Optimal. Leaf size=69 \[ \frac{\sqrt [4]{e} \sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)-i\right )\right )}{4 b}+\frac{\sqrt [4]{e} \sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)+i\right )\right )}{4 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0518688, antiderivative size = 69, 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 = {4836, 4473, 2234, 2204} \[ \frac{\sqrt [4]{e} \sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)-i\right )\right )}{4 b}+\frac{\sqrt [4]{e} \sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)+i\right )\right )}{4 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4836
Rule 4473
Rule 2234
Rule 2204
Rubi steps
\begin{align*} \int e^{\sin ^{-1}(a+b x)^2} \, dx &=\frac{\operatorname{Subst}\left (\int e^{x^2} \cos (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=\frac{\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}\\ &=\frac{\operatorname{Subst}\left (\int e^{-i x+x^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b}+\frac{\operatorname{Subst}\left (\int e^{i x+x^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b}\\ &=\frac{\sqrt [4]{e} \operatorname{Subst}\left (\int e^{\frac{1}{4} (-i+2 x)^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b}+\frac{\sqrt [4]{e} \operatorname{Subst}\left (\int e^{\frac{1}{4} (i+2 x)^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b}\\ &=\frac{\sqrt [4]{e} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} \left (-i+2 \sin ^{-1}(a+b x)\right )\right )}{4 b}+\frac{\sqrt [4]{e} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} \left (i+2 \sin ^{-1}(a+b x)\right )\right )}{4 b}\\ \end{align*}
Mathematica [A] time = 0.0324684, size = 52, normalized size = 0.75 \[ \frac{\sqrt [4]{e} \sqrt{\pi } \left (\text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)-i\right )\right )+\text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)+i\right )\right )\right )}{4 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.006, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{ \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\left (\arcsin \left (b x + a\right )^{2}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (e^{\left (\arcsin \left (b x + a\right )^{2}\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\operatorname{asin}^{2}{\left (a + b x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\left (\arcsin \left (b x + a\right )^{2}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]