Optimal. Leaf size=123 \[ \frac{e \sqrt{\pi } \text{Erf}\left (1-i \sin ^{-1}(a+b x)\right )}{8 b^2}+\frac{e \sqrt{\pi } \text{Erf}\left (1+i \sin ^{-1}(a+b x)\right )}{8 b^2}-\frac{\sqrt [4]{e} \sqrt{\pi } a \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)-i\right )\right )}{4 b^2}-\frac{\sqrt [4]{e} \sqrt{\pi } a \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)+i\right )\right )}{4 b^2} \]
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Rubi [A] time = 0.272831, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {4836, 6741, 12, 6742, 4473, 2234, 2204, 4474} \[ \frac{e \sqrt{\pi } \text{Erf}\left (1-i \sin ^{-1}(a+b x)\right )}{8 b^2}+\frac{e \sqrt{\pi } \text{Erf}\left (1+i \sin ^{-1}(a+b x)\right )}{8 b^2}-\frac{\sqrt [4]{e} \sqrt{\pi } a \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)-i\right )\right )}{4 b^2}-\frac{\sqrt [4]{e} \sqrt{\pi } a \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)+i\right )\right )}{4 b^2} \]
Antiderivative was successfully verified.
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Rule 4836
Rule 6741
Rule 12
Rule 6742
Rule 4473
Rule 2234
Rule 2204
Rule 4474
Rubi steps
\begin{align*} \int e^{\sin ^{-1}(a+b x)^2} x \, dx &=\frac{\operatorname{Subst}\left (\int e^{x^2} \cos (x) \left (-\frac{a}{b}+\frac{\sin (x)}{b}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{e^{x^2} \cos (x) (-a+\sin (x))}{b} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int e^{x^2} \cos (x) (-a+\sin (x)) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^2}\\ &=\frac{\operatorname{Subst}\left (\int \left (-a e^{x^2} \cos (x)+e^{x^2} \cos (x) \sin (x)\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^2}\\ &=\frac{\operatorname{Subst}\left (\int e^{x^2} \cos (x) \sin (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^2}-\frac{a \operatorname{Subst}\left (\int e^{x^2} \cos (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^2}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{4} i e^{-2 i x+x^2}-\frac{1}{4} i e^{2 i x+x^2}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^2}-\frac{a \operatorname{Subst}\left (\int \left (\frac{1}{2} e^{-i x+x^2}+\frac{1}{2} e^{i x+x^2}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^2}\\ &=\frac{i \operatorname{Subst}\left (\int e^{-2 i x+x^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{4 b^2}-\frac{i \operatorname{Subst}\left (\int e^{2 i x+x^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{4 b^2}-\frac{a \operatorname{Subst}\left (\int e^{-i x+x^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^2}-\frac{a \operatorname{Subst}\left (\int e^{i x+x^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^2}\\ &=-\frac{\left (a \sqrt [4]{e}\right ) \operatorname{Subst}\left (\int e^{\frac{1}{4} (-i+2 x)^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^2}-\frac{\left (a \sqrt [4]{e}\right ) \operatorname{Subst}\left (\int e^{\frac{1}{4} (i+2 x)^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^2}+\frac{(i e) \operatorname{Subst}\left (\int e^{\frac{1}{4} (-2 i+2 x)^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{4 b^2}-\frac{(i e) \operatorname{Subst}\left (\int e^{\frac{1}{4} (2 i+2 x)^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{4 b^2}\\ &=\frac{e \sqrt{\pi } \text{erf}\left (1-i \sin ^{-1}(a+b x)\right )}{8 b^2}+\frac{e \sqrt{\pi } \text{erf}\left (1+i \sin ^{-1}(a+b x)\right )}{8 b^2}-\frac{a \sqrt [4]{e} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} \left (-i+2 \sin ^{-1}(a+b x)\right )\right )}{4 b^2}-\frac{a \sqrt [4]{e} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} \left (i+2 \sin ^{-1}(a+b x)\right )\right )}{4 b^2}\\ \end{align*}
Mathematica [A] time = 0.1061, size = 93, normalized size = 0.76 \[ \frac{\sqrt{\pi } \left (e \text{Erf}\left (1-i \sin ^{-1}(a+b x)\right )+e \text{Erf}\left (1+i \sin ^{-1}(a+b x)\right )-2 \sqrt [4]{e} a \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)-i\right )\right )-2 \sqrt [4]{e} a \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)+i\right )\right )\right )}{8 b^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.009, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{ \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}}}x\, 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 e^{\left (\arcsin \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 e^{\left (\arcsin \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 e^{\operatorname{asin}^{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 e^{\left (\arcsin \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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