Optimal. Leaf size=55 \[ \frac{B x}{b}-\frac{2 (A b-a B) \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{b \sqrt{a^2+b^2}} \]
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Rubi [A] time = 0.0750333, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {2735, 2660, 618, 206} \[ \frac{B x}{b}-\frac{2 (A b-a B) \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{b \sqrt{a^2+b^2}} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2660
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{A+B \sinh (x)}{a+b \sinh (x)} \, dx &=\frac{B x}{b}-\frac{(i (i A b-i a B)) \int \frac{1}{a+b \sinh (x)} \, dx}{b}\\ &=\frac{B x}{b}-\frac{(2 i (i A b-i a B)) \operatorname{Subst}\left (\int \frac{1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{b}\\ &=\frac{B x}{b}+\frac{(4 i (i A b-i a B)) \operatorname{Subst}\left (\int \frac{1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tanh \left (\frac{x}{2}\right )\right )}{b}\\ &=\frac{B x}{b}-\frac{2 (A b-a B) \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{b \sqrt{a^2+b^2}}\\ \end{align*}
Mathematica [A] time = 0.102054, size = 61, normalized size = 1.11 \[ \frac{\frac{2 (A b-a B) \tan ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{-a^2-b^2}}\right )}{\sqrt{-a^2-b^2}}+B x}{b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 101, normalized size = 1.8 \begin{align*}{\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 ) }+2\,{\frac{A}{\sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,a\tanh \left ( x/2 \right ) -2\,b}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) }-2\,{\frac{aB}{b\sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,a\tanh \left ( x/2 \right ) -2\,b}{\sqrt{{a}^{2}+{b}^{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 [B] time = 1.87912, size = 387, normalized size = 7.04 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 97.3601, size = 469, normalized size = 8.53 \begin{align*} \begin{cases} \tilde{\infty } \left (A \log{\left (\tanh{\left (\frac{x}{2} \right )} \right )} + B x\right ) & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2 A b^{2} \tanh{\left (\frac{x}{2} \right )}}{- b^{3} \tanh{\left (\frac{x}{2} \right )} + i b^{2} \sqrt{b^{2}}} - \frac{B b^{2} x \tanh{\left (\frac{x}{2} \right )}}{- b^{3} \tanh{\left (\frac{x}{2} \right )} + i b^{2} \sqrt{b^{2}}} + \frac{i B b x \sqrt{b^{2}}}{- b^{3} \tanh{\left (\frac{x}{2} \right )} + i b^{2} \sqrt{b^{2}}} - \frac{2 i B b \sqrt{b^{2}} \tanh{\left (\frac{x}{2} \right )}}{- b^{3} \tanh{\left (\frac{x}{2} \right )} + i b^{2} \sqrt{b^{2}}} & \text{for}\: a = - \sqrt{- b^{2}} \\\frac{2 A b^{2} \tanh{\left (\frac{x}{2} \right )}}{b^{3} \tanh{\left (\frac{x}{2} \right )} + i b^{2} \sqrt{b^{2}}} + \frac{B b^{2} x \tanh{\left (\frac{x}{2} \right )}}{b^{3} \tanh{\left (\frac{x}{2} \right )} + i b^{2} \sqrt{b^{2}}} + \frac{i B b x \sqrt{b^{2}}}{b^{3} \tanh{\left (\frac{x}{2} \right )} + i b^{2} \sqrt{b^{2}}} - \frac{2 i B b \sqrt{b^{2}} \tanh{\left (\frac{x}{2} \right )}}{b^{3} \tanh{\left (\frac{x}{2} \right )} + i b^{2} \sqrt{b^{2}}} & \text{for}\: a = \sqrt{- b^{2}} \\\frac{A x + B \cosh{\left (x \right )}}{a} & \text{for}\: b = 0 \\\frac{A \log{\left (\tanh{\left (\frac{x}{2} \right )} \right )} + B x}{b} & \text{for}\: a = 0 \\- \frac{A \log{\left (\tanh{\left (\frac{x}{2} \right )} - \frac{b}{a} - \frac{\sqrt{a^{2} + b^{2}}}{a} \right )}}{\sqrt{a^{2} + b^{2}}} + \frac{A \log{\left (\tanh{\left (\frac{x}{2} \right )} - \frac{b}{a} + \frac{\sqrt{a^{2} + b^{2}}}{a} \right )}}{\sqrt{a^{2} + b^{2}}} + \frac{B a \log{\left (\tanh{\left (\frac{x}{2} \right )} - \frac{b}{a} - \frac{\sqrt{a^{2} + b^{2}}}{a} \right )}}{b \sqrt{a^{2} + b^{2}}} - \frac{B a \log{\left (\tanh{\left (\frac{x}{2} \right )} - \frac{b}{a} + \frac{\sqrt{a^{2} + b^{2}}}{a} \right )}}{b \sqrt{a^{2} + b^{2}}} + \frac{B x}{b} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22866, size = 101, normalized size = 1.84 \begin{align*} \frac{B x}{b} - \frac{{\left (B a - A b\right )} \log \left (\frac{{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{\sqrt{a^{2} + b^{2}} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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