Optimal. Leaf size=65 \[ \frac{\sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sinh ^{-1}(a+b x)-1\right )\right )}{4 \sqrt [4]{e} b}+\frac{\sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sinh ^{-1}(a+b x)+1\right )\right )}{4 \sqrt [4]{e} b} \]
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Rubi [A] time = 0.055106, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5896, 5513, 2234, 2204} \[ \frac{\sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sinh ^{-1}(a+b x)-1\right )\right )}{4 \sqrt [4]{e} b}+\frac{\sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sinh ^{-1}(a+b x)+1\right )\right )}{4 \sqrt [4]{e} b} \]
Antiderivative was successfully verified.
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Rule 5896
Rule 5513
Rule 2234
Rule 2204
Rubi steps
\begin{align*} \int e^{\sinh ^{-1}(a+b x)^2} \, dx &=\frac{\operatorname{Subst}\left (\int e^{x^2} \cosh (x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=\frac{\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}\\ &=\frac{\operatorname{Subst}\left (\int e^{-x+x^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b}+\frac{\operatorname{Subst}\left (\int e^{x+x^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b}\\ &=\frac{\operatorname{Subst}\left (\int e^{\frac{1}{4} (-1+2 x)^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b \sqrt [4]{e}}+\frac{\operatorname{Subst}\left (\int e^{\frac{1}{4} (1+2 x)^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b \sqrt [4]{e}}\\ &=\frac{\sqrt{\pi } \text{erfi}\left (\frac{1}{2} \left (-1+2 \sinh ^{-1}(a+b x)\right )\right )}{4 b \sqrt [4]{e}}+\frac{\sqrt{\pi } \text{erfi}\left (\frac{1}{2} \left (1+2 \sinh ^{-1}(a+b x)\right )\right )}{4 b \sqrt [4]{e}}\\ \end{align*}
Mathematica [A] time = 0.0324645, size = 44, normalized size = 0.68 \[ \frac{\sqrt{\pi } \left (\text{Erfi}\left (\sinh ^{-1}(a+b x)+\frac{1}{2}\right )+\text{Erfi}\left (\frac{1}{2} \left (2 \sinh ^{-1}(a+b x)-1\right )\right )\right )}{4 \sqrt [4]{e} b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.005, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{ \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{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 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 (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 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 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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