Optimal. Leaf size=56 \[ \frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{\sqrt{a-b} \sqrt{a+b}}+\frac{B \log (a+b \cosh (x))}{b} \]
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Rubi [A] time = 0.127799, antiderivative size = 56, 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} \[ \frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{\sqrt{a-b} \sqrt{a+b}}+\frac{B \log (a+b \cosh (x))}{b} \]
Antiderivative was successfully verified.
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Rule 4401
Rule 2659
Rule 208
Rule 2668
Rule 31
Rubi steps
\begin{align*} \int \frac{A+B \sinh (x)}{a+b \cosh (x)} \, dx &=\int \left (\frac{A}{a+b \cosh (x)}+\frac{B \sinh (x)}{a+b \cosh (x)}\right ) \, dx\\ &=A \int \frac{1}{a+b \cosh (x)} \, dx+B \int \frac{\sinh (x)}{a+b \cosh (x)} \, dx\\ &=(2 A) \operatorname{Subst}\left (\int \frac{1}{a+b-(a-b) x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )+\frac{B \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \cosh (x)\right )}{b}\\ &=\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{\sqrt{a-b} \sqrt{a+b}}+\frac{B \log (a+b \cosh (x))}{b}\\ \end{align*}
Mathematica [A] time = 0.0822838, size = 55, normalized size = 0.98 \[ \frac{B \log (a+b \cosh (x))}{b}-\frac{2 A \tan ^{-1}\left (\frac{(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{b^2-a^2}}\right )}{\sqrt{b^2-a^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.017, size = 137, normalized size = 2.5 \begin{align*}{\frac{aB}{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{B}{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{A}{\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 ) }-{\frac{B}{b}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{B}{b}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -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 = 1.91991, size = 717, normalized size = 12.8 \begin{align*} \left [\frac{\sqrt{a^{2} - b^{2}} A b \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 (B a^{2} - B b^{2}\right )} x +{\left (B a^{2} - B 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}} A b \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 (B a^{2} - B b^{2}\right )} x -{\left (B a^{2} - B 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 [A] time = 138.828, size = 741, normalized size = 13.23 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21904, size = 81, normalized size = 1.45 \begin{align*} \frac{2 \, A \arctan \left (\frac{b e^{x} + a}{\sqrt{-a^{2} + b^{2}}}\right )}{\sqrt{-a^{2} + b^{2}}} - \frac{B x}{b} + \frac{B \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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