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

Optimal. Leaf size=82 \[ \frac{4 x \sqrt{x^2-1} \left (3 x^2-19 x+83\right )}{105 \sqrt{x^2} \sqrt{x-1}}+\frac{4 x \tanh ^{-1}\left (\frac{\sqrt{x^2-1}}{\sqrt{x-1}}\right )}{7 \sqrt{x^2}}+\frac{2}{7} (x-1)^{7/2} \csc ^{-1}(x) \]

[Out]

(4*x*Sqrt[-1 + x^2]*(83 - 19*x + 3*x^2))/(105*Sqrt[-1 + x]*Sqrt[x^2]) + (2*(-1 + x)^(7/2)*ArcCsc[x])/7 + (4*x*
ArcTanh[Sqrt[-1 + x^2]/Sqrt[-1 + x]])/(7*Sqrt[x^2])

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Rubi [A]  time = 0.0784756, antiderivative size = 140, normalized size of antiderivative = 1.71, number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {5227, 1574, 892, 88, 63, 207} \[ \frac{4 (x+1)^3 \sqrt{x-1}}{35 \sqrt{1-\frac{1}{x^2}} x}-\frac{20 (x+1)^2 \sqrt{x-1}}{21 \sqrt{1-\frac{1}{x^2}} x}+\frac{4 (x+1) \sqrt{x-1}}{\sqrt{1-\frac{1}{x^2}} x}+\frac{4 \sqrt{x+1} \sqrt{x-1} \tanh ^{-1}\left (\sqrt{x+1}\right )}{7 \sqrt{1-\frac{1}{x^2}} x}+\frac{2}{7} (x-1)^{7/2} \csc ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(-1 + x)^(5/2)*ArcCsc[x],x]

[Out]

(4*Sqrt[-1 + x]*(1 + x))/(Sqrt[1 - x^(-2)]*x) - (20*Sqrt[-1 + x]*(1 + x)^2)/(21*Sqrt[1 - x^(-2)]*x) + (4*Sqrt[
-1 + x]*(1 + x)^3)/(35*Sqrt[1 - x^(-2)]*x) + (2*(-1 + x)^(7/2)*ArcCsc[x])/7 + (4*Sqrt[-1 + x]*Sqrt[1 + x]*ArcT
anh[Sqrt[1 + x]])/(7*Sqrt[1 - x^(-2)]*x)

Rule 5227

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(a + b
*ArcCsc[c*x]))/(e*(m + 1)), x] + Dist[b/(c*e*(m + 1)), Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x],
x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]

Rule 1574

Int[(x_)^(m_.)*((a_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Dist[(x^(2*n*Fr
acPart[p])*(a + c/x^(2*n))^FracPart[p])/(c + a*x^(2*n))^FracPart[p], Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + a*x^
(2*n))^p, x], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[mn2, -2*n] &&  !IntegerQ[p] &&  !IntegerQ[q] &&
PosQ[n]

Rule 892

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + c*x^
2)^FracPart[p]/((d + e*x)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p]), Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + (
c*x)/e)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] &&  !Int
egerQ[p] &&  !IGtQ[m, 0] &&  !IGtQ[n, 0]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int (-1+x)^{5/2} \csc ^{-1}(x) \, dx &=\frac{2}{7} (-1+x)^{7/2} \csc ^{-1}(x)+\frac{2}{7} \int \frac{(-1+x)^{7/2}}{\sqrt{1-\frac{1}{x^2}} x^2} \, dx\\ &=\frac{2}{7} (-1+x)^{7/2} \csc ^{-1}(x)+\frac{\left (2 \sqrt{-1+x^2}\right ) \int \frac{(-1+x)^{7/2}}{x \sqrt{-1+x^2}} \, dx}{7 \sqrt{1-\frac{1}{x^2}} x}\\ &=\frac{2}{7} (-1+x)^{7/2} \csc ^{-1}(x)+\frac{\left (2 \sqrt{-1+x} \sqrt{1+x}\right ) \int \frac{(-1+x)^3}{x \sqrt{1+x}} \, dx}{7 \sqrt{1-\frac{1}{x^2}} x}\\ &=\frac{2}{7} (-1+x)^{7/2} \csc ^{-1}(x)+\frac{\left (2 \sqrt{-1+x} \sqrt{1+x}\right ) \int \left (\frac{7}{\sqrt{1+x}}-\frac{1}{x \sqrt{1+x}}-5 \sqrt{1+x}+(1+x)^{3/2}\right ) \, dx}{7 \sqrt{1-\frac{1}{x^2}} x}\\ &=\frac{4 \sqrt{-1+x} (1+x)}{\sqrt{1-\frac{1}{x^2}} x}-\frac{20 \sqrt{-1+x} (1+x)^2}{21 \sqrt{1-\frac{1}{x^2}} x}+\frac{4 \sqrt{-1+x} (1+x)^3}{35 \sqrt{1-\frac{1}{x^2}} x}+\frac{2}{7} (-1+x)^{7/2} \csc ^{-1}(x)-\frac{\left (2 \sqrt{-1+x} \sqrt{1+x}\right ) \int \frac{1}{x \sqrt{1+x}} \, dx}{7 \sqrt{1-\frac{1}{x^2}} x}\\ &=\frac{4 \sqrt{-1+x} (1+x)}{\sqrt{1-\frac{1}{x^2}} x}-\frac{20 \sqrt{-1+x} (1+x)^2}{21 \sqrt{1-\frac{1}{x^2}} x}+\frac{4 \sqrt{-1+x} (1+x)^3}{35 \sqrt{1-\frac{1}{x^2}} x}+\frac{2}{7} (-1+x)^{7/2} \csc ^{-1}(x)-\frac{\left (4 \sqrt{-1+x} \sqrt{1+x}\right ) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sqrt{1+x}\right )}{7 \sqrt{1-\frac{1}{x^2}} x}\\ &=\frac{4 \sqrt{-1+x} (1+x)}{\sqrt{1-\frac{1}{x^2}} x}-\frac{20 \sqrt{-1+x} (1+x)^2}{21 \sqrt{1-\frac{1}{x^2}} x}+\frac{4 \sqrt{-1+x} (1+x)^3}{35 \sqrt{1-\frac{1}{x^2}} x}+\frac{2}{7} (-1+x)^{7/2} \csc ^{-1}(x)+\frac{4 \sqrt{-1+x} \sqrt{1+x} \tanh ^{-1}\left (\sqrt{1+x}\right )}{7 \sqrt{1-\frac{1}{x^2}} x}\\ \end{align*}

Mathematica [A]  time = 0.0811706, size = 72, normalized size = 0.88 \[ \frac{4 \sqrt{1-\frac{1}{x^2}} x \left (3 x^2-19 x+83\right )}{105 \sqrt{x-1}}+\frac{4}{7} \tanh ^{-1}\left (\frac{\sqrt{1-\frac{1}{x^2}} x}{\sqrt{x-1}}\right )+\frac{2}{7} (x-1)^{7/2} \csc ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x)^(5/2)*ArcCsc[x],x]

[Out]

(4*Sqrt[1 - x^(-2)]*x*(83 - 19*x + 3*x^2))/(105*Sqrt[-1 + x]) + (2*(-1 + x)^(7/2)*ArcCsc[x])/7 + (4*ArcTanh[(S
qrt[1 - x^(-2)]*x)/Sqrt[-1 + x]])/7

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Maple [A]  time = 0.015, size = 76, normalized size = 0.9 \begin{align*}{\frac{2\,{\rm arccsc} \left (x\right )}{7} \left ( -1+x \right ) ^{{\frac{7}{2}}}}+{\frac{4}{105\,x}\sqrt{-1+x}\sqrt{1+x} \left ( 3\, \left ( -1+x \right ) ^{2}\sqrt{1+x}-13\, \left ( -1+x \right ) \sqrt{1+x}+15\,{\it Artanh} \left ( \sqrt{1+x} \right ) +67\,\sqrt{1+x} \right ){\frac{1}{\sqrt{{\frac{ \left ( 1+x \right ) \left ( -1+x \right ) }{{x}^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*(-1+x)^(7/2)*arccsc(x)+4/105*(-1+x)^(1/2)*(1+x)^(1/2)*(3*(-1+x)^2*(1+x)^(1/2)-13*(-1+x)*(1+x)^(1/2)+15*arc
tanh((1+x)^(1/2))+67*(1+x)^(1/2))/((-1+x)*(1+x)/x^2)^(1/2)/x

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Maxima [A]  time = 3.69916, size = 157, normalized size = 1.91 \begin{align*} \frac{4}{35} \,{\left (x + 1\right )}^{\frac{5}{2}} - \frac{20}{21} \,{\left (x + 1\right )}^{\frac{3}{2}} + \frac{2}{7} \,{\left (x^{3} \arctan \left (1, \sqrt{x + 1} \sqrt{x - 1}\right ) - 3 \, x^{2} \arctan \left (1, \sqrt{x + 1} \sqrt{x - 1}\right ) + 3 \, x \arctan \left (1, \sqrt{x + 1} \sqrt{x - 1}\right ) - \arctan \left (1, \sqrt{x + 1} \sqrt{x - 1}\right )\right )} \sqrt{x - 1} + 4 \, \sqrt{x + 1} + \frac{2}{7} \, \log \left (\sqrt{x + 1} + 1\right ) - \frac{2}{7} \, \log \left (\sqrt{x + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/35*(x + 1)^(5/2) - 20/21*(x + 1)^(3/2) + 2/7*(x^3*arctan2(1, sqrt(x + 1)*sqrt(x - 1)) - 3*x^2*arctan2(1, sqr
t(x + 1)*sqrt(x - 1)) + 3*x*arctan2(1, sqrt(x + 1)*sqrt(x - 1)) - arctan2(1, sqrt(x + 1)*sqrt(x - 1)))*sqrt(x
- 1) + 4*sqrt(x + 1) + 2/7*log(sqrt(x + 1) + 1) - 2/7*log(sqrt(x + 1) - 1)

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Fricas [B]  time = 2.47565, size = 347, normalized size = 4.23 \begin{align*} \frac{2 \,{\left (15 \,{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )} \sqrt{x - 1} \operatorname{arccsc}\left (x\right ) + 2 \,{\left (3 \, x^{2} - 19 \, x + 83\right )} \sqrt{x^{2} - 1} \sqrt{x - 1} + 15 \,{\left (x - 1\right )} \log \left (\frac{x^{2} + \sqrt{x^{2} - 1} \sqrt{x - 1} - 1}{x^{2} - 1}\right ) - 15 \,{\left (x - 1\right )} \log \left (-\frac{x^{2} - \sqrt{x^{2} - 1} \sqrt{x - 1} - 1}{x^{2} - 1}\right )\right )}}{105 \,{\left (x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)*sqrt(x - 1)*arccsc(x) + 2*(3*x^2 - 19*x + 83)*sqrt(x^2 - 1)*sqrt(x -
1) + 15*(x - 1)*log((x^2 + sqrt(x^2 - 1)*sqrt(x - 1) - 1)/(x^2 - 1)) - 15*(x - 1)*log(-(x^2 - sqrt(x^2 - 1)*s
qrt(x - 1) - 1)/(x^2 - 1)))/(x - 1)

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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((-1+x)**(5/2)*acsc(x),x)

[Out]

Timed out

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Giac [B]  time = 1.24695, size = 266, normalized size = 3.24 \begin{align*} \frac{2}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} \arcsin \left (\frac{1}{x}\right ) + \frac{2}{105} \,{\left (15 \,{\left (x - 1\right )}^{\frac{7}{2}} + 42 \,{\left (x - 1\right )}^{\frac{5}{2}} + 35 \,{\left (x - 1\right )}^{\frac{3}{2}}\right )} \arcsin \left (\frac{1}{x}\right ) - \frac{4}{15} \,{\left (3 \,{\left (x - 1\right )}^{\frac{5}{2}} + 5 \,{\left (x - 1\right )}^{\frac{3}{2}}\right )} \arcsin \left (\frac{1}{x}\right ) + \frac{4 \,{\left (3 \,{\left (x + 1\right )}^{\frac{5}{2}} - 11 \,{\left (x + 1\right )}^{\frac{3}{2}} + 14 \, \sqrt{x + 1}\right )}}{105 \, \mathrm{sgn}\left ({\left (x - 1\right )}^{\frac{3}{2}} + \sqrt{x - 1}\right )} - \frac{8 \,{\left ({\left (x + 1\right )}^{\frac{3}{2}} - 4 \, \sqrt{x + 1}\right )}}{15 \, \mathrm{sgn}\left ({\left (x - 1\right )}^{\frac{3}{2}} + \sqrt{x - 1}\right )} + \frac{2 \, \log \left (\sqrt{x + 1} + 1\right )}{7 \, \mathrm{sgn}\left ({\left (x - 1\right )}^{\frac{3}{2}} + \sqrt{x - 1}\right )} - \frac{2 \, \log \left (\sqrt{x + 1} - 1\right )}{7 \, \mathrm{sgn}\left ({\left (x - 1\right )}^{\frac{3}{2}} + \sqrt{x - 1}\right )} + \frac{4 \, \sqrt{x + 1}}{3 \, \mathrm{sgn}\left ({\left (x - 1\right )}^{\frac{3}{2}} + \sqrt{x - 1}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x - 1)^(3/2)*arcsin(1/x) + 2/105*(15*(x - 1)^(7/2) + 42*(x - 1)^(5/2) + 35*(x - 1)^(3/2))*arcsin(1/x) - 4
/15*(3*(x - 1)^(5/2) + 5*(x - 1)^(3/2))*arcsin(1/x) + 4/105*(3*(x + 1)^(5/2) - 11*(x + 1)^(3/2) + 14*sqrt(x +
1))/sgn((x - 1)^(3/2) + sqrt(x - 1)) - 8/15*((x + 1)^(3/2) - 4*sqrt(x + 1))/sgn((x - 1)^(3/2) + sqrt(x - 1)) +
2/7*log(sqrt(x + 1) + 1)/sgn((x - 1)^(3/2) + sqrt(x - 1)) - 2/7*log(sqrt(x + 1) - 1)/sgn((x - 1)^(3/2) + sqrt
(x - 1)) + 4/3*sqrt(x + 1)/sgn((x - 1)^(3/2) + sqrt(x - 1))