Optimal. Leaf size=44 \[ \frac {1}{10} e^{\sin ^{-1}(x)} \left (3 x+x^3-3 \sqrt {1-x^2}-3 x^2 \sqrt {1-x^2}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.47, antiderivative size = 62, normalized size of antiderivative = 1.41, number of steps
used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {4920, 6873,
6852, 4519, 4517} \begin {gather*} \frac {1}{10} x^3 e^{\text {ArcSin}(x)}-\frac {3}{10} \sqrt {1-x^2} x^2 e^{\text {ArcSin}(x)}-\frac {3}{10} \sqrt {1-x^2} e^{\text {ArcSin}(x)}+\frac {3}{10} x e^{\text {ArcSin}(x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 4517
Rule 4519
Rule 4920
Rule 6852
Rule 6873
Rubi steps
\begin {align*} \int \frac {e^{\sin ^{-1}(x)} x^3}{\sqrt {1-x^2}} \, dx &=\text {Subst}\left (\int \frac {e^x \cos (x) \sin ^3(x)}{\sqrt {1-\sin ^2(x)}} \, dx,x,\sin ^{-1}(x)\right )\\ &=\text {Subst}\left (\int \frac {e^x \cos (x) \sin ^3(x)}{\sqrt {\cos ^2(x)}} \, dx,x,\sin ^{-1}(x)\right )\\ &=1 \text {Subst}\left (\int e^x \sin ^3(x) \, dx,x,\sin ^{-1}(x)\right )\\ &=\frac {1}{10} e^{\sin ^{-1}(x)} x^3-\frac {3}{10} e^{\sin ^{-1}(x)} x^2 \sqrt {1-x^2}+\frac {3}{5} \text {Subst}\left (\int e^x \sin (x) \, dx,x,\sin ^{-1}(x)\right )\\ &=\frac {3}{10} e^{\sin ^{-1}(x)} x+\frac {1}{10} e^{\sin ^{-1}(x)} x^3-\frac {3}{10} e^{\sin ^{-1}(x)} \sqrt {1-x^2}-\frac {3}{10} e^{\sin ^{-1}(x)} x^2 \sqrt {1-x^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.13, size = 38, normalized size = 0.86 \begin {gather*} -\frac {1}{40} e^{\sin ^{-1}(x)} \left (15 \left (-x+\sqrt {1-x^2}\right )-3 \cos \left (3 \sin ^{-1}(x)\right )+\sin \left (3 \sin ^{-1}(x)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{\arcsin \left (x \right )} x^{3}}{\sqrt {-x^{2}+1}}\, dx\]
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 [A]
time = 0.78, size = 28, normalized size = 0.64 \begin {gather*} \frac {1}{10} \, {\left (x^{3} - 3 \, {\left (x^{2} + 1\right )} \sqrt {-x^{2} + 1} + 3 \, x\right )} e^{\arcsin \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.38, size = 56, normalized size = 1.27 \begin {gather*} \frac {x^{3} e^{\operatorname {asin}{\left (x \right )}}}{10} - \frac {3 x^{2} \sqrt {1 - x^{2}} e^{\operatorname {asin}{\left (x \right )}}}{10} + \frac {3 x e^{\operatorname {asin}{\left (x \right )}}}{10} - \frac {3 \sqrt {1 - x^{2}} e^{\operatorname {asin}{\left (x \right )}}}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.45, size = 46, normalized size = 1.05 \begin {gather*} \frac {1}{10} \, {\left (x^{2} - 1\right )} x e^{\arcsin \left (x\right )} + \frac {3}{10} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} e^{\arcsin \left (x\right )} + \frac {2}{5} \, x e^{\arcsin \left (x\right )} - \frac {3}{5} \, \sqrt {-x^{2} + 1} e^{\arcsin \left (x\right )} \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 \frac {x^3\,{\mathrm {e}}^{\mathrm {asin}\left (x\right )}}{\sqrt {1-x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________