Optimal. Leaf size=102 \[ -\frac{2 i \sqrt{a+b \cosh (x)+c \sinh (x)} E\left (\frac{1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac{2 \sqrt{b^2-c^2}}{a+\sqrt{b^2-c^2}}\right )}{\sqrt{\frac{a+b \cosh (x)+c \sinh (x)}{a+\sqrt{b^2-c^2}}}} \]
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Rubi [A] time = 0.0714839, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3119, 2653} \[ -\frac{2 i \sqrt{a+b \cosh (x)+c \sinh (x)} E\left (\frac{1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac{2 \sqrt{b^2-c^2}}{a+\sqrt{b^2-c^2}}\right )}{\sqrt{\frac{a+b \cosh (x)+c \sinh (x)}{a+\sqrt{b^2-c^2}}}} \]
Antiderivative was successfully verified.
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Rule 3119
Rule 2653
Rubi steps
\begin{align*} \int \sqrt{a+b \cosh (x)+c \sinh (x)} \, dx &=\frac{\sqrt{a+b \cosh (x)+c \sinh (x)} \int \sqrt{\frac{a}{a+\sqrt{b^2-c^2}}+\frac{\sqrt{b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}{a+\sqrt{b^2-c^2}}} \, dx}{\sqrt{\frac{a+b \cosh (x)+c \sinh (x)}{a+\sqrt{b^2-c^2}}}}\\ &=-\frac{2 i E\left (\frac{1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac{2 \sqrt{b^2-c^2}}{a+\sqrt{b^2-c^2}}\right ) \sqrt{a+b \cosh (x)+c \sinh (x)}}{\sqrt{\frac{a+b \cosh (x)+c \sinh (x)}{a+\sqrt{b^2-c^2}}}}\\ \end{align*}
Mathematica [C] time = 6.10253, size = 1401, normalized size = 13.74 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.373, size = 314, normalized size = 3.1 \begin{align*}{ \left ( -{b}^{2}+{c}^{2} \right ) \cosh \left ( x \right ){\frac{1}{\sqrt{{b}^{2}-{c}^{2}}}}{\frac{1}{\sqrt{{ \left ( -\sinh \left ( x \right ){b}^{2}+\sinh \left ( x \right ){c}^{2}+a\sqrt{{b}^{2}-{c}^{2}} \right ){\frac{1}{\sqrt{{b}^{2}-{c}^{2}}}}}}}}}+{\frac{a}{\sinh \left ( x \right ) }\sqrt{{ \left ( \sinh \left ( x \right ) \right ) ^{2} \left ( -\sinh \left ( x \right ){b}^{2}+\sinh \left ( x \right ){c}^{2}+a\sqrt{{b}^{2}-{c}^{2}} \right ){\frac{1}{\sqrt{{b}^{2}-{c}^{2}}}}}}\ln \left ({ \left ( \cosh \left ( x \right ) \sinh \left ( x \right ) \left ( -{b}^{2}+{c}^{2} \right ) +\cosh \left ( x \right ) \sqrt{{b}^{2}-{c}^{2}}a+\sqrt{{ \left ( -{b}^{2}+{c}^{2} \right ) \left ( \sinh \left ( x \right ) \right ) ^{3}{\frac{1}{\sqrt{{b}^{2}-{c}^{2}}}}}+a \left ( \sinh \left ( x \right ) \right ) ^{2}}\sqrt{{b}^{2}-{c}^{2}}\sqrt{{ \left ( -{b}^{2}+{c}^{2} \right ) \sinh \left ( x \right ){\frac{1}{\sqrt{{b}^{2}-{c}^{2}}}}}+a} \right ){\frac{1}{\sqrt{{b}^{2}-{c}^{2}}}}{\frac{1}{\sqrt{{ \left ( -\sinh \left ( x \right ){b}^{2}+\sinh \left ( x \right ){c}^{2}+a\sqrt{{b}^{2}-{c}^{2}} \right ){\frac{1}{\sqrt{{b}^{2}-{c}^{2}}}}}}}}} \right ) \sqrt{{b}^{2}-{c}^{2}} \left ( -\sinh \left ( x \right ){b}^{2}+\sinh \left ( x \right ){c}^{2}+a\sqrt{{b}^{2}-{c}^{2}} \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*} \int \sqrt{b \cosh \left (x\right ) + c \sinh \left (x\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{b \cosh \left (x\right ) + c \sinh \left (x\right ) + a}, x\right ) \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 \sqrt{a + b \cosh{\left (x \right )} + c \sinh{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \cosh \left (x\right ) + c \sinh \left (x\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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