Optimal. Leaf size=12 \[ \frac{\cosh (x)}{b-a \sinh (x)} \]
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Rubi [A] time = 0.0906189, antiderivative size = 12, 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 = {3288, 2754, 8} \[ \frac{\cosh (x)}{b-a \sinh (x)} \]
Antiderivative was successfully verified.
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Rule 3288
Rule 2754
Rule 8
Rubi steps
\begin{align*} \int \frac{a+b \sinh (x)}{b^2-2 a b \sinh (x)+a^2 \sinh ^2(x)} \, dx &=-\left (\left (4 a^2\right ) \int \frac{a+b \sinh (x)}{\left (2 i a b-2 i a^2 \sinh (x)\right )^2} \, dx\right )\\ &=\frac{\cosh (x)}{b-a \sinh (x)}-\frac{\int 0 \, dx}{a^2+b^2}\\ &=\frac{\cosh (x)}{b-a \sinh (x)}\\ \end{align*}
Mathematica [A] time = 0.0337063, size = 14, normalized size = 1.17 \[ -\frac{\cosh (x)}{a \sinh (x)-b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 36, normalized size = 3. \begin{align*} -2\,{\frac{1}{ \left ( \tanh \left ( x/2 \right ) \right ) ^{2}b+2\,a\tanh \left ( x/2 \right ) -b} \left ( -{\frac{a\tanh \left ( x/2 \right ) }{b}}+1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [B] time = 1.74691, size = 159, normalized size = 13.25 \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.
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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.
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Giac [A] time = 1.19291, size = 38, normalized size = 3.17 \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.
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