Optimal. Leaf size=206 \[ -\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )}{d x}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )}{d x}+\frac{2 \cosh ^2\left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{d x} \]
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Rubi [A] time = 0.0263156, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {5879} \[ -\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )}{d x}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )}{d x}+\frac{2 \cosh ^2\left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{d x} \]
Antiderivative was successfully verified.
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Rule 5879
Rubi steps
\begin{align*} \int \sqrt{a+b \cosh ^{-1}\left (-1+d x^2\right )} \, dx &=\frac{2 \sqrt{a+b \cosh ^{-1}\left (-1+d x^2\right )} \cosh ^2\left (\frac{1}{2} \cosh ^{-1}\left (-1+d x^2\right )\right )}{d x}-\frac{\sqrt{b} \sqrt{\frac{\pi }{2}} \cosh \left (\frac{1}{2} \cosh ^{-1}\left (-1+d x^2\right )\right ) \text{erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (-1+d x^2\right )}}{\sqrt{2} \sqrt{b}}\right ) \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right )}{d x}-\frac{\sqrt{b} \sqrt{\frac{\pi }{2}} \cosh \left (\frac{1}{2} \cosh ^{-1}\left (-1+d x^2\right )\right ) \text{erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (-1+d x^2\right )}}{\sqrt{2} \sqrt{b}}\right ) \left (\cosh \left (\frac{a}{2 b}\right )+\sinh \left (\frac{a}{2 b}\right )\right )}{d x}\\ \end{align*}
Mathematica [A] time = 0.250142, size = 178, normalized size = 0.86 \[ \frac{\cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (-\sqrt{2 \pi } \sqrt{b} \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )+\sqrt{2 \pi } \sqrt{b} \left (\sinh \left (\frac{a}{2 b}\right )-\cosh \left (\frac{a}{2 b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )+4 \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}\right )}{2 d x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.063, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a+b{\rm arccosh} \left (d{x}^{2}-1\right )}\, dx \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 \operatorname{arcosh}\left (d x^{2} - 1\right ) + a}\,{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: UnboundLocalError} \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 \operatorname{acosh}{\left (d x^{2} - 1 \right )}}\, 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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