3.7.92 \(\int \frac {\csc ^{-1}(x)^4}{x^2 \sqrt {-1+x^2}} \, dx\) [692]

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

[Out]

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

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Rubi [A]
time = 0.13, 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 = {5351, 4767, 4715, 267} \begin {gather*} \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}} \end {gather*}

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 267

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]

Rule 4715

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 4767

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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 5351

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]

Rubi steps

\begin {align*} \int \frac {\csc ^{-1}(x)^4}{x^2 \sqrt {-1+x^2}} \, dx &=-\frac {\sqrt {x^2} \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.04, size = 76, normalized size = 1.03 \begin {gather*} \frac {24 \left (-1+x^2\right )+24 \sqrt {1-\frac {1}{x^2}} x \csc ^{-1}(x)-12 \left (-1+x^2\right ) \csc ^{-1}(x)^2-4 \sqrt {1-\frac {1}{x^2}} x \csc ^{-1}(x)^3+\left (-1+x^2\right ) \csc ^{-1}(x)^4}{x \sqrt {-1+x^2}} \end {gather*}

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] Result contains complex when optimal does not.
time = 0.53, size = 330, normalized size = 4.46

method	result	size
default	\(\frac {\left (i x^{2}-2 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -2 i\right ) \mathrm {arccsc}\left (x \right )^{3}}{\sqrt {x^{2}-1}\, x}+\frac {\left (x^{2}-2+2 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x \right ) \mathrm {arccsc}\left (x \right )^{4}}{4 \sqrt {x^{2}-1}\, x}-\frac {6 \left (i x^{2}-2 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -2 i\right ) \mathrm {arccsc}\left (x \right )}{\sqrt {x^{2}-1}\, x}-\frac {3 \left (x^{2}-2+2 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x \right ) \mathrm {arccsc}\left (x \right )^{2}}{\sqrt {x^{2}-1}\, x}+\frac {6 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}-24 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -18 x^{2}+24}{\sqrt {x^{2}-1}\, \left (i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -1\right ) x}+\frac {\left (-i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +x^{2}-1\right ) \left (3 \mathrm {arccsc}\left (x \right )^{4}-36 \mathrm {arccsc}\left (x \right )^{2}-4 i \mathrm {arccsc}\left (x \right )^{3}+72+24 i \mathrm {arccsc}\left (x \right )\right )}{4 \sqrt {x^{2}-1}\, x}+\frac {\left (i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +1\right ) \left (\mathrm {arccsc}\left (x \right )^{4}+4 i \mathrm {arccsc}\left (x \right )^{3}-12 \mathrm {arccsc}\left (x \right )^{2}-24 i \mathrm {arccsc}\left (x \right )+24\right )}{4 \sqrt {x^{2}-1}\, x}\)	\(330\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)^4-6/(x^2-1)^(1/2)/x*(I*x^2-2*((x^2-1)/x^2)^(1/2)*x-2*I)*arccsc(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*(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+1/4/(x^2-1)^(1/2)*(-I*((x^2-1)/x^2)^(1/2)*x+x^2-1)*(3*arccsc(x)^4-36*arc
csc(x)^2-4*I*arccsc(x)^3+72+24*I*arccsc(x))/x+1/4/(x^2-1)^(1/2)*(I*((x^2-1)/x^2)^(1/2)*x+1)*(arccsc(x)^4+4*I*a
rccsc(x)^3-12*arccsc(x)^2-24*I*arccsc(x)+24)/x

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Maxima [A]
time = 3.63, size = 58, normalized size = 0.78 \begin {gather*} \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 {gather*}

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 = 0.59, size = 37, normalized size = 0.50 \begin {gather*} -\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 {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acsc}^{4}{\left (x \right )}}{x^{2} \sqrt {\left (x - 1\right ) \left (x + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acsc(x)**4/(x**2*sqrt((x - 1)*(x + 1))), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {asin}\left (\frac {1}{x}\right )}^4}{x^2\,\sqrt {x^2-1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(1/x)^4/(x^2*(x^2 - 1)^(1/2)),x)

[Out]

int(asin(1/x)^4/(x^2*(x^2 - 1)^(1/2)), x)

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