Optimal. Leaf size=51 \[ -\frac{2 \tanh ^{-1}\left (\frac{c-(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2+c^2}}\right )}{\sqrt{a^2-b^2+c^2}} \]
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Rubi [A] time = 0.0687801, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3124, 618, 206} \[ -\frac{2 \tanh ^{-1}\left (\frac{c-(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2+c^2}}\right )}{\sqrt{a^2-b^2+c^2}} \]
Antiderivative was successfully verified.
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Rule 3124
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{a+b \cosh (x)+c \sinh (x)} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{a+b+2 c x-(a-b) x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )\\ &=-\left (4 \operatorname{Subst}\left (\int \frac{1}{4 \left (a^2-b^2+c^2\right )-x^2} \, dx,x,2 c+2 (-a+b) \tanh \left (\frac{x}{2}\right )\right )\right )\\ &=-\frac{2 \tanh ^{-1}\left (\frac{c-(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2+c^2}}\right )}{\sqrt{a^2-b^2+c^2}}\\ \end{align*}
Mathematica [A] time = 0.0729275, size = 54, normalized size = 1.06 \[ \frac{2 \tan ^{-1}\left (\frac{(b-a) \tanh \left (\frac{x}{2}\right )+c}{\sqrt{-a^2+b^2-c^2}}\right )}{\sqrt{-a^2+b^2-c^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 53, normalized size = 1. \begin{align*} -2\,{\frac{1}{\sqrt{-{a}^{2}+{b}^{2}-{c}^{2}}}\arctan \left ( 1/2\,{\frac{2\, \left ( a-b \right ) \tanh \left ( x/2 \right ) -2\,c}{\sqrt{-{a}^{2}+{b}^{2}-{c}^{2}}}} \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 [A] time = 2.09105, size = 660, normalized size = 12.94 \begin{align*} \left [\frac{\log \left (\frac{{\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{2} +{\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \left (x\right )^{2} + 2 \, a^{2} - b^{2} + c^{2} + 2 \,{\left (a b + a c\right )} \cosh \left (x\right ) + 2 \,{\left (a b + a c +{\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) - 2 \, \sqrt{a^{2} - b^{2} + c^{2}}{\left ({\left (b + c\right )} \cosh \left (x\right ) +{\left (b + c\right )} \sinh \left (x\right ) + a\right )}}{{\left (b + c\right )} \cosh \left (x\right )^{2} +{\left (b + c\right )} \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \,{\left ({\left (b + c\right )} \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + b - c}\right )}{\sqrt{a^{2} - b^{2} + c^{2}}}, \frac{2 \, \sqrt{-a^{2} + b^{2} - c^{2}} \arctan \left (\frac{\sqrt{-a^{2} + b^{2} - c^{2}}{\left ({\left (b + c\right )} \cosh \left (x\right ) +{\left (b + c\right )} \sinh \left (x\right ) + a\right )}}{a^{2} - b^{2} + c^{2}}\right )}{a^{2} - b^{2} + c^{2}}\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 = 1.1667, size = 62, normalized size = 1.22 \begin{align*} \frac{2 \, \arctan \left (\frac{b e^{x} + c e^{x} + a}{\sqrt{-a^{2} + b^{2} - c^{2}}}\right )}{\sqrt{-a^{2} + b^{2} - c^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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