Optimal. Leaf size=64 \[ \frac{b}{\left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))}+\frac{a \tan ^{-1}\left (\frac{a \sinh (x)+b \cosh (x)}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}} \]
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Rubi [A] time = 0.0550593, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3155, 3074, 206} \[ \frac{b}{\left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))}+\frac{a \tan ^{-1}\left (\frac{a \sinh (x)+b \cosh (x)}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3155
Rule 3074
Rule 206
Rubi steps
\begin{align*} \int \frac{\cosh (x)}{(a \cosh (x)+b \sinh (x))^2} \, dx &=\frac{b}{\left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))}+\frac{a \int \frac{1}{a \cosh (x)+b \sinh (x)} \, dx}{a^2-b^2}\\ &=\frac{b}{\left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))}+\frac{(i a) \operatorname{Subst}\left (\int \frac{1}{a^2-b^2-x^2} \, dx,x,-i b \cosh (x)-i a \sinh (x)\right )}{a^2-b^2}\\ &=\frac{a \tan ^{-1}\left (\frac{b \cosh (x)+a \sinh (x)}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+\frac{b}{\left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))}\\ \end{align*}
Mathematica [A] time = 0.135574, size = 124, normalized size = 1.94 \[ \frac{2 a^2 \sqrt{a+b} \cosh (x) \tan ^{-1}\left (\frac{a \tanh \left (\frac{x}{2}\right )+b}{\sqrt{a-b} \sqrt{a+b}}\right )+2 a b \sqrt{a+b} \sinh (x) \tan ^{-1}\left (\frac{a \tanh \left (\frac{x}{2}\right )+b}{\sqrt{a-b} \sqrt{a+b}}\right )+b \sqrt{a-b} (a+b)}{(a-b)^{3/2} (a+b)^2 (a \cosh (x)+b \sinh (x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 98, normalized size = 1.5 \begin{align*} 2\,{\frac{1}{a+2\,\tanh \left ( x/2 \right ) b+a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}} \left ({\frac{{b}^{2}\tanh \left ( x/2 \right ) }{a \left ({a}^{2}-{b}^{2} \right ) }}+{\frac{b}{{a}^{2}-{b}^{2}}} \right ) }+2\,{\frac{a}{ \left ({a}^{2}-{b}^{2} \right ) ^{3/2}}\arctan \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 = 1.92698, size = 1474, normalized size = 23.03 \begin{align*} \text{result too large to display} \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 = 1.15309, size = 97, normalized size = 1.52 \begin{align*} \frac{2 \, a \arctan \left (\frac{a e^{x} + b e^{x}}{\sqrt{a^{2} - b^{2}}}\right )}{{\left (a^{2} - b^{2}\right )}^{\frac{3}{2}}} + \frac{2 \, b e^{x}}{{\left (a^{2} - b^{2}\right )}{\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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