3.684 \(\int \frac{(-1+x^2)^{5/2} \csc ^{-1}(x)}{x^3} \, dx\)

Optimal. Leaf size=106 \[ \frac{2 x^4+3}{12 x \sqrt{x^2}}-\frac{7 x \log (x)}{3 \sqrt{x^2}}+\frac{\left (x^2-1\right )^{5/2} \csc ^{-1}(x)}{3 x^2}-\frac{5 \left (x^2-1\right )^{3/2} \csc ^{-1}(x)}{3 x^2}-\frac{5 \sqrt{x^2-1} \csc ^{-1}(x)}{2 x^2}-\frac{5 x \csc ^{-1}(x)^2}{4 \sqrt{x^2}} \]

[Out]

(3 + 2*x^4)/(12*x*Sqrt[x^2]) - (5*Sqrt[-1 + x^2]*ArcCsc[x])/(2*x^2) - (5*(-1 + x^2)^(3/2)*ArcCsc[x])/(3*x^2) +
((-1 + x^2)^(5/2)*ArcCsc[x])/(3*x^2) - (5*x*ArcCsc[x]^2)/(4*Sqrt[x^2]) - (7*x*Log[x])/(3*Sqrt[x^2])

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Rubi [A]  time = 0.19574, antiderivative size = 133, normalized size of antiderivative = 1.25, number of steps used = 11, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533, Rules used = {5243, 4695, 4647, 4641, 30, 14, 266, 43} \[ \frac{x \sqrt{x^2}}{6}+\frac{\sqrt{x^2}}{4 x^3}-\frac{7 \sqrt{x^2} \log (x)}{3 x}+\frac{1}{3} \left (x^2\right )^{3/2} \left (1-\frac{1}{x^2}\right )^{5/2} \csc ^{-1}(x)-\frac{5}{3} \sqrt{x^2} \left (1-\frac{1}{x^2}\right )^{3/2} \csc ^{-1}(x)-\frac{5 \sqrt{1-\frac{1}{x^2}} \csc ^{-1}(x)}{2 \sqrt{x^2}}-\frac{5 \sqrt{x^2} \csc ^{-1}(x)^2}{4 x} \]

Antiderivative was successfully verified.

[In]

Int[((-1 + x^2)^(5/2)*ArcCsc[x])/x^3,x]

[Out]

Sqrt[x^2]/(4*x^3) + (x*Sqrt[x^2])/6 - (5*Sqrt[1 - x^(-2)]*ArcCsc[x])/(2*Sqrt[x^2]) - (5*(1 - x^(-2))^(3/2)*Sqr
t[x^2]*ArcCsc[x])/3 + ((1 - x^(-2))^(5/2)*(x^2)^(3/2)*ArcCsc[x])/3 - (5*Sqrt[x^2]*ArcCsc[x]^2)/(4*x) - (7*Sqrt
[x^2]*Log[x])/(3*x)

Rule 5243

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_), x_Symbol] :> -Dist[Sqrt[x
^2]/x, Subst[Int[((e + d*x^2)^p*(a + b*ArcSin[x/c])^n)/x^(m + 2*(p + 1)), x], x, 1/x], x] /; FreeQ[{a, b, c, d
, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p + 1/2] && GtQ[e, 0] && LtQ[d, 0]

Rule 4695

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*(a + b*ArcSin[c*x])^n)/(f*(m + 1)), x] + (-Dist[(2*e*p)/(f^2*(m + 1)), Int[(f*x)^
(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], 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*ArcSin[c*x])^(n - 1),
x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]

Rule 4647

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(x*Sqrt[d + e*x^2]*(
a + b*ArcSin[c*x])^n)/2, x] + (Dist[Sqrt[d + e*x^2]/(2*Sqrt[1 - c^2*x^2]), Int[(a + b*ArcSin[c*x])^n/Sqrt[1 -
c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(2*Sqrt[1 - c^2*x^2]), Int[x*(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]

Rule 4641

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSin[c*x])^
(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && NeQ[n,
-1]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 )^{5/2} \csc ^{-1}(x)}{x^3} \, dx &=-\frac{\sqrt{x^2} \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^{5/2} \sin ^{-1}(x)}{x^4} \, dx,x,\frac{1}{x}\right )}{x}\\ &=\frac{1}{3} \left (1-\frac{1}{x^2}\right )^{5/2} \left (x^2\right )^{3/2} \csc ^{-1}(x)-\frac{\sqrt{x^2} \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2}{x^3} \, dx,x,\frac{1}{x}\right )}{3 x}+\frac{\left (5 \sqrt{x^2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^{3/2} \sin ^{-1}(x)}{x^2} \, dx,x,\frac{1}{x}\right )}{3 x}\\ &=-\frac{5}{3} \left (1-\frac{1}{x^2}\right )^{3/2} \sqrt{x^2} \csc ^{-1}(x)+\frac{1}{3} \left (1-\frac{1}{x^2}\right )^{5/2} \left (x^2\right )^{3/2} \csc ^{-1}(x)-\frac{\sqrt{x^2} \operatorname{Subst}\left (\int \frac{(1-x)^2}{x^2} \, dx,x,\frac{1}{x^2}\right )}{6 x}+\frac{\left (5 \sqrt{x^2}\right ) \operatorname{Subst}\left (\int \frac{1-x^2}{x} \, dx,x,\frac{1}{x}\right )}{3 x}-\frac{\left (5 \sqrt{x^2}\right ) \operatorname{Subst}\left (\int \sqrt{1-x^2} \sin ^{-1}(x) \, dx,x,\frac{1}{x}\right )}{x}\\ &=-\frac{5 \sqrt{1-\frac{1}{x^2}} \csc ^{-1}(x)}{2 \sqrt{x^2}}-\frac{5}{3} \left (1-\frac{1}{x^2}\right )^{3/2} \sqrt{x^2} \csc ^{-1}(x)+\frac{1}{3} \left (1-\frac{1}{x^2}\right )^{5/2} \left (x^2\right )^{3/2} \csc ^{-1}(x)-\frac{\sqrt{x^2} \operatorname{Subst}\left (\int \left (1+\frac{1}{x^2}-\frac{2}{x}\right ) \, dx,x,\frac{1}{x^2}\right )}{6 x}+\frac{\left (5 \sqrt{x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x}-x\right ) \, dx,x,\frac{1}{x}\right )}{3 x}+\frac{\left (5 \sqrt{x^2}\right ) \operatorname{Subst}\left (\int x \, dx,x,\frac{1}{x}\right )}{2 x}-\frac{\left (5 \sqrt{x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin ^{-1}(x)}{\sqrt{1-x^2}} \, dx,x,\frac{1}{x}\right )}{2 x}\\ &=\frac{\sqrt{x^2}}{4 x^3}+\frac{x \sqrt{x^2}}{6}-\frac{5 \sqrt{1-\frac{1}{x^2}} \csc ^{-1}(x)}{2 \sqrt{x^2}}-\frac{5}{3} \left (1-\frac{1}{x^2}\right )^{3/2} \sqrt{x^2} \csc ^{-1}(x)+\frac{1}{3} \left (1-\frac{1}{x^2}\right )^{5/2} \left (x^2\right )^{3/2} \csc ^{-1}(x)-\frac{5 \sqrt{x^2} \csc ^{-1}(x)^2}{4 x}-\frac{7 \sqrt{x^2} \log (x)}{3 x}\\ \end{align*}

Mathematica [A]  time = 0.294714, size = 86, normalized size = 0.81 \[ \frac{\sqrt{x^2-1} \left (4 x^2+\csc ^{-1}(x) \left (8 \sqrt{1-\frac{1}{x^2}} x \left (x^2-7\right )-6 \sin \left (2 \csc ^{-1}(x)\right )\right )+48 \log \left (\frac{1}{x}\right )-8 \log (x)-30 \csc ^{-1}(x)^2-3 \cos \left (2 \csc ^{-1}(x)\right )\right )}{24 \sqrt{1-\frac{1}{x^2}} x} \]

Antiderivative was successfully verified.

[In]

Integrate[((-1 + x^2)^(5/2)*ArcCsc[x])/x^3,x]

[Out]

(Sqrt[-1 + x^2]*(4*x^2 - 30*ArcCsc[x]^2 - 3*Cos[2*ArcCsc[x]] + 48*Log[x^(-1)] - 8*Log[x] + ArcCsc[x]*(8*Sqrt[1
- x^(-2)]*x*(-7 + x^2) - 6*Sin[2*ArcCsc[x]])))/(24*Sqrt[1 - x^(-2)]*x)

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Maple [C]  time = 0.322, size = 305, normalized size = 2.9 \begin{align*} -{\frac{5\,x \left ({\rm arccsc} \left (x\right ) \right ) ^{2}}{4}\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{\frac{1}{\sqrt{{x}^{2}-1}}}}+{\frac{2\,{\rm arccsc} \left (x\right )+i}{16\,{x}^{2}} \left ( i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{x}^{3}-2\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x-2\,{x}^{2}+2 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}-{\frac{-i+2\,{\rm arccsc} \left (x\right )}{16\,{x}^{2}} \left ( i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{x}^{3}-2\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x+2\,{x}^{2}-2 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}-{{\frac{14\,i}{3}}x{\rm arccsc} \left (x\right )\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{\frac{1}{\sqrt{{x}^{2}-1}}}}+{\frac{1}{6\,{x}^{4}-90\,{x}^{2}+378} \left ({x}^{4}+7\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x-8\,{x}^{2}+7 \right ) \left ( 2\,{\rm arccsc} \left (x\right ){x}^{4}+\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{x}^{3}-30\,{\rm arccsc} \left (x\right ){x}^{2}-7\,\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x+126\,{\rm arccsc} \left (x\right )-7\,i \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}+{\frac{7\,x}{3}\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}\ln \left ( \left ({\frac{i}{x}}+\sqrt{1-{x}^{-2}} \right ) ^{2}-1 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-1)^(5/2)*arccsc(x)/x^3,x)

[Out]

-5/4/(x^2-1)^(1/2)*((x^2-1)/x^2)^(1/2)*x*arccsc(x)^2+1/16/(x^2-1)^(1/2)/x^2*(I*((x^2-1)/x^2)^(1/2)*x^3-2*I*((x
^2-1)/x^2)^(1/2)*x-2*x^2+2)*(2*arccsc(x)+I)-1/16/(x^2-1)^(1/2)/x^2*(I*((x^2-1)/x^2)^(1/2)*x^3-2*I*((x^2-1)/x^2
)^(1/2)*x+2*x^2-2)*(-I+2*arccsc(x))-14/3*I/(x^2-1)^(1/2)*((x^2-1)/x^2)^(1/2)*x*arccsc(x)+1/6/(x^2-1)^(1/2)*(x^
4+7*I*((x^2-1)/x^2)^(1/2)*x-8*x^2+7)*(2*arccsc(x)*x^4+((x^2-1)/x^2)^(1/2)*x^3-30*arccsc(x)*x^2-7*((x^2-1)/x^2)
^(1/2)*x+126*arccsc(x)-7*I)/(x^4-15*x^2+63)+7/3/(x^2-1)^(1/2)*((x^2-1)/x^2)^(1/2)*x*ln((I/x+(1-1/x^2)^(1/2))^2
-1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (x^{2} - 1\right )}^{\frac{5}{2}} \operatorname{arccsc}\left (x\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)^(5/2)*arccsc(x)/x^3,x, algorithm="maxima")

[Out]

integrate((x^2 - 1)^(5/2)*arccsc(x)/x^3, x)

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Fricas [A]  time = 2.75193, size = 146, normalized size = 1.38 \begin{align*} \frac{2 \, x^{4} - 15 \, x^{2} \operatorname{arccsc}\left (x\right )^{2} - 28 \, x^{2} \log \left (x\right ) + 2 \,{\left (2 \, x^{4} - 14 \, x^{2} - 3\right )} \sqrt{x^{2} - 1} \operatorname{arccsc}\left (x\right ) + 3}{12 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)^(5/2)*arccsc(x)/x^3,x, algorithm="fricas")

[Out]

1/12*(2*x^4 - 15*x^2*arccsc(x)^2 - 28*x^2*log(x) + 2*(2*x^4 - 14*x^2 - 3)*sqrt(x^2 - 1)*arccsc(x) + 3)/x^2

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2-1)**(5/2)*acsc(x)/x**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (x^{2} - 1\right )}^{\frac{5}{2}} \operatorname{arccsc}\left (x\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)^(5/2)*arccsc(x)/x^3,x, algorithm="giac")

[Out]

integrate((x^2 - 1)^(5/2)*arccsc(x)/x^3, x)