Integrand size = 14, antiderivative size = 62 \[ \int \frac {\arcsin (x)}{\left (1-x^2\right )^{5/2}} \, dx=-\frac {1}{6 \left (1-x^2\right )}+\frac {x \arcsin (x)}{3 \left (1-x^2\right )^{3/2}}+\frac {2 x \arcsin (x)}{3 \sqrt {1-x^2}}+\frac {1}{3} \log \left (1-x^2\right ) \] Output:
-1/6/(-x^2+1)+1/3*x*arcsin(x)/(-x^2+1)^(3/2)+1/3*ln(-x^2+1)+2/3*x*arcsin(x )/(-x^2+1)^(1/2)
Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.73 \[ \int \frac {\arcsin (x)}{\left (1-x^2\right )^{5/2}} \, dx=\frac {1}{6} \left (\frac {1}{-1+x^2}-\frac {2 x \left (-3+2 x^2\right ) \arcsin (x)}{\left (1-x^2\right )^{3/2}}+2 \log \left (1-x^2\right )\right ) \] Input:
Integrate[ArcSin[x]/(1 - x^2)^(5/2),x]
Output:
((-1 + x^2)^(-1) - (2*x*(-3 + 2*x^2)*ArcSin[x])/(1 - x^2)^(3/2) + 2*Log[1 - x^2])/6
Time = 0.24 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5162, 241, 5160, 240}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\arcsin (x)}{\left (1-x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 5162 |
\(\displaystyle \frac {2}{3} \int \frac {\arcsin (x)}{\left (1-x^2\right )^{3/2}}dx-\frac {1}{3} \int \frac {x}{\left (1-x^2\right )^2}dx+\frac {x \arcsin (x)}{3 \left (1-x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 241 |
\(\displaystyle \frac {2}{3} \int \frac {\arcsin (x)}{\left (1-x^2\right )^{3/2}}dx+\frac {x \arcsin (x)}{3 \left (1-x^2\right )^{3/2}}-\frac {1}{6 \left (1-x^2\right )}\) |
\(\Big \downarrow \) 5160 |
\(\displaystyle \frac {2}{3} \left (\frac {x \arcsin (x)}{\sqrt {1-x^2}}-\int \frac {x}{1-x^2}dx\right )+\frac {x \arcsin (x)}{3 \left (1-x^2\right )^{3/2}}-\frac {1}{6 \left (1-x^2\right )}\) |
\(\Big \downarrow \) 240 |
\(\displaystyle \frac {x \arcsin (x)}{3 \left (1-x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \arcsin (x)}{\sqrt {1-x^2}}+\frac {1}{2} \log \left (1-x^2\right )\right )-\frac {1}{6 \left (1-x^2\right )}\) |
Input:
Int[ArcSin[x]/(1 - x^2)^(5/2),x]
Output:
-1/6*1/(1 - x^2) + (x*ArcSin[x])/(3*(1 - x^2)^(3/2)) + (2*((x*ArcSin[x])/S qrt[1 - x^2] + Log[1 - x^2]/2))/3
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x _Symbol] :> Simp[x*((a + b*ArcSin[c*x])^n/(d*Sqrt[d + e*x^2])), x] - Simp[b *c*(n/d)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]] Int[x*((a + b*ArcSin[c*x ])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_ Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*d*(p + 1 ))), x] + (Simp[(2*p + 3)/(2*d*(p + 1)) Int[(d + e*x^2)^(p + 1)*(a + b*Ar cSin[c*x])^n, x], x] + Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2 *x^2)^p] Int[x*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x ]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
Time = 0.30 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {1}{6 x^{2}-6}+\frac {\sqrt {-x^{2}+1}\, \arcsin \left (x \right ) x}{3 \left (x^{2}-1\right )^{2}}+\frac {\ln \left (-x^{2}+1\right )}{3}-\frac {2 \sqrt {-x^{2}+1}\, \arcsin \left (x \right ) x}{3 \left (x^{2}-1\right )}\) | \(63\) |
Input:
int(arcsin(x)/(-x^2+1)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/6/(x^2-1)+1/3*(-x^2+1)^(1/2)/(x^2-1)^2*arcsin(x)*x+1/3*ln(-x^2+1)-2/3*(- x^2+1)^(1/2)/(x^2-1)*arcsin(x)*x
Time = 0.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.98 \[ \int \frac {\arcsin (x)}{\left (1-x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (2 \, x^{3} - 3 \, x\right )} \sqrt {-x^{2} + 1} \arcsin \left (x\right ) - x^{2} - 2 \, {\left (x^{4} - 2 \, x^{2} + 1\right )} \log \left (x^{2} - 1\right ) + 1}{6 \, {\left (x^{4} - 2 \, x^{2} + 1\right )}} \] Input:
integrate(arcsin(x)/(-x^2+1)^(5/2),x, algorithm="fricas")
Output:
-1/6*(2*(2*x^3 - 3*x)*sqrt(-x^2 + 1)*arcsin(x) - x^2 - 2*(x^4 - 2*x^2 + 1) *log(x^2 - 1) + 1)/(x^4 - 2*x^2 + 1)
Time = 12.42 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.26 \[ \int \frac {\arcsin (x)}{\left (1-x^2\right )^{5/2}} \, dx=\left (\begin {cases} \frac {x^{3}}{3 \left (1 - x^{2}\right )^{\frac {3}{2}}} + \frac {x}{\sqrt {1 - x^{2}}} & \text {for}\: x > -1 \wedge x < 1 \end {cases}\right ) \operatorname {asin}{\left (x \right )} - \begin {cases} \text {NaN} & \text {for}\: x < -1 \\- \frac {2 x^{2} \log {\left (1 - x^{2} \right )}}{6 x^{2} - 6} - \frac {x^{2}}{6 x^{2} - 6} + \frac {2 \log {\left (1 - x^{2} \right )}}{6 x^{2} - 6} & \text {for}\: x < 1 \\\text {NaN} & \text {otherwise} \end {cases} \] Input:
integrate(asin(x)/(-x**2+1)**(5/2),x)
Output:
Piecewise((x**3/(3*(1 - x**2)**(3/2)) + x/sqrt(1 - x**2), (x > -1) & (x < 1)))*asin(x) - Piecewise((nan, x < -1), (-2*x**2*log(1 - x**2)/(6*x**2 - 6 ) - x**2/(6*x**2 - 6) + 2*log(1 - x**2)/(6*x**2 - 6), x < 1), (nan, True))
Time = 0.11 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.77 \[ \int \frac {\arcsin (x)}{\left (1-x^2\right )^{5/2}} \, dx=\frac {1}{3} \, {\left (\frac {2 \, x}{\sqrt {-x^{2} + 1}} + \frac {x}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}\right )} \arcsin \left (x\right ) + \frac {1}{6 \, {\left (x^{2} - 1\right )}} + \frac {1}{3} \, \log \left (-3 \, x^{2} + 3\right ) \] Input:
integrate(arcsin(x)/(-x^2+1)^(5/2),x, algorithm="maxima")
Output:
1/3*(2*x/sqrt(-x^2 + 1) + x/(-x^2 + 1)^(3/2))*arcsin(x) + 1/6/(x^2 - 1) + 1/3*log(-3*x^2 + 3)
Time = 0.14 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.87 \[ \int \frac {\arcsin (x)}{\left (1-x^2\right )^{5/2}} \, dx=-\frac {{\left (2 \, x^{2} - 3\right )} \sqrt {-x^{2} + 1} x \arcsin \left (x\right )}{3 \, {\left (x^{2} - 1\right )}^{2}} - \frac {2 \, x^{2} - 3}{6 \, {\left (x^{2} - 1\right )}} + \frac {1}{3} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \] Input:
integrate(arcsin(x)/(-x^2+1)^(5/2),x, algorithm="giac")
Output:
-1/3*(2*x^2 - 3)*sqrt(-x^2 + 1)*x*arcsin(x)/(x^2 - 1)^2 - 1/6*(2*x^2 - 3)/ (x^2 - 1) + 1/3*log(abs(x^2 - 1))
Timed out. \[ \int \frac {\arcsin (x)}{\left (1-x^2\right )^{5/2}} \, dx=\int \frac {\mathrm {asin}\left (x\right )}{{\left (1-x^2\right )}^{5/2}} \,d x \] Input:
int(asin(x)/(1 - x^2)^(5/2),x)
Output:
int(asin(x)/(1 - x^2)^(5/2), x)
\[ \int \frac {\arcsin (x)}{\left (1-x^2\right )^{5/2}} \, dx=\int \frac {\mathit {asin} \left (x \right )}{\sqrt {-x^{2}+1}\, x^{4}-2 \sqrt {-x^{2}+1}\, x^{2}+\sqrt {-x^{2}+1}}d x \] Input:
int(asin(x)/(-x^2+1)^(5/2),x)
Output:
int(asin(x)/(sqrt( - x**2 + 1)*x**4 - 2*sqrt( - x**2 + 1)*x**2 + sqrt( - x **2 + 1)),x)