Optimal. Leaf size=205 \[ \frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )}{d x}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )}{d x}+\frac{2 \sinh ^2\left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{d x} \]
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Rubi [A] time = 0.0264593, antiderivative size = 205, 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 = {5878} \[ \frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )}{d x}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )}{d x}+\frac{2 \sinh ^2\left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{d x} \]
Antiderivative was successfully verified.
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Rule 5878
Rubi steps
\begin{align*} \int \sqrt{a+b \cosh ^{-1}\left (1+d x^2\right )} \, dx &=-\frac{\sqrt{b} \sqrt{\frac{\pi }{2}} \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 ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (1+d x^2\right )\right )}{d x}+\frac{\sqrt{b} \sqrt{\frac{\pi }{2}} \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 ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (1+d x^2\right )\right )}{d x}+\frac{2 \sqrt{a+b \cosh ^{-1}\left (1+d x^2\right )} \sinh ^2\left (\frac{1}{2} \cosh ^{-1}\left (1+d x^2\right )\right )}{d x}\\ \end{align*}
Mathematica [A] time = 0.291669, size = 210, normalized size = 1.02 \[ \frac{x \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \left (\sqrt{2 \pi } \sqrt{b} \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )+\sqrt{2 \pi } \sqrt{b} \left (\sinh \left (\frac{a}{2 b}\right )-\cosh \left (\frac{a}{2 b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )+4 \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}\right )}{2 \sqrt{d x^2} \sqrt{\frac{d x^2}{d x^2+2}} \sqrt{d x^2+2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.068, 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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