3.658 \(\int \frac{(1-x^2)^{3/2} \sin ^{-1}(x)}{x^6} \, dx\)

Optimal. Leaf size=41 \[ \frac{1}{5 x^2}-\frac{1}{20 x^4}-\frac{\left (1-x^2\right )^{5/2} \sin ^{-1}(x)}{5 x^5}+\frac{\log (x)}{5} \]

[Out]

-1/(20*x^4) + 1/(5*x^2) - ((1 - x^2)^(5/2)*ArcSin[x])/(5*x^5) + Log[x]/5

________________________________________________________________________________________

Rubi [A]  time = 0.0591848, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {4681, 266, 43} \[ \frac{1}{5 x^2}-\frac{1}{20 x^4}-\frac{\left (1-x^2\right )^{5/2} \sin ^{-1}(x)}{5 x^5}+\frac{\log (x)}{5} \]

Antiderivative was successfully verified.

[In]

Int[((1 - x^2)^(3/2)*ArcSin[x])/x^6,x]

[Out]

-1/(20*x^4) + 1/(5*x^2) - ((1 - x^2)^(5/2)*ArcSin[x])/(5*x^5) + Log[x]/5

Rule 4681

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(
(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n)/(d*f*(m + 1)), x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x
^2)^FracPart[p])/(f*(m + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSi
n[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p
+ 3, 0] && NeQ[m, -1]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\left (1-x^2\right )^{3/2} \sin ^{-1}(x)}{x^6} \, dx &=-\frac{\left (1-x^2\right )^{5/2} \sin ^{-1}(x)}{5 x^5}+\frac{1}{5} \int \frac{\left (1-x^2\right )^2}{x^5} \, dx\\ &=-\frac{\left (1-x^2\right )^{5/2} \sin ^{-1}(x)}{5 x^5}+\frac{1}{10} \operatorname{Subst}\left (\int \frac{(1-x)^2}{x^3} \, dx,x,x^2\right )\\ &=-\frac{\left (1-x^2\right )^{5/2} \sin ^{-1}(x)}{5 x^5}+\frac{1}{10} \operatorname{Subst}\left (\int \left (\frac{1}{x^3}-\frac{2}{x^2}+\frac{1}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{20 x^4}+\frac{1}{5 x^2}-\frac{\left (1-x^2\right )^{5/2} \sin ^{-1}(x)}{5 x^5}+\frac{\log (x)}{5}\\ \end{align*}

Mathematica [A]  time = 0.0436426, size = 36, normalized size = 0.88 \[ -\frac{-4 x^3-4 x^5 \log (x)+4 \left (1-x^2\right )^{5/2} \sin ^{-1}(x)+x}{20 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - x^2)^(3/2)*ArcSin[x])/x^6,x]

[Out]

-(x - 4*x^3 + 4*(1 - x^2)^(5/2)*ArcSin[x] - 4*x^5*Log[x])/(20*x^5)

________________________________________________________________________________________

Maple [C]  time = 0.532, size = 201, normalized size = 4.9 \begin{align*} -{\frac{2\,i}{5}}\arcsin \left ( x \right ) +{\frac{1}{ \left ( 100\,{x}^{8}-200\,{x}^{6}+200\,{x}^{4}-100\,{x}^{2}+20 \right ){x}^{5}} \left ( -\sqrt{-{x}^{2}+1}{x}^{4}+i{x}^{5}+2\,\sqrt{-{x}^{2}+1}{x}^{2}-\sqrt{-{x}^{2}+1} \right ) \left ( 20\,\arcsin \left ( x \right ){x}^{8}-4\,i{x}^{8}-4\,\sqrt{-{x}^{2}+1}{x}^{7}-40\,\arcsin \left ( x \right ){x}^{6}+i{x}^{6}+9\,\sqrt{-{x}^{2}+1}{x}^{5}+40\,\arcsin \left ( x \right ){x}^{4}-6\,\sqrt{-{x}^{2}+1}{x}^{3}-20\,{x}^{2}\arcsin \left ( x \right ) +x\sqrt{-{x}^{2}+1}+4\,\arcsin \left ( x \right ) \right ) }+{\frac{1}{5}\ln \left ( \left ( ix+\sqrt{-{x}^{2}+1} \right ) ^{2}-1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2+1)^(3/2)*arcsin(x)/x^6,x)

[Out]

-2/5*I*arcsin(x)+1/20*(-(-x^2+1)^(1/2)*x^4+I*x^5+2*(-x^2+1)^(1/2)*x^2-(-x^2+1)^(1/2))*(20*arcsin(x)*x^8-4*I*x^
8-4*(-x^2+1)^(1/2)*x^7-40*arcsin(x)*x^6+I*x^6+9*(-x^2+1)^(1/2)*x^5+40*arcsin(x)*x^4-6*(-x^2+1)^(1/2)*x^3-20*x^
2*arcsin(x)+x*(-x^2+1)^(1/2)+4*arcsin(x))/(5*x^8-10*x^6+10*x^4-5*x^2+1)/x^5+1/5*ln((I*x+(-x^2+1)^(1/2))^2-1)

________________________________________________________________________________________

Maxima [A]  time = 1.45164, size = 47, normalized size = 1.15 \begin{align*} -\frac{{\left (-x^{2} + 1\right )}^{\frac{5}{2}} \arcsin \left (x\right )}{5 \, x^{5}} + \frac{4 \, x^{2} - 1}{20 \, x^{4}} + \frac{1}{10} \, \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)^(3/2)*arcsin(x)/x^6,x, algorithm="maxima")

[Out]

-1/5*(-x^2 + 1)^(5/2)*arcsin(x)/x^5 + 1/20*(4*x^2 - 1)/x^4 + 1/10*log(x^2)

________________________________________________________________________________________

Fricas [A]  time = 2.78533, size = 113, normalized size = 2.76 \begin{align*} \frac{4 \, x^{5} \log \left (x\right ) + 4 \, x^{3} - 4 \,{\left (x^{4} - 2 \, x^{2} + 1\right )} \sqrt{-x^{2} + 1} \arcsin \left (x\right ) - x}{20 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)^(3/2)*arcsin(x)/x^6,x, algorithm="fricas")

[Out]

1/20*(4*x^5*log(x) + 4*x^3 - 4*(x^4 - 2*x^2 + 1)*sqrt(-x^2 + 1)*arcsin(x) - x)/x^5

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- \left (x - 1\right ) \left (x + 1\right )\right )^{\frac{3}{2}} \operatorname{asin}{\left (x \right )}}{x^{6}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**2+1)**(3/2)*asin(x)/x**6,x)

[Out]

Integral((-(x - 1)*(x + 1))**(3/2)*asin(x)/x**6, x)

________________________________________________________________________________________

Giac [B]  time = 1.09303, size = 182, normalized size = 4.44 \begin{align*} -\frac{1}{160} \,{\left (\frac{x^{5}{\left (\frac{5 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - \frac{10 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}^{4}}{x^{4}} - 1\right )}}{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{5}} + \frac{10 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}{x} - \frac{5 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}^{3}}{x^{3}} + \frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{5}}{x^{5}}\right )} \arcsin \left (x\right ) - \frac{3 \, x^{4} - 4 \, x^{2} + 1}{20 \, x^{4}} + \frac{1}{10} \, \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)^(3/2)*arcsin(x)/x^6,x, algorithm="giac")

[Out]

-1/160*(x^5*(5*(sqrt(-x^2 + 1) - 1)^2/x^2 - 10*(sqrt(-x^2 + 1) - 1)^4/x^4 - 1)/(sqrt(-x^2 + 1) - 1)^5 + 10*(sq
rt(-x^2 + 1) - 1)/x - 5*(sqrt(-x^2 + 1) - 1)^3/x^3 + (sqrt(-x^2 + 1) - 1)^5/x^5)*arcsin(x) - 1/20*(3*x^4 - 4*x
^2 + 1)/x^4 + 1/10*log(x^2)