Optimal. Leaf size=38 \[ \frac{\sinh (x)}{b}-\frac{a \tan ^{-1}\left (\frac{\sqrt{b} \sinh (x)}{\sqrt{a+b}}\right )}{b^{3/2} \sqrt{a+b}} \]
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Rubi [A] time = 0.0592605, antiderivative size = 38, 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 = {3186, 388, 205} \[ \frac{\sinh (x)}{b}-\frac{a \tan ^{-1}\left (\frac{\sqrt{b} \sinh (x)}{\sqrt{a+b}}\right )}{b^{3/2} \sqrt{a+b}} \]
Antiderivative was successfully verified.
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Rule 3186
Rule 388
Rule 205
Rubi steps
\begin{align*} \int \frac{\cosh ^3(x)}{a+b \cosh ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1+x^2}{a+b+b x^2} \, dx,x,\sinh (x)\right )\\ &=\frac{\sinh (x)}{b}-\frac{a \operatorname{Subst}\left (\int \frac{1}{a+b+b x^2} \, dx,x,\sinh (x)\right )}{b}\\ &=-\frac{a \tan ^{-1}\left (\frac{\sqrt{b} \sinh (x)}{\sqrt{a+b}}\right )}{b^{3/2} \sqrt{a+b}}+\frac{\sinh (x)}{b}\\ \end{align*}
Mathematica [A] time = 0.028578, size = 38, normalized size = 1. \[ \frac{\sinh (x)}{b}-\frac{a \tan ^{-1}\left (\frac{\sqrt{b} \sinh (x)}{\sqrt{a+b}}\right )}{b^{3/2} \sqrt{a+b}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.033, size = 96, normalized size = 2.5 \begin{align*} -{a\arctan \left ({\frac{1}{2} \left ( 2\,\tanh \left ( x/2 \right ) \sqrt{a+b}+2\,\sqrt{a} \right ){\frac{1}{\sqrt{b}}}} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{a+b}}}}-{a\arctan \left ({\frac{1}{2} \left ( 2\,\tanh \left ( x/2 \right ) \sqrt{a+b}-2\,\sqrt{a} \right ){\frac{1}{\sqrt{b}}}} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{a+b}}}}-{\frac{1}{b} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}-{\frac{1}{b} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}}{2 \, b} - \frac{1}{8} \, \int \frac{16 \,{\left (a e^{\left (3 \, x\right )} + a e^{x}\right )}}{b^{2} e^{\left (4 \, x\right )} + b^{2} + 2 \,{\left (2 \, a b + b^{2}\right )} e^{\left (2 \, x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.98394, size = 1443, normalized size = 37.97 \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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