Optimal. Leaf size=25 \[ \sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcSin}(x)}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4809, 3386,
3432} \begin {gather*} \sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcSin}(x)}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3386
Rule 3432
Rule 4809
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {1-x^2} \sqrt {\sin ^{-1}(x)}} \, dx &=\text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(x)\right )\\ &=2 \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(x)}\right )\\ &=\sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(x)}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.05, size = 53, normalized size = 2.12 \begin {gather*} -\frac {\sqrt {-i \text {ArcSin}(x)} \text {Gamma}\left (\frac {1}{2},-i \text {ArcSin}(x)\right )+\sqrt {i \text {ArcSin}(x)} \text {Gamma}\left (\frac {1}{2},i \text {ArcSin}(x)\right )}{2 \sqrt {\text {ArcSin}(x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.29, size = 20, normalized size = 0.80
method | result | size |
default | \(\mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi }\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 \frac {x}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \sqrt {\operatorname {asin}{\left (x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [C] Result contains complex when optimal does not.
time = 0.42, size = 37, normalized size = 1.48 \begin {gather*} \left (\frac {1}{4} i - \frac {1}{4}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (x\right )}\right ) - \left (\frac {1}{4} i + \frac {1}{4}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (x\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x}{\sqrt {\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]
________________________________________________________________________________________