Optimal. Leaf size=60 \[ \frac{2 (A b-a B) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{b \sqrt{a-b} \sqrt{a+b}}+\frac{B x}{b} \]
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Rubi [A] time = 0.0676142, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2735, 2659, 208} \[ \frac{2 (A b-a B) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{b \sqrt{a-b} \sqrt{a+b}}+\frac{B x}{b} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2659
Rule 208
Rubi steps
\begin{align*} \int \frac{A+B \cosh (x)}{a+b \cosh (x)} \, dx &=\frac{B x}{b}-\frac{(-A b+a B) \int \frac{1}{a+b \cosh (x)} \, dx}{b}\\ &=\frac{B x}{b}-\frac{(2 (-A b+a B)) \operatorname{Subst}\left (\int \frac{1}{a+b-(a-b) x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{b}\\ &=\frac{B x}{b}+\frac{2 (A b-a B) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{\sqrt{a-b} b \sqrt{a+b}}\\ \end{align*}
Mathematica [A] time = 0.0912761, size = 59, normalized size = 0.98 \[ \frac{2 (a B-A b) \tan ^{-1}\left (\frac{(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{b^2-a^2}}\right )}{b \sqrt{b^2-a^2}}+\frac{B x}{b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 103, normalized size = 1.7 \begin{align*} 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 ) }-2\,{\frac{aB}{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 ) }+{\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 [A] time = 2.27611, size = 574, normalized size = 9.57 \begin{align*} \left [-\frac{{\left (B a - A b\right )} \sqrt{a^{2} - b^{2}} \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}{a^{2} b - b^{3}}, \frac{2 \,{\left (B a - A b\right )} \sqrt{-a^{2} + b^{2}} \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}{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 = 154.892, size = 403, normalized size = 6.72 \begin{align*} \begin{cases} \tilde{\infty } \left (2 A \operatorname{atan}{\left (\tanh{\left (\frac{x}{2} \right )} \right )} + B x\right ) & \text{for}\: a = 0 \wedge b = 0 \\\frac{A \tanh{\left (\frac{x}{2} \right )}}{b} + \frac{B x}{b} - \frac{B \tanh{\left (\frac{x}{2} \right )}}{b} & \text{for}\: a = b \\- \frac{A}{b \tanh{\left (\frac{x}{2} \right )}} + \frac{B x}{b} - \frac{B}{b \tanh{\left (\frac{x}{2} \right )}} & \text{for}\: a = - b \\\frac{A x + B \sinh{\left (x \right )}}{a} & \text{for}\: b = 0 \\- \frac{A b \log{\left (- \sqrt{\frac{a}{a - b} + \frac{b}{a - b}} + \tanh{\left (\frac{x}{2} \right )} \right )}}{a b \sqrt{\frac{a}{a - b} + \frac{b}{a - b}} - b^{2} \sqrt{\frac{a}{a - b} + \frac{b}{a - b}}} + \frac{A b \log{\left (\sqrt{\frac{a}{a - b} + \frac{b}{a - b}} + \tanh{\left (\frac{x}{2} \right )} \right )}}{a b \sqrt{\frac{a}{a - b} + \frac{b}{a - b}} - b^{2} \sqrt{\frac{a}{a - b} + \frac{b}{a - b}}} + \frac{B a x \sqrt{\frac{a}{a - b} + \frac{b}{a - b}}}{a b \sqrt{\frac{a}{a - b} + \frac{b}{a - b}} - b^{2} \sqrt{\frac{a}{a - b} + \frac{b}{a - b}}} + \frac{B a \log{\left (- \sqrt{\frac{a}{a - b} + \frac{b}{a - b}} + \tanh{\left (\frac{x}{2} \right )} \right )}}{a b \sqrt{\frac{a}{a - b} + \frac{b}{a - b}} - b^{2} \sqrt{\frac{a}{a - b} + \frac{b}{a - b}}} - \frac{B a \log{\left (\sqrt{\frac{a}{a - b} + \frac{b}{a - b}} + \tanh{\left (\frac{x}{2} \right )} \right )}}{a b \sqrt{\frac{a}{a - b} + \frac{b}{a - b}} - b^{2} \sqrt{\frac{a}{a - b} + \frac{b}{a - b}}} - \frac{B b x \sqrt{\frac{a}{a - b} + \frac{b}{a - b}}}{a b \sqrt{\frac{a}{a - b} + \frac{b}{a - b}} - b^{2} \sqrt{\frac{a}{a - b} + \frac{b}{a - b}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17967, size = 68, normalized size = 1.13 \begin{align*} \frac{B x}{b} - \frac{2 \,{\left (B a - A b\right )} \arctan \left (\frac{b e^{x} + a}{\sqrt{-a^{2} + b^{2}}}\right )}{\sqrt{-a^{2} + b^{2}} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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