Optimal. Leaf size=59 \[ \frac{2 (b+c) \tanh ^{-1}\left (\frac{\sqrt{a+b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a-b}}\right )}{\sqrt{a-b} \sqrt{a+b}}-\frac{\log (a-b \cosh (x))}{b} \]
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Rubi [A] time = 0.13998, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4401, 2659, 208, 2668, 31} \[ \frac{2 (b+c) \tanh ^{-1}\left (\frac{\sqrt{a+b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a-b}}\right )}{\sqrt{a-b} \sqrt{a+b}}-\frac{\log (a-b \cosh (x))}{b} \]
Antiderivative was successfully verified.
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Rule 4401
Rule 2659
Rule 208
Rule 2668
Rule 31
Rubi steps
\begin{align*} \int \frac{b+c+\sinh (x)}{a-b \cosh (x)} \, dx &=\int \left (\frac{-b-c}{-a+b \cosh (x)}+\frac{\sinh (x)}{a-b \cosh (x)}\right ) \, dx\\ &=(-b-c) \int \frac{1}{-a+b \cosh (x)} \, dx+\int \frac{\sinh (x)}{a-b \cosh (x)} \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,-b \cosh (x)\right )}{b}-(2 (b+c)) \operatorname{Subst}\left (\int \frac{1}{-a+b-(-a-b) x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )\\ &=\frac{2 (b+c) \tanh ^{-1}\left (\frac{\sqrt{a+b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a-b}}\right )}{\sqrt{a-b} \sqrt{a+b}}-\frac{\log (a-b \cosh (x))}{b}\\ \end{align*}
Mathematica [A] time = 0.0947193, size = 56, normalized size = 0.95 \[ -\frac{2 (b+c) \tan ^{-1}\left (\frac{(a+b) \tanh \left (\frac{x}{2}\right )}{\sqrt{b^2-a^2}}\right )}{\sqrt{b^2-a^2}}-\frac{\log (a-b \cosh (x))}{b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.021, size = 154, normalized size = 2.6 \begin{align*}{\frac{1}{b}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+{\frac{1}{b}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) }-{\frac{a}{b \left ( a+b \right ) }\ln \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+ \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b-a+b \right ) }-{\frac{1}{a+b}\ln \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+ \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b-a+b \right ) }+2\,{\frac{b}{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}{\it Artanh} \left ({\frac{ \left ( a+b \right ) \tanh \left ( x/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) }+2\,{\frac{c}{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}{\it Artanh} \left ({\frac{ \left ( a+b \right ) \tanh \left ( x/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \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 = 2.49708, size = 716, normalized size = 12.14 \begin{align*} \left [\frac{\sqrt{a^{2} - b^{2}}{\left (b^{2} + b c\right )} \log \left (\frac{b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} - 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} - b^{2} + 2 \,{\left (b^{2} \cosh \left (x\right ) - a b\right )} \sinh \left (x\right ) + 2 \, \sqrt{a^{2} - b^{2}}{\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) - a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} - 2 \, a \cosh \left (x\right ) + 2 \,{\left (b \cosh \left (x\right ) - a\right )} \sinh \left (x\right ) + b}\right ) +{\left (a^{2} - b^{2}\right )} x -{\left (a^{2} - b^{2}\right )} \log \left (\frac{2 \,{\left (b \cosh \left (x\right ) - a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a^{2} b - b^{3}}, \frac{2 \, \sqrt{-a^{2} + b^{2}}{\left (b^{2} + b c\right )} \arctan \left (-\frac{\sqrt{-a^{2} + b^{2}}{\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) - a\right )}}{a^{2} - b^{2}}\right ) +{\left (a^{2} - b^{2}\right )} x -{\left (a^{2} - b^{2}\right )} \log \left (\frac{2 \,{\left (b \cosh \left (x\right ) - a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a^{2} b - b^{3}}\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 = 1.15017, size = 84, normalized size = 1.42 \begin{align*} -\frac{2 \,{\left (b + c\right )} \arctan \left (\frac{b e^{x} - a}{\sqrt{-a^{2} + b^{2}}}\right )}{\sqrt{-a^{2} + b^{2}}} + \frac{x}{b} - \frac{\log \left (b e^{\left (2 \, x\right )} - 2 \, a e^{x} + b\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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