Optimal. Leaf size=251 \[ \frac{\sqrt{\pi } a^2 \text{Erfi}\left (\frac{1}{2} \left (2 \sinh ^{-1}(a+b x)-1\right )\right )}{4 \sqrt [4]{e} b^3}+\frac{\sqrt{\pi } a^2 \text{Erfi}\left (\frac{1}{2} \left (2 \sinh ^{-1}(a+b x)+1\right )\right )}{4 \sqrt [4]{e} b^3}-\frac{\sqrt{\pi } a \text{Erfi}\left (1-\sinh ^{-1}(a+b x)\right )}{4 e b^3}-\frac{\sqrt{\pi } a \text{Erfi}\left (\sinh ^{-1}(a+b x)+1\right )}{4 e b^3}+\frac{\sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sinh ^{-1}(a+b x)-3\right )\right )}{16 e^{9/4} b^3}-\frac{\sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sinh ^{-1}(a+b x)-1\right )\right )}{16 \sqrt [4]{e} b^3}-\frac{\sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sinh ^{-1}(a+b x)+1\right )\right )}{16 \sqrt [4]{e} b^3}+\frac{\sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sinh ^{-1}(a+b x)+3\right )\right )}{16 e^{9/4} b^3} \]
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Rubi [A] time = 0.516042, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {5898, 6741, 12, 6742, 5513, 2234, 2204, 5514} \[ \frac{\sqrt{\pi } a^2 \text{Erfi}\left (\frac{1}{2} \left (2 \sinh ^{-1}(a+b x)-1\right )\right )}{4 \sqrt [4]{e} b^3}+\frac{\sqrt{\pi } a^2 \text{Erfi}\left (\frac{1}{2} \left (2 \sinh ^{-1}(a+b x)+1\right )\right )}{4 \sqrt [4]{e} b^3}-\frac{\sqrt{\pi } a \text{Erfi}\left (1-\sinh ^{-1}(a+b x)\right )}{4 e b^3}-\frac{\sqrt{\pi } a \text{Erfi}\left (\sinh ^{-1}(a+b x)+1\right )}{4 e b^3}+\frac{\sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sinh ^{-1}(a+b x)-3\right )\right )}{16 e^{9/4} b^3}-\frac{\sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sinh ^{-1}(a+b x)-1\right )\right )}{16 \sqrt [4]{e} b^3}-\frac{\sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sinh ^{-1}(a+b x)+1\right )\right )}{16 \sqrt [4]{e} b^3}+\frac{\sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sinh ^{-1}(a+b x)+3\right )\right )}{16 e^{9/4} b^3} \]
Antiderivative was successfully verified.
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Rule 5898
Rule 6741
Rule 12
Rule 6742
Rule 5513
Rule 2234
Rule 2204
Rule 5514
Rubi steps
\begin{align*} \int e^{\sinh ^{-1}(a+b x)^2} x^2 \, dx &=\frac{\operatorname{Subst}\left (\int e^{x^2} \cosh (x) \left (-\frac{a}{b}+\frac{\sinh (x)}{b}\right )^2 \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{e^{x^2} \cosh (x) (a-\sinh (x))^2}{b^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int e^{x^2} \cosh (x) (a-\sinh (x))^2 \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^2 e^{x^2} \cosh (x)-2 a e^{x^2} \cosh (x) \sinh (x)+e^{x^2} \cosh (x) \sinh ^2(x)\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}\\ &=\frac{\operatorname{Subst}\left (\int e^{x^2} \cosh (x) \sinh ^2(x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}-\frac{(2 a) \operatorname{Subst}\left (\int e^{x^2} \cosh (x) \sinh (x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}+\frac{a^2 \operatorname{Subst}\left (\int e^{x^2} \cosh (x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{8} e^{-3 x+x^2}-\frac{1}{8} e^{-x+x^2}-\frac{e^{x+x^2}}{8}+\frac{1}{8} e^{3 x+x^2}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}-\frac{(2 a) \operatorname{Subst}\left (\int \left (-\frac{1}{4} e^{-2 x+x^2}+\frac{1}{4} e^{2 x+x^2}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}+\frac{a^2 \operatorname{Subst}\left (\int \left (\frac{1}{2} e^{-x+x^2}+\frac{e^{x+x^2}}{2}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}\\ &=\frac{\operatorname{Subst}\left (\int e^{-3 x+x^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{8 b^3}-\frac{\operatorname{Subst}\left (\int e^{-x+x^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{8 b^3}-\frac{\operatorname{Subst}\left (\int e^{x+x^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{8 b^3}+\frac{\operatorname{Subst}\left (\int e^{3 x+x^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{8 b^3}+\frac{a \operatorname{Subst}\left (\int e^{-2 x+x^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b^3}-\frac{a \operatorname{Subst}\left (\int e^{2 x+x^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b^3}+\frac{a^2 \operatorname{Subst}\left (\int e^{-x+x^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b^3}+\frac{a^2 \operatorname{Subst}\left (\int e^{x+x^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b^3}\\ &=\frac{\operatorname{Subst}\left (\int e^{\frac{1}{4} (-3+2 x)^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{8 b^3 e^{9/4}}+\frac{\operatorname{Subst}\left (\int e^{\frac{1}{4} (3+2 x)^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{8 b^3 e^{9/4}}+\frac{a \operatorname{Subst}\left (\int e^{\frac{1}{4} (-2+2 x)^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b^3 e}-\frac{a \operatorname{Subst}\left (\int e^{\frac{1}{4} (2+2 x)^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b^3 e}-\frac{\operatorname{Subst}\left (\int e^{\frac{1}{4} (-1+2 x)^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{8 b^3 \sqrt [4]{e}}-\frac{\operatorname{Subst}\left (\int e^{\frac{1}{4} (1+2 x)^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{8 b^3 \sqrt [4]{e}}+\frac{a^2 \operatorname{Subst}\left (\int e^{\frac{1}{4} (-1+2 x)^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b^3 \sqrt [4]{e}}+\frac{a^2 \operatorname{Subst}\left (\int e^{\frac{1}{4} (1+2 x)^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b^3 \sqrt [4]{e}}\\ &=-\frac{a \sqrt{\pi } \text{erfi}\left (1-\sinh ^{-1}(a+b x)\right )}{4 b^3 e}-\frac{a \sqrt{\pi } \text{erfi}\left (1+\sinh ^{-1}(a+b x)\right )}{4 b^3 e}+\frac{\sqrt{\pi } \text{erfi}\left (\frac{1}{2} \left (-3+2 \sinh ^{-1}(a+b x)\right )\right )}{16 b^3 e^{9/4}}-\frac{\sqrt{\pi } \text{erfi}\left (\frac{1}{2} \left (-1+2 \sinh ^{-1}(a+b x)\right )\right )}{16 b^3 \sqrt [4]{e}}+\frac{a^2 \sqrt{\pi } \text{erfi}\left (\frac{1}{2} \left (-1+2 \sinh ^{-1}(a+b x)\right )\right )}{4 b^3 \sqrt [4]{e}}-\frac{\sqrt{\pi } \text{erfi}\left (\frac{1}{2} \left (1+2 \sinh ^{-1}(a+b x)\right )\right )}{16 b^3 \sqrt [4]{e}}+\frac{a^2 \sqrt{\pi } \text{erfi}\left (\frac{1}{2} \left (1+2 \sinh ^{-1}(a+b x)\right )\right )}{4 b^3 \sqrt [4]{e}}+\frac{\sqrt{\pi } \text{erfi}\left (\frac{1}{2} \left (3+2 \sinh ^{-1}(a+b x)\right )\right )}{16 b^3 e^{9/4}}\\ \end{align*}
Mathematica [A] time = 0.183048, size = 138, normalized size = 0.55 \[ -\frac{\sqrt{\pi } \left (-4 e^2 a^2 \text{Erfi}\left (\sinh ^{-1}(a+b x)+\frac{1}{2}\right )+e^2 \left (4 a^2-1\right ) \text{Erfi}\left (\frac{1}{2}-\sinh ^{-1}(a+b x)\right )+4 e^{5/4} a \text{Erfi}\left (1-\sinh ^{-1}(a+b x)\right )+4 e^{5/4} a \text{Erfi}\left (\sinh ^{-1}(a+b x)+1\right )+\text{Erfi}\left (\frac{3}{2}-\sinh ^{-1}(a+b x)\right )+e^2 \text{Erfi}\left (\sinh ^{-1}(a+b x)+\frac{1}{2}\right )-\text{Erfi}\left (\sinh ^{-1}(a+b x)+\frac{3}{2}\right )\right )}{16 e^{9/4} b^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.01, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{ \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{2}}}{x}^{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 x^{2} e^{\left (\operatorname{arsinh}\left (b x + a\right )^{2}\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 (x^{2} e^{\left (\operatorname{arsinh}\left (b x + a\right )^{2}\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 x^{2} e^{\operatorname{asinh}^{2}{\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 x^{2} e^{\left (\operatorname{arsinh}\left (b x + a\right )^{2}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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