Optimal. Leaf size=49 \[ \frac {e \sqrt {\pi } \text {Erf}(1-i \text {ArcSin}(a x))}{8 a^2}+\frac {e \sqrt {\pi } \text {Erf}(1+i \text {ArcSin}(a x))}{8 a^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4920, 12, 4562,
2266, 2235} \begin {gather*} \frac {e \sqrt {\pi } \text {Erf}(1-i \text {ArcSin}(a x))}{8 a^2}+\frac {e \sqrt {\pi } \text {Erf}(1+i \text {ArcSin}(a x))}{8 a^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2235
Rule 2266
Rule 4562
Rule 4920
Rubi steps
\begin {align*} \int e^{\sin ^{-1}(a x)^2} x \, dx &=\frac {\text {Subst}\left (\int \frac {e^{x^2} \cos (x) \sin (x)}{a} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {\text {Subst}\left (\int e^{x^2} \cos (x) \sin (x) \, dx,x,\sin ^{-1}(a x)\right )}{a^2}\\ &=\frac {\text {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 x)\right )}{a^2}\\ &=\frac {i \text {Subst}\left (\int e^{-2 i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{4 a^2}-\frac {i \text {Subst}\left (\int e^{2 i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{4 a^2}\\ &=\frac {(i e) \text {Subst}\left (\int e^{\frac {1}{4} (-2 i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{4 a^2}-\frac {(i e) \text {Subst}\left (\int e^{\frac {1}{4} (2 i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{4 a^2}\\ &=\frac {e \sqrt {\pi } \text {erf}\left (1-i \sin ^{-1}(a x)\right )}{8 a^2}+\frac {e \sqrt {\pi } \text {erf}\left (1+i \sin ^{-1}(a x)\right )}{8 a^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 36, normalized size = 0.73 \begin {gather*} \frac {e \sqrt {\pi } (\text {Erf}(1-i \text {ArcSin}(a x))+\text {Erf}(1+i \text {ArcSin}(a x)))}{8 a^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{\arcsin \left (a x \right )^{2}} x\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x e^{\operatorname {asin}^{2}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x\,{\mathrm {e}}^{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________