3.8.2 \(\int (-1+x)^{5/2} \csc ^{-1}(x) \, dx\) [702]

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

[Out]

2/7*(-1+x)^(7/2)*arccsc(x)+4/7*x*arctanh((x^2-1)^(1/2)/(-1+x)^(1/2))/(x^2)^(1/2)+4/105*x*(3*x^2-19*x+83)*(x^2-
1)^(1/2)/(-1+x)^(1/2)/(x^2)^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 140, normalized size of antiderivative = 1.71, number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5335, 1588, 906, 90, 65, 213} \begin {gather*} \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) \end {gather*}

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 65

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 90

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 213

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

Rule 906

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/e)*x)^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 1588

Int[(x_)^(m_.)*((a_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Dist[x^(2*n*Fra
cPart[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 5335

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]

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 ) \text {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*}

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Mathematica [A]
time = 0.06, size = 72, normalized size = 0.88 \begin {gather*} \frac {4 \sqrt {1-\frac {1}{x^2}} x \left (83-19 x+3 x^2\right )}{105 \sqrt {-1+x}}+\frac {2}{7} (-1+x)^{7/2} \csc ^{-1}(x)+\frac {4}{7} \tanh ^{-1}\left (\frac {\sqrt {1-\frac {1}{x^2}} x}{\sqrt {-1+x}}\right ) \end {gather*}

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 = 0.08, size = 76, normalized size = 0.93

method	result	size
derivativedivides	\(\frac {2 \left (-1+x \right )^{\frac {7}{2}} \mathrm {arccsc}\left (x \right )}{7}+\frac {4 \sqrt {-1+x}\, \sqrt {1+x}\, \left (3 \left (-1+x \right )^{2} \sqrt {1+x}-13 \left (-1+x \right ) \sqrt {1+x}+15 \arctanh \left (\sqrt {1+x}\right )+67 \sqrt {1+x}\right )}{105 \sqrt {\frac {\left (-1+x \right ) \left (1+x \right )}{x^{2}}}\, x}\)	\(76\)
default	\(\frac {2 \left (-1+x \right )^{\frac {7}{2}} \mathrm {arccsc}\left (x \right )}{7}+\frac {4 \sqrt {-1+x}\, \sqrt {1+x}\, \left (3 \left (-1+x \right )^{2} \sqrt {1+x}-13 \left (-1+x \right ) \sqrt {1+x}+15 \arctanh \left (\sqrt {1+x}\right )+67 \sqrt {1+x}\right )}{105 \sqrt {\frac {\left (-1+x \right ) \left (1+x \right )}{x^{2}}}\, x}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 2.81, size = 116, normalized size = 1.41 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (62) = 124\).
time = 0.78, size = 125, normalized size = 1.52 \begin {gather*} \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 {gather*}

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3061 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (62) = 124\).
time = 0.97, size = 228, normalized size = 2.78 \begin {gather*} \frac {2}{35} \, {\left (5 \, {\left (x - 1\right )}^{\frac {7}{2}} + 21 \, {\left (x - 1\right )}^{\frac {5}{2}} + 35 \, {\left (x - 1\right )}^{\frac {3}{2}} + 35 \, \sqrt {x - 1}\right )} \arcsin \left (\frac {1}{x}\right ) - \frac {2}{5} \, {\left (3 \, {\left (x - 1\right )}^{\frac {5}{2}} + 10 \, {\left (x - 1\right )}^{\frac {3}{2}} + 15 \, \sqrt {x - 1}\right )} \arcsin \left (\frac {1}{x}\right ) + 2 \, {\left ({\left (x - 1\right )}^{\frac {3}{2}} + 3 \, \sqrt {x - 1}\right )} \arcsin \left (\frac {1}{x}\right ) - 2 \, \sqrt {x - 1} \arcsin \left (\frac {1}{x}\right ) + \frac {4 \, {\left (3 \, {\left (x + 1\right )}^{\frac {5}{2}} - 4 \, {\left (x + 1\right )}^{\frac {3}{2}} + 21 \, \sqrt {x + 1}\right )}}{105 \, \mathrm {sgn}\left ({\left (x - 1\right )}^{\frac {3}{2}} + \sqrt {x - 1}\right )} - \frac {4 \, {\left ({\left (x + 1\right )}^{\frac {3}{2}} + \sqrt {x + 1}\right )}}{5 \, \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}}{\mathrm {sgn}\left ({\left (x - 1\right )}^{\frac {3}{2}} + \sqrt {x - 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(5*(x - 1)^(7/2) + 21*(x - 1)^(5/2) + 35*(x - 1)^(3/2) + 35*sqrt(x - 1))*arcsin(1/x) - 2/5*(3*(x - 1)^(5/
2) + 10*(x - 1)^(3/2) + 15*sqrt(x - 1))*arcsin(1/x) + 2*((x - 1)^(3/2) + 3*sqrt(x - 1))*arcsin(1/x) - 2*sqrt(x
- 1)*arcsin(1/x) + 4/105*(3*(x + 1)^(5/2) - 4*(x + 1)^(3/2) + 21*sqrt(x + 1))/sgn((x - 1)^(3/2) + sqrt(x - 1)
) - 4/5*((x + 1)^(3/2) + 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*sqrt(x + 1)/sgn((x - 1)^(
3/2) + sqrt(x - 1))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \mathrm {asin}\left (\frac {1}{x}\right )\,{\left (x-1\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(1/x)*(x - 1)^(5/2),x)

[Out]

int(asin(1/x)*(x - 1)^(5/2), x)

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