Optimal. Leaf size=100 \[ \frac{b \sinh (x)+c \cosh (x)}{3 \sqrt{b^2-c^2} \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2}-\frac{\sqrt{b^2-c^2} \sinh (x)+c}{3 c \sqrt{b^2-c^2} (b \sinh (x)+c \cosh (x))} \]
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Rubi [A] time = 0.0806866, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3116, 3114} \[ \frac{b \sinh (x)+c \cosh (x)}{3 \sqrt{b^2-c^2} \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2}-\frac{\sqrt{b^2-c^2} \sinh (x)+c}{3 c \sqrt{b^2-c^2} (b \sinh (x)+c \cosh (x))} \]
Antiderivative was successfully verified.
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Rule 3116
Rule 3114
Rubi steps
\begin{align*} \int \frac{1}{\left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2} \, dx &=\frac{c \cosh (x)+b \sinh (x)}{3 \sqrt{b^2-c^2} \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2}+\frac{\int \frac{1}{\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)} \, dx}{3 \sqrt{b^2-c^2}}\\ &=\frac{c \cosh (x)+b \sinh (x)}{3 \sqrt{b^2-c^2} \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2}-\frac{c+\sqrt{b^2-c^2} \sinh (x)}{3 c \sqrt{b^2-c^2} (c \cosh (x)+b \sinh (x))}\\ \end{align*}
Mathematica [A] time = 0.143531, size = 68, normalized size = 0.68 \[ -\frac{-2 c \sqrt{b^2-c^2}+b^2 \sinh ^3(x)+2 b c \cosh ^3(x)+2 c^2 \sinh (x)+c^2 \sinh (x) \cosh ^2(x)}{3 c (b \sinh (x)+c \cosh (x))^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.086, size = 217, normalized size = 2.2 \begin{align*} 2\,{\frac{\sqrt{{b}^{2}-{c}^{2}}+b}{{c}^{2}} \left ({\frac{ \left ( \sqrt{{b}^{2}-{c}^{2}}+b \right ) \left ( \tanh \left ( x/2 \right ) \right ) ^{2}}{{c}^{2}}}+{\frac{ \left ( 2\,{b}^{2}-{c}^{2}+2\,\sqrt{{b}^{2}-{c}^{2}}b \right ) \tanh \left ( x/2 \right ) }{{c}^{3}}}+2/3\,{\frac{2\,\sqrt{{b}^{2}-{c}^{2}}{b}^{2}-\sqrt{{b}^{2}-{c}^{2}}{c}^{2}+2\,{b}^{3}-2\,b{c}^{2}}{{c}^{4}}} \right ) \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+2\,{\frac{\sqrt{ \left ( b-c \right ) \left ( b+c \right ) }\tanh \left ( x/2 \right ) }{c}}+2\,{\frac{\tanh \left ( x/2 \right ) b}{c}}+2\,{\frac{\sqrt{ \left ( b-c \right ) \left ( b+c \right ) }b}{{c}^{2}}}+2\,{\frac{{b}^{2}}{{c}^{2}}}-1 \right ) ^{-1} \left ( \tanh \left ( x/2 \right ) +{\frac{\sqrt{ \left ( b-c \right ) \left ( b+c \right ) }}{c}}+{\frac{b}{c}} \right ) ^{-1}} \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: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.05014, size = 1639, normalized size = 16.39 \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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