Optimal. Leaf size=238 \[ \frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )}{d x}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )}{d x}-\frac{3 b \left (d x^4+2 x^2\right ) \sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{x \sqrt{d x^2} \sqrt{d x^2+2}}+x \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^{3/2} \]
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Rubi [A] time = 0.0967674, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {5880, 5883} \[ \frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )}{d x}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )}{d x}-\frac{3 b \left (d x^4+2 x^2\right ) \sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{x \sqrt{d x^2} \sqrt{d x^2+2}}+x \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 5880
Rule 5883
Rubi steps
\begin{align*} \int \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^{3/2} \, dx &=-\frac{3 b \left (2 x^2+d x^4\right ) \sqrt{a+b \cosh ^{-1}\left (1+d x^2\right )}}{x \sqrt{d x^2} \sqrt{2+d x^2}}+x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^{3/2}+\left (3 b^2\right ) \int \frac{1}{\sqrt{a+b \cosh ^{-1}\left (1+d x^2\right )}} \, dx\\ &=-\frac{3 b \left (2 x^2+d x^4\right ) \sqrt{a+b \cosh ^{-1}\left (1+d x^2\right )}}{x \sqrt{d x^2} \sqrt{2+d x^2}}+x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^{3/2}+\frac{3 b^{3/2} \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{3 b^{3/2} \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}\\ \end{align*}
Mathematica [A] time = 0.657109, size = 254, normalized size = 1.07 \[ \frac{x \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \left (3 \sqrt{2 \pi } b^{3/2} \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )+3 \sqrt{2 \pi } b^{3/2} \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )+4 \sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )} \left (a \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right )-3 b \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right )+b \cosh ^{-1}\left (d x^2+1\right ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right )\right )\right )}{2 \sqrt{d x^2} \sqrt{\frac{d x^2}{d x^2+2}} \sqrt{d x^2+2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.066, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\rm arccosh} \left (d{x}^{2}+1\right ) \right ) ^{{\frac{3}{2}}}\, 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{\left (b \operatorname{arcosh}\left (d x^{2} + 1\right ) + a\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: 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 \left (a + b \operatorname{acosh}{\left (d x^{2} + 1 \right )}\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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