3.649 \(\int x^3 \csc ^{-1}(x)^2 \, dx\)

Optimal. Leaf size=63 \[ \frac{x^2}{12}+\frac{1}{4} x^4 \csc ^{-1}(x)^2+\frac{1}{6} \sqrt{1-\frac{1}{x^2}} x^3 \csc ^{-1}(x)+\frac{1}{3} \sqrt{1-\frac{1}{x^2}} x \csc ^{-1}(x)+\frac{\log (x)}{3} \]

[Out]

x^2/12 + (Sqrt[1 - x^(-2)]*x*ArcCsc[x])/3 + (Sqrt[1 - x^(-2)]*x^3*ArcCsc[x])/6 + (x^4*ArcCsc[x]^2)/4 + Log[x]/
3

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Rubi [A]  time = 0.0671453, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {5223, 3758, 4185, 4184, 3475} \[ \frac{x^2}{12}+\frac{1}{4} x^4 \csc ^{-1}(x)^2+\frac{1}{6} \sqrt{1-\frac{1}{x^2}} x^3 \csc ^{-1}(x)+\frac{1}{3} \sqrt{1-\frac{1}{x^2}} x \csc ^{-1}(x)+\frac{\log (x)}{3} \]

Antiderivative was successfully verified.

[In]

Int[x^3*ArcCsc[x]^2,x]

[Out]

x^2/12 + (Sqrt[1 - x^(-2)]*x*ArcCsc[x])/3 + (Sqrt[1 - x^(-2)]*x^3*ArcCsc[x])/6 + (x^4*ArcCsc[x]^2)/4 + Log[x]/
3

Rule 5223

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b*
x)^n*Csc[x]^(m + 1)*Cot[x], x], x, ArcCsc[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (G
tQ[n, 0] || LtQ[m, -1])

Rule 3758

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

Rule 4185

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> -Simp[(b^2*(c + d*x)*Cot[e + f*x]*
(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2
), x], x] - Simp[(b^2*d*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[{b, c, d, e, f}, x] && G
tQ[n, 1] && NeQ[n, 2]

Rule 4184

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int x^3 \csc ^{-1}(x)^2 \, dx &=-\operatorname{Subst}\left (\int x^2 \cot (x) \csc ^4(x) \, dx,x,\csc ^{-1}(x)\right )\\ &=\frac{1}{4} x^4 \csc ^{-1}(x)^2-\frac{1}{2} \operatorname{Subst}\left (\int x \csc ^4(x) \, dx,x,\csc ^{-1}(x)\right )\\ &=\frac{x^2}{12}+\frac{1}{6} \sqrt{1-\frac{1}{x^2}} x^3 \csc ^{-1}(x)+\frac{1}{4} x^4 \csc ^{-1}(x)^2-\frac{1}{3} \operatorname{Subst}\left (\int x \csc ^2(x) \, dx,x,\csc ^{-1}(x)\right )\\ &=\frac{x^2}{12}+\frac{1}{3} \sqrt{1-\frac{1}{x^2}} x \csc ^{-1}(x)+\frac{1}{6} \sqrt{1-\frac{1}{x^2}} x^3 \csc ^{-1}(x)+\frac{1}{4} x^4 \csc ^{-1}(x)^2-\frac{1}{3} \operatorname{Subst}\left (\int \cot (x) \, dx,x,\csc ^{-1}(x)\right )\\ &=\frac{x^2}{12}+\frac{1}{3} \sqrt{1-\frac{1}{x^2}} x \csc ^{-1}(x)+\frac{1}{6} \sqrt{1-\frac{1}{x^2}} x^3 \csc ^{-1}(x)+\frac{1}{4} x^4 \csc ^{-1}(x)^2+\frac{\log (x)}{3}\\ \end{align*}

Mathematica [A]  time = 0.0475762, size = 42, normalized size = 0.67 \[ \frac{1}{12} \left (x^2+3 x^4 \csc ^{-1}(x)^2+2 \sqrt{1-\frac{1}{x^2}} \left (x^2+2\right ) x \csc ^{-1}(x)+4 \log (x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*ArcCsc[x]^2,x]

[Out]

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

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Maple [A]  time = 0.034, size = 56, normalized size = 0.9 \begin{align*}{\frac{{x}^{4} \left ({\rm arccsc} \left (x\right ) \right ) ^{2}}{4}}+{\frac{{\rm arccsc} \left (x\right ){x}^{3}}{6}\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}}+{\frac{{x}^{2}}{12}}+{\frac{{\rm arccsc} \left (x\right )x}{3}\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}}-{\frac{\ln \left ({x}^{-1} \right ) }{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arccsc(x)^2,x)

[Out]

1/4*x^4*arccsc(x)^2+1/6*arccsc(x)*x^3*((x^2-1)/x^2)^(1/2)+1/12*x^2+1/3*((x^2-1)/x^2)^(1/2)*arccsc(x)*x-1/3*ln(
1/x)

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Maxima [A]  time = 1.66807, size = 128, normalized size = 2.03 \begin{align*} \frac{1}{4} \, x^{4} \operatorname{arccsc}\left (x\right )^{2} + \frac{2 \, x^{4} \arctan \left (1, \sqrt{x + 1} \sqrt{x - 1}\right ) + 2 \, x^{2} \arctan \left (1, \sqrt{x + 1} \sqrt{x - 1}\right ) +{\left (x^{2} + 2 \, \log \left (x^{2}\right )\right )} \sqrt{x + 1} \sqrt{x - 1} - 4 \, \arctan \left (1, \sqrt{x + 1} \sqrt{x - 1}\right )}{12 \, \sqrt{x + 1} \sqrt{x - 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccsc(x)^2,x, algorithm="maxima")

[Out]

1/4*x^4*arccsc(x)^2 + 1/12*(2*x^4*arctan2(1, sqrt(x + 1)*sqrt(x - 1)) + 2*x^2*arctan2(1, sqrt(x + 1)*sqrt(x -
1)) + (x^2 + 2*log(x^2))*sqrt(x + 1)*sqrt(x - 1) - 4*arctan2(1, sqrt(x + 1)*sqrt(x - 1)))/(sqrt(x + 1)*sqrt(x
- 1))

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Fricas [A]  time = 2.62521, size = 115, normalized size = 1.83 \begin{align*} \frac{1}{4} \, x^{4} \operatorname{arccsc}\left (x\right )^{2} + \frac{1}{6} \,{\left (x^{2} + 2\right )} \sqrt{x^{2} - 1} \operatorname{arccsc}\left (x\right ) + \frac{1}{12} \, x^{2} + \frac{1}{3} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccsc(x)^2,x, algorithm="fricas")

[Out]

1/4*x^4*arccsc(x)^2 + 1/6*(x^2 + 2)*sqrt(x^2 - 1)*arccsc(x) + 1/12*x^2 + 1/3*log(x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*acsc(x)**2,x)

[Out]

Integral(x**3*acsc(x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccsc(x)^2,x, algorithm="giac")

[Out]

sage2