Optimal. Leaf size=52 \[ \frac{\log (a+b \sinh (x))}{b}-\frac{2 (b+c) \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\sqrt{a^2+b^2}} \]
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Rubi [A] time = 0.132436, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4401, 2660, 618, 206, 2668, 31} \[ \frac{\log (a+b \sinh (x))}{b}-\frac{2 (b+c) \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\sqrt{a^2+b^2}} \]
Antiderivative was successfully verified.
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Rule 4401
Rule 2660
Rule 618
Rule 206
Rule 2668
Rule 31
Rubi steps
\begin{align*} \int \frac{b+c+\cosh (x)}{a+b \sinh (x)} \, dx &=\int \left (\frac{\left (1+\frac{b}{c}\right ) c}{a+b \sinh (x)}+\frac{\cosh (x)}{a+b \sinh (x)}\right ) \, dx\\ &=(b+c) \int \frac{1}{a+b \sinh (x)} \, dx+\int \frac{\cosh (x)}{a+b \sinh (x)} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \sinh (x)\right )}{b}+(2 (b+c)) \operatorname{Subst}\left (\int \frac{1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )\\ &=\frac{\log (a+b \sinh (x))}{b}-(4 (b+c)) \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 )\\ &=-\frac{2 (b+c) \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\sqrt{a^2+b^2}}+\frac{\log (a+b \sinh (x))}{b}\\ \end{align*}
Mathematica [A] time = 0.106482, size = 60, normalized size = 1.15 \[ \frac{2 (b+c) \tan ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{-a^2-b^2}}\right )}{\sqrt{-a^2-b^2}}+\frac{\log (a+b \sinh (x))}{b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.025, size = 120, normalized size = 2.3 \begin{align*} -{\frac{1}{b}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+{\frac{1}{b}\ln \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) b-a \right ) }+2\,{\frac{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 ) }+2\,{\frac{c}{\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 ) }-{\frac{1}{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 = 2.62594, size = 458, normalized size = 8.81 \begin{align*} \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 \sinh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a^{2} b + b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 94.0366, size = 770, normalized size = 14.81 \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 = 1.15197, size = 117, normalized size = 2.25 \begin{align*} \frac{{\left (b + c\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}}} - \frac{x}{b} + \frac{\log \left ({\left | b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b \right |}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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