Optimal. Leaf size=101 \[ \frac {e \sqrt {\pi } \text {Erf}(1-i \text {ArcSin}(a x))}{16 a^4}-\frac {e^4 \sqrt {\pi } \text {Erf}(2-i \text {ArcSin}(a x))}{32 a^4}+\frac {e \sqrt {\pi } \text {Erf}(1+i \text {ArcSin}(a x))}{16 a^4}-\frac {e^4 \sqrt {\pi } \text {Erf}(2+i \text {ArcSin}(a x))}{32 a^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4920, 12,
4562, 2266, 2235} \begin {gather*} \frac {e \sqrt {\pi } \text {Erf}(1-i \text {ArcSin}(a x))}{16 a^4}-\frac {e^4 \sqrt {\pi } \text {Erf}(2-i \text {ArcSin}(a x))}{32 a^4}+\frac {e \sqrt {\pi } \text {Erf}(1+i \text {ArcSin}(a x))}{16 a^4}-\frac {e^4 \sqrt {\pi } \text {Erf}(2+i \text {ArcSin}(a x))}{32 a^4} \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^3 \, dx &=\frac {\text {Subst}\left (\int \frac {e^{x^2} \cos (x) \sin ^3(x)}{a^3} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {\text {Subst}\left (\int e^{x^2} \cos (x) \sin ^3(x) \, dx,x,\sin ^{-1}(a x)\right )}{a^4}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{8} i e^{-2 i x+x^2}-\frac {1}{8} i e^{2 i x+x^2}-\frac {1}{16} i e^{-4 i x+x^2}+\frac {1}{16} i e^{4 i x+x^2}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^4}\\ &=-\frac {i \text {Subst}\left (\int e^{-4 i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{16 a^4}+\frac {i \text {Subst}\left (\int e^{4 i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{16 a^4}+\frac {i \text {Subst}\left (\int e^{-2 i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^4}-\frac {i \text {Subst}\left (\int e^{2 i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^4}\\ &=\frac {(i e) \text {Subst}\left (\int e^{\frac {1}{4} (-2 i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^4}-\frac {(i e) \text {Subst}\left (\int e^{\frac {1}{4} (2 i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^4}-\frac {\left (i e^4\right ) \text {Subst}\left (\int e^{\frac {1}{4} (-4 i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{16 a^4}+\frac {\left (i e^4\right ) \text {Subst}\left (\int e^{\frac {1}{4} (4 i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{16 a^4}\\ &=\frac {e \sqrt {\pi } \text {erf}\left (1-i \sin ^{-1}(a x)\right )}{16 a^4}-\frac {e^4 \sqrt {\pi } \text {erf}\left (2-i \sin ^{-1}(a x)\right )}{32 a^4}+\frac {e \sqrt {\pi } \text {erf}\left (1+i \sin ^{-1}(a x)\right )}{16 a^4}-\frac {e^4 \sqrt {\pi } \text {erf}\left (2+i \sin ^{-1}(a x)\right )}{32 a^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 67, normalized size = 0.66 \begin {gather*} \frac {e \sqrt {\pi } \left (2 (\text {Erf}(1-i \text {ArcSin}(a x))+\text {Erf}(1+i \text {ArcSin}(a x)))-e^3 (\text {Erf}(2-i \text {ArcSin}(a x))+\text {Erf}(2+i \text {ArcSin}(a x)))\right )}{32 a^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{\arcsin \left (a x \right )^{2}} x^{3}\, dx\]
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^{3} 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 = -0.01 \begin {gather*} \int x^3\,{\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]
________________________________________________________________________________________