 26, normalized size = 2.36 \[ -\frac {2 \, {\left (b e^{x} + a\right )}}{{\left (a e^{\left (2 \, x\right )} + 2 \, b e^{x} + a\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.16, size = 29, normalized size = 2.64 \[ \frac {2 \tanh \left (\frac {x}{2}\right )}{a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b +a +b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.93, size = 51, normalized size = 4.64 \[ -\frac {\frac {2\,{\mathrm {e}}^x\,\left (a\,b^3-a^3\,b\right )}{a\,\left (a\,b^2-a^3\right )}+2}{a+2\,b\,{\mathrm {e}}^x+a\,{\mathrm {e}}^{2\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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