3.692 \(\int \frac{\csc ^{-1}(x)^4}{x^2 \sqrt{-1+x^2}} \, dx\)

Optimal. Leaf size=74 \[ \frac{24 \sqrt{x^2-1}}{x}+\frac{\sqrt{x^2-1} \csc ^{-1}(x)^4}{x}-\frac{4 \csc ^{-1}(x)^3}{\sqrt{x^2}}-\frac{12 \sqrt{x^2-1} \csc ^{-1}(x)^2}{x}+\frac{24 \csc ^{-1}(x)}{\sqrt{x^2}} \]

[Out]

(24*Sqrt[-1 + x^2])/x + (24*ArcCsc[x])/Sqrt[x^2] - (12*Sqrt[-1 + x^2]*ArcCsc[x]^2)/x - (4*ArcCsc[x]^3)/Sqrt[x^
2] + (Sqrt[-1 + x^2]*ArcCsc[x]^4)/x

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Rubi [A]  time = 0.178698, antiderivative size = 101, normalized size of antiderivative = 1.36, number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {5243, 4677, 4619, 261} \[ \frac{24 \sqrt{1-\frac{1}{x^2}} \sqrt{x^2}}{x}+\frac{\sqrt{1-\frac{1}{x^2}} \sqrt{x^2} \csc ^{-1}(x)^4}{x}-\frac{4 \csc ^{-1}(x)^3}{\sqrt{x^2}}-\frac{12 \sqrt{1-\frac{1}{x^2}} \sqrt{x^2} \csc ^{-1}(x)^2}{x}+\frac{24 \csc ^{-1}(x)}{\sqrt{x^2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCsc[x]^4/(x^2*Sqrt[-1 + x^2]),x]

[Out]

(24*Sqrt[1 - x^(-2)]*Sqrt[x^2])/x + (24*ArcCsc[x])/Sqrt[x^2] - (12*Sqrt[1 - x^(-2)]*Sqrt[x^2]*ArcCsc[x]^2)/x -
(4*ArcCsc[x]^3)/Sqrt[x^2] + (Sqrt[1 - x^(-2)]*Sqrt[x^2]*ArcCsc[x]^4)/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 4677

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

Rule 4619

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

Rule 261

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

Rubi steps

\begin{align*} \int \frac{\csc ^{-1}(x)^4}{x^2 \sqrt{-1+x^2}} \, dx &=-\frac{\sqrt{x^2} \operatorname{Subst}\left (\int \frac{x \sin ^{-1}(x)^4}{\sqrt{1-x^2}} \, dx,x,\frac{1}{x}\right )}{x}\\ &=\frac{\sqrt{1-\frac{1}{x^2}} \sqrt{x^2} \csc ^{-1}(x)^4}{x}-\frac{\left (4 \sqrt{x^2}\right ) \operatorname{Subst}\left (\int \sin ^{-1}(x)^3 \, dx,x,\frac{1}{x}\right )}{x}\\ &=-\frac{4 \csc ^{-1}(x)^3}{\sqrt{x^2}}+\frac{\sqrt{1-\frac{1}{x^2}} \sqrt{x^2} \csc ^{-1}(x)^4}{x}+\frac{\left (12 \sqrt{x^2}\right ) \operatorname{Subst}\left (\int \frac{x \sin ^{-1}(x)^2}{\sqrt{1-x^2}} \, dx,x,\frac{1}{x}\right )}{x}\\ &=-\frac{12 \sqrt{1-\frac{1}{x^2}} \sqrt{x^2} \csc ^{-1}(x)^2}{x}-\frac{4 \csc ^{-1}(x)^3}{\sqrt{x^2}}+\frac{\sqrt{1-\frac{1}{x^2}} \sqrt{x^2} \csc ^{-1}(x)^4}{x}+\frac{\left (24 \sqrt{x^2}\right ) \operatorname{Subst}\left (\int \sin ^{-1}(x) \, dx,x,\frac{1}{x}\right )}{x}\\ &=\frac{24 \csc ^{-1}(x)}{\sqrt{x^2}}-\frac{12 \sqrt{1-\frac{1}{x^2}} \sqrt{x^2} \csc ^{-1}(x)^2}{x}-\frac{4 \csc ^{-1}(x)^3}{\sqrt{x^2}}+\frac{\sqrt{1-\frac{1}{x^2}} \sqrt{x^2} \csc ^{-1}(x)^4}{x}-\frac{\left (24 \sqrt{x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-x^2}} \, dx,x,\frac{1}{x}\right )}{x}\\ &=\frac{24 \sqrt{1-\frac{1}{x^2}} \sqrt{x^2}}{x}+\frac{24 \csc ^{-1}(x)}{\sqrt{x^2}}-\frac{12 \sqrt{1-\frac{1}{x^2}} \sqrt{x^2} \csc ^{-1}(x)^2}{x}-\frac{4 \csc ^{-1}(x)^3}{\sqrt{x^2}}+\frac{\sqrt{1-\frac{1}{x^2}} \sqrt{x^2} \csc ^{-1}(x)^4}{x}\\ \end{align*}

Mathematica [A]  time = 0.0588532, size = 76, normalized size = 1.03 \[ \frac{24 \left (x^2-1\right )+\left (x^2-1\right ) \csc ^{-1}(x)^4-4 \sqrt{1-\frac{1}{x^2}} x \csc ^{-1}(x)^3-12 \left (x^2-1\right ) \csc ^{-1}(x)^2+24 \sqrt{1-\frac{1}{x^2}} x \csc ^{-1}(x)}{x \sqrt{x^2-1}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCsc[x]^4/(x^2*Sqrt[-1 + x^2]),x]

[Out]

(24*(-1 + x^2) + 24*Sqrt[1 - x^(-2)]*x*ArcCsc[x] - 12*(-1 + x^2)*ArcCsc[x]^2 - 4*Sqrt[1 - x^(-2)]*x*ArcCsc[x]^
3 + (-1 + x^2)*ArcCsc[x]^4)/(x*Sqrt[-1 + x^2])

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Maple [C]  time = 0.405, size = 443, normalized size = 6. \begin{align*}{\frac{ \left ({\rm arccsc} \left (x\right ) \right ) ^{3}}{x} \left ( i{x}^{2}-2\,\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x-2\,i \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}+{\frac{ \left ({\rm arccsc} \left (x\right ) \right ) ^{4}}{4\,x} \left ({x}^{2}-2+2\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}+{\frac{x \left ({\rm arccsc} \left (x\right ) \right ) ^{4}}{2}{\frac{1}{\sqrt{{x}^{2}-1}}}}-6\,{\frac{{\rm arccsc} \left (x\right )}{\sqrt{{x}^{2}-1}x} \left ( i{x}^{2}-2\,\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x-2\,i \right ) }-3\,{\frac{ \left ({\rm arccsc} \left (x\right ) \right ) ^{2}}{\sqrt{{x}^{2}-1}x} \left ({x}^{2}-2+2\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x \right ) }-6\,{\frac{x \left ({\rm arccsc} \left (x\right ) \right ) ^{2}}{\sqrt{{x}^{2}-1}}}+6\,{\frac{1}{\sqrt{{x}^{2}-1}x} \left ( i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{x}^{3}-4\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x-3\,{x}^{2}+4 \right ) \left ( i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x-1 \right ) ^{-1}}+12\,{\frac{x}{\sqrt{{x}^{2}-1}}}-{\frac{ \left ({\rm arccsc} \left (x\right ) \right ) ^{3}}{x} \left ( i{x}^{2}+2\,\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x-2\,i \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}+{\frac{ \left ({\rm arccsc} \left (x\right ) \right ) ^{4}}{4\,x} \left ({x}^{2}-2-2\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}+6\,{\frac{{\rm arccsc} \left (x\right )}{\sqrt{{x}^{2}-1}x} \left ( i{x}^{2}+2\,\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x-2\,i \right ) }-3\,{\frac{ \left ({\rm arccsc} \left (x\right ) \right ) ^{2}}{\sqrt{{x}^{2}-1}x} \left ({x}^{2}-2-2\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x \right ) }+{6\,i{x}^{3}{\frac{1}{\sqrt{{x}^{2}-1}}} \left ( i{x}^{2}-2\,\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x-2\,i \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccsc(x)^4/x^2/(x^2-1)^(1/2),x)

[Out]

1/(x^2-1)^(1/2)/x*(I*x^2-2*((x^2-1)/x^2)^(1/2)*x-2*I)*arccsc(x)^3+1/4/(x^2-1)^(1/2)/x*(x^2-2+2*I*((x^2-1)/x^2)
^(1/2)*x)*arccsc(x)^4+1/2/(x^2-1)^(1/2)*x*arccsc(x)^4-6/(x^2-1)^(1/2)/x*(I*x^2-2*((x^2-1)/x^2)^(1/2)*x-2*I)*ar
ccsc(x)-3/(x^2-1)^(1/2)/x*(x^2-2+2*I*((x^2-1)/x^2)^(1/2)*x)*arccsc(x)^2-6/(x^2-1)^(1/2)*x*arccsc(x)^2+6*(I*((x
^2-1)/x^2)^(1/2)*x^3-4*I*((x^2-1)/x^2)^(1/2)*x-3*x^2+4)/(x^2-1)^(1/2)/(I*((x^2-1)/x^2)^(1/2)*x-1)/x+12*x/(x^2-
1)^(1/2)-1/(x^2-1)^(1/2)*(I*x^2+2*((x^2-1)/x^2)^(1/2)*x-2*I)*arccsc(x)^3/x+1/4/(x^2-1)^(1/2)*(x^2-2-2*I*((x^2-
1)/x^2)^(1/2)*x)*arccsc(x)^4/x+6/(x^2-1)^(1/2)*(I*x^2+2*((x^2-1)/x^2)^(1/2)*x-2*I)*arccsc(x)/x-3/(x^2-1)^(1/2)
*(x^2-2-2*I*((x^2-1)/x^2)^(1/2)*x)*arccsc(x)^2/x+6*I*x^3/(x^2-1)^(1/2)/(I*x^2-2*((x^2-1)/x^2)^(1/2)*x-2*I)

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Maxima [A]  time = 1.5069, size = 78, normalized size = 1.05 \begin{align*} \frac{\sqrt{x^{2} - 1} \operatorname{arccsc}\left (x\right )^{4}}{x} - 12 \, \sqrt{-\frac{1}{x^{2}} + 1} \operatorname{arccsc}\left (x\right )^{2} - \frac{4 \, \operatorname{arccsc}\left (x\right )^{3}}{x} + 24 \, \sqrt{-\frac{1}{x^{2}} + 1} + \frac{24 \, \operatorname{arccsc}\left (x\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsc(x)^4/x^2/(x^2-1)^(1/2),x, algorithm="maxima")

[Out]

sqrt(x^2 - 1)*arccsc(x)^4/x - 12*sqrt(-1/x^2 + 1)*arccsc(x)^2 - 4*arccsc(x)^3/x + 24*sqrt(-1/x^2 + 1) + 24*arc
csc(x)/x

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Fricas [A]  time = 2.58062, size = 117, normalized size = 1.58 \begin{align*} -\frac{4 \, \operatorname{arccsc}\left (x\right )^{3} -{\left (\operatorname{arccsc}\left (x\right )^{4} - 12 \, \operatorname{arccsc}\left (x\right )^{2} + 24\right )} \sqrt{x^{2} - 1} - 24 \, \operatorname{arccsc}\left (x\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsc(x)^4/x^2/(x^2-1)^(1/2),x, algorithm="fricas")

[Out]

-(4*arccsc(x)^3 - (arccsc(x)^4 - 12*arccsc(x)^2 + 24)*sqrt(x^2 - 1) - 24*arccsc(x))/x

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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(acsc(x)**4/x**2/(x**2-1)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arccsc}\left (x\right )^{4}}{\sqrt{x^{2} - 1} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsc(x)^4/x^2/(x^2-1)^(1/2),x, algorithm="giac")

[Out]

integrate(arccsc(x)^4/(sqrt(x^2 - 1)*x^2), x)