Optimal. Leaf size=47 \[ \frac{\text{Chi}\left (2 \sinh ^{-1}(a+b x)\right )}{2 b}+\frac{\text{Chi}\left (4 \sinh ^{-1}(a+b x)\right )}{8 b}+\frac{3 \log \left (\sinh ^{-1}(a+b x)\right )}{8 b} \]
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Rubi [A] time = 0.14352, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {5867, 5699, 3312, 3301} \[ \frac{\text{Chi}\left (2 \sinh ^{-1}(a+b x)\right )}{2 b}+\frac{\text{Chi}\left (4 \sinh ^{-1}(a+b x)\right )}{8 b}+\frac{3 \log \left (\sinh ^{-1}(a+b x)\right )}{8 b} \]
Antiderivative was successfully verified.
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Rule 5867
Rule 5699
Rule 3312
Rule 3301
Rubi steps
\begin{align*} \int \frac{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{\sinh ^{-1}(a+b x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^{3/2}}{\sinh ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\cosh ^4(x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{3}{8 x}+\frac{\cosh (2 x)}{2 x}+\frac{\cosh (4 x)}{8 x}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=\frac{3 \log \left (\sinh ^{-1}(a+b x)\right )}{8 b}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (4 x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{8 b}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b}\\ &=\frac{\text{Chi}\left (2 \sinh ^{-1}(a+b x)\right )}{2 b}+\frac{\text{Chi}\left (4 \sinh ^{-1}(a+b x)\right )}{8 b}+\frac{3 \log \left (\sinh ^{-1}(a+b x)\right )}{8 b}\\ \end{align*}
Mathematica [A] time = 0.286448, size = 37, normalized size = 0.79 \[ \frac{4 \text{Chi}\left (2 \sinh ^{-1}(a+b x)\right )+\text{Chi}\left (4 \sinh ^{-1}(a+b x)\right )+3 \log \left (\sinh ^{-1}(a+b x)\right )}{8 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 42, normalized size = 0.9 \begin{align*}{\frac{{\it Chi} \left ( 2\,{\it Arcsinh} \left ( bx+a \right ) \right ) }{2\,b}}+{\frac{{\it Chi} \left ( 4\,{\it Arcsinh} \left ( bx+a \right ) \right ) }{8\,b}}+{\frac{3\,\ln \left ({\it Arcsinh} \left ( bx+a \right ) \right ) }{8\,b}} \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{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac{3}{2}}}{\operatorname{arsinh}\left (b x + a\right )}\,{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 (\frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac{3}{2}}}{\operatorname{arsinh}\left (b x + a\right )}, 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 \frac{\left (a^{2} + 2 a b x + b^{2} x^{2} + 1\right )^{\frac{3}{2}}}{\operatorname{asinh}{\left (a + b 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 \frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac{3}{2}}}{\operatorname{arsinh}\left (b x + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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