Optimal. Leaf size=155 \[ \frac{b \sinh (x)+c \cosh (x)}{2 \sqrt{b^2-c^2} \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}+\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt{2} \sqrt{\sqrt{b^2-c^2}+\sqrt{b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}}\right )}{2 \sqrt{2} \left (b^2-c^2\right )^{3/4}} \]
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Rubi [A] time = 0.132054, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3116, 3115, 2649, 206} \[ \frac{b \sinh (x)+c \cosh (x)}{2 \sqrt{b^2-c^2} \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}+\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt{2} \sqrt{\sqrt{b^2-c^2}+\sqrt{b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}}\right )}{2 \sqrt{2} \left (b^2-c^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 3116
Rule 3115
Rule 2649
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}} \, dx &=\frac{c \cosh (x)+b \sinh (x)}{2 \sqrt{b^2-c^2} \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}+\frac{\int \frac{1}{\sqrt{\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)}} \, dx}{4 \sqrt{b^2-c^2}}\\ &=\frac{c \cosh (x)+b \sinh (x)}{2 \sqrt{b^2-c^2} \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}+\frac{\int \frac{1}{\sqrt{\sqrt{b^2-c^2}+\sqrt{b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}} \, dx}{4 \sqrt{b^2-c^2}}\\ &=\frac{c \cosh (x)+b \sinh (x)}{2 \sqrt{b^2-c^2} \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}+\frac{i \operatorname{Subst}\left (\int \frac{1}{2 \sqrt{b^2-c^2}-x^2} \, dx,x,-\frac{i \sqrt{b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt{\sqrt{b^2-c^2}+\sqrt{b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}}\right )}{2 \sqrt{b^2-c^2}}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt{2} \sqrt{\sqrt{b^2-c^2}+\sqrt{b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}}\right )}{2 \sqrt{2} \left (b^2-c^2\right )^{3/4}}+\frac{c \cosh (x)+b \sinh (x)}{2 \sqrt{b^2-c^2} \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}\\ \end{align*}
Mathematica [F] time = 180.002, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
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Maple [B] time = 0.785, size = 417, normalized size = 2.7 \begin{align*}{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\frac{\cosh \left ( x \right ) \sqrt{2}}{2}} \right ){\frac{1}{\sqrt{{b}^{2}-{c}^{2}}}}{\frac{1}{\sqrt{-{(\sinh \left ( x \right ){b}^{2}-\sinh \left ( x \right ){c}^{2}-{b}^{2}+{c}^{2}){\frac{1}{\sqrt{{b}^{2}-{c}^{2}}}}}}}}}+{\frac{\sqrt{2}}{ \left ( 4\,b-4\,c \right ) \left ( b+c \right ) \sinh \left ( x \right ) }\sqrt{-\sqrt{{b}^{2}-{c}^{2}} \left ( \sinh \left ( x \right ) -1 \right ) \left ( \sinh \left ( x \right ) \right ) ^{2}}\sqrt{{b}^{2}-{c}^{2}} \left ( \ln \left ( -2\,{\frac{\cosh \left ( x \right ) \sqrt{{b}^{2}-{c}^{2}}\sqrt{2}\sinh \left ( x \right ) -\sinh \left ( x \right ) \sqrt{{b}^{2}-{c}^{2}}-\cosh \left ( x \right ) \sqrt{{b}^{2}-{c}^{2}}\sqrt{2}+\sqrt{{b}^{2}-{c}^{2}}-\sqrt{-\sqrt{{b}^{2}-{c}^{2}} \left ( \sinh \left ( x \right ) -1 \right ) }\sqrt{-\sqrt{{b}^{2}-{c}^{2}} \left ( \sinh \left ( x \right ) -1 \right ) \left ( \sinh \left ( x \right ) \right ) ^{2}}}{\cosh \left ( x \right ) -\sqrt{2}}} \right ) -\ln \left ( 2\,{\frac{\cosh \left ( x \right ) \sqrt{{b}^{2}-{c}^{2}}\sqrt{2}\sinh \left ( x \right ) +\sinh \left ( x \right ) \sqrt{{b}^{2}-{c}^{2}}-\cosh \left ( x \right ) \sqrt{{b}^{2}-{c}^{2}}\sqrt{2}-\sqrt{{b}^{2}-{c}^{2}}+\sqrt{-\sqrt{{b}^{2}-{c}^{2}} \left ( \sinh \left ( x \right ) -1 \right ) }\sqrt{-\sqrt{{b}^{2}-{c}^{2}} \left ( \sinh \left ( x \right ) -1 \right ) \left ( \sinh \left ( x \right ) \right ) ^{2}}}{\cosh \left ( x \right ) +\sqrt{2}}} \right ) \right ){\frac{1}{\sqrt{-\sqrt{{b}^{2}-{c}^{2}} \left ( \sinh \left ( x \right ) -1 \right ) }}}{\frac{1}{\sqrt{-{(\sinh \left ( x \right ){b}^{2}-\sinh \left ( x \right ){c}^{2}-{b}^{2}+{c}^{2}){\frac{1}{\sqrt{{b}^{2}-{c}^{2}}}}}}}}} \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*} \int \frac{1}{{\left (b \cosh \left (x\right ) + c \sinh \left (x\right ) + \sqrt{b^{2} - c^{2}}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b \cosh{\left (x \right )} + c \sinh{\left (x \right )} + \sqrt{b^{2} - c^{2}}\right )^{\frac{3}{2}}}\, dx \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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