\(\int \frac {\csc ^{-1}(\sqrt {x})}{x^3} \, dx\) [8]

Optimal result

Integrand size = 10, antiderivative size = 54 \[ \int \frac {\csc ^{-1}\left (\sqrt {x}\right )}{x^3} \, dx=-\frac {\sqrt {-1+x}}{8 x^2}-\frac {3 \sqrt {-1+x}}{16 x}-\frac {\csc ^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {3}{16} \arctan \left (\sqrt {-1+x}\right ) \]

[Out]

-1/2*arccsc(x^(1/2))/x^2-3/16*arctan((-1+x)^(1/2))-1/8*(-1+x)^(1/2)/x^2-3/16*(-1+x)^(1/2)/x

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5379, 12, 44, 65, 209} \[ \int \frac {\csc ^{-1}\left (\sqrt {x}\right )}{x^3} \, dx=-\frac {3}{16} \arctan \left (\sqrt {x-1}\right )-\frac {\sqrt {x-1}}{8 x^2}-\frac {\csc ^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {3 \sqrt {x-1}}{16 x} \]

[In]

Int[ArcCsc[Sqrt[x]]/x^3,x]

[Out]

-1/8*Sqrt[-1 + x]/x^2 - (3*Sqrt[-1 + x])/(16*x) - ArcCsc[Sqrt[x]]/(2*x^2) - (3*ArcTan[Sqrt[-1 + x]])/16

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

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 209

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

Rule 5379

Int[((a_.) + ArcCsc[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*((a + b*ArcCsc[
u])/(d*(m + 1))), x] + Dist[b*(u/(d*(m + 1)*Sqrt[u^2])), Int[SimplifyIntegrand[(c + d*x)^(m + 1)*(D[u, x]/(u*S
qrt[u^2 - 1])), x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !Funct
ionOfQ[(c + d*x)^(m + 1), u, x] &&  !FunctionOfExponentialQ[u, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {\csc ^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {1}{2} \int \frac {1}{2 \sqrt {-1+x} x^3} \, dx \\ & = -\frac {\csc ^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {1}{4} \int \frac {1}{\sqrt {-1+x} x^3} \, dx \\ & = -\frac {\sqrt {-1+x}}{8 x^2}-\frac {\csc ^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {3}{16} \int \frac {1}{\sqrt {-1+x} x^2} \, dx \\ & = -\frac {\sqrt {-1+x}}{8 x^2}-\frac {3 \sqrt {-1+x}}{16 x}-\frac {\csc ^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {3}{32} \int \frac {1}{\sqrt {-1+x} x} \, dx \\ & = -\frac {\sqrt {-1+x}}{8 x^2}-\frac {3 \sqrt {-1+x}}{16 x}-\frac {\csc ^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {3}{16} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x}\right ) \\ & = -\frac {\sqrt {-1+x}}{8 x^2}-\frac {3 \sqrt {-1+x}}{16 x}-\frac {\csc ^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {3}{16} \arctan \left (\sqrt {-1+x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.02 \[ \int \frac {\csc ^{-1}\left (\sqrt {x}\right )}{x^3} \, dx=\left (-\frac {1}{8 x^{3/2}}-\frac {3}{16 \sqrt {x}}\right ) \sqrt {\frac {-1+x}{x}}-\frac {\csc ^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {3}{16} \arcsin \left (\frac {1}{\sqrt {x}}\right ) \]

[In]

Integrate[ArcCsc[Sqrt[x]]/x^3,x]

[Out]

(-1/8*1/x^(3/2) - 3/(16*Sqrt[x]))*Sqrt[(-1 + x)/x] - ArcCsc[Sqrt[x]]/(2*x^2) + (3*ArcSin[1/Sqrt[x]])/16

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.06

method	result	size
derivativedivides	\(-\frac {\operatorname {arccsc}\left (\sqrt {x}\right )}{2 x^{2}}+\frac {\sqrt {x -1}\, \left (3 \arctan \left (\frac {1}{\sqrt {x -1}}\right ) x^{2}-3 \sqrt {x -1}\, x -2 \sqrt {x -1}\right )}{16 \sqrt {\frac {x -1}{x}}\, x^{\frac {5}{2}}}\)	\(57\)
default	\(-\frac {\operatorname {arccsc}\left (\sqrt {x}\right )}{2 x^{2}}+\frac {\sqrt {x -1}\, \left (3 \arctan \left (\frac {1}{\sqrt {x -1}}\right ) x^{2}-3 \sqrt {x -1}\, x -2 \sqrt {x -1}\right )}{16 \sqrt {\frac {x -1}{x}}\, x^{\frac {5}{2}}}\)	\(57\)
parts	\(-\frac {\operatorname {arccsc}\left (\sqrt {x}\right )}{2 x^{2}}-\frac {\sqrt {\frac {x -1}{x}}\, \left (3 \arctan \left (\sqrt {x -1}\right ) x^{2}+3 \sqrt {x -1}\, x +2 \sqrt {x -1}\right )}{16 x^{\frac {3}{2}} \sqrt {x -1}}\)	\(57\)

[In]

int(arccsc(x^(1/2))/x^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*arccsc(x^(1/2))/x^2+1/16*(x-1)^(1/2)*(3*arctan(1/(x-1)^(1/2))*x^2-3*(x-1)^(1/2)*x-2*(x-1)^(1/2))/((x-1)/x
)^(1/2)/x^(5/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.56 \[ \int \frac {\csc ^{-1}\left (\sqrt {x}\right )}{x^3} \, dx=\frac {{\left (3 \, x^{2} - 8\right )} \operatorname {arccsc}\left (\sqrt {x}\right ) - {\left (3 \, x + 2\right )} \sqrt {x - 1}}{16 \, x^{2}} \]

[In]

integrate(arccsc(x^(1/2))/x^3,x, algorithm="fricas")

[Out]

1/16*((3*x^2 - 8)*arccsc(sqrt(x)) - (3*x + 2)*sqrt(x - 1))/x^2

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 42.49 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.70 \[ \int \frac {\csc ^{-1}\left (\sqrt {x}\right )}{x^3} \, dx=- \frac {\begin {cases} \frac {3 i \operatorname {acosh}{\left (\frac {1}{\sqrt {x}} \right )}}{4} - \frac {3 i}{4 \sqrt {x} \sqrt {-1 + \frac {1}{x}}} + \frac {i}{4 x^{\frac {3}{2}} \sqrt {-1 + \frac {1}{x}}} + \frac {i}{2 x^{\frac {5}{2}} \sqrt {-1 + \frac {1}{x}}} & \text {for}\: \frac {1}{\left |{x}\right |} > 1 \\- \frac {3 \operatorname {asin}{\left (\frac {1}{\sqrt {x}} \right )}}{4} + \frac {3}{4 \sqrt {x} \sqrt {1 - \frac {1}{x}}} - \frac {1}{4 x^{\frac {3}{2}} \sqrt {1 - \frac {1}{x}}} - \frac {1}{2 x^{\frac {5}{2}} \sqrt {1 - \frac {1}{x}}} & \text {otherwise} \end {cases}}{4} - \frac {\operatorname {acsc}{\left (\sqrt {x} \right )}}{2 x^{2}} \]

[In]

integrate(acsc(x**(1/2))/x**3,x)

[Out]

-Piecewise((3*I*acosh(1/sqrt(x))/4 - 3*I/(4*sqrt(x)*sqrt(-1 + 1/x)) + I/(4*x**(3/2)*sqrt(-1 + 1/x)) + I/(2*x**
(5/2)*sqrt(-1 + 1/x)), 1/Abs(x) > 1), (-3*asin(1/sqrt(x))/4 + 3/(4*sqrt(x)*sqrt(1 - 1/x)) - 1/(4*x**(3/2)*sqrt
(1 - 1/x)) - 1/(2*x**(5/2)*sqrt(1 - 1/x)), True))/4 - acsc(sqrt(x))/(2*x**2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (38) = 76\).

Time = 0.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.48 \[ \int \frac {\csc ^{-1}\left (\sqrt {x}\right )}{x^3} \, dx=-\frac {3 \, x^{\frac {3}{2}} {\left (-\frac {1}{x} + 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {x} \sqrt {-\frac {1}{x} + 1}}{16 \, {\left (x^{2} {\left (\frac {1}{x} - 1\right )}^{2} - 2 \, x {\left (\frac {1}{x} - 1\right )} + 1\right )}} - \frac {\operatorname {arccsc}\left (\sqrt {x}\right )}{2 \, x^{2}} - \frac {3}{16} \, \arctan \left (\sqrt {x} \sqrt {-\frac {1}{x} + 1}\right ) \]

[In]

integrate(arccsc(x^(1/2))/x^3,x, algorithm="maxima")

[Out]

-1/16*(3*x^(3/2)*(-1/x + 1)^(3/2) + 5*sqrt(x)*sqrt(-1/x + 1))/(x^2*(1/x - 1)^2 - 2*x*(1/x - 1) + 1) - 1/2*arcc
sc(sqrt(x))/x^2 - 3/16*arctan(sqrt(x)*sqrt(-1/x + 1))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.09 \[ \int \frac {\csc ^{-1}\left (\sqrt {x}\right )}{x^3} \, dx=-\frac {1}{2} \, {\left (\frac {1}{x} - 1\right )}^{2} \arcsin \left (\frac {1}{\sqrt {x}}\right ) - {\left (\frac {1}{x} - 1\right )} \arcsin \left (\frac {1}{\sqrt {x}}\right ) + \frac {{\left (-\frac {1}{x} + 1\right )}^{\frac {3}{2}}}{8 \, \sqrt {x}} - \frac {5 \, \sqrt {-\frac {1}{x} + 1}}{16 \, \sqrt {x}} - \frac {5}{16} \, \arcsin \left (\frac {1}{\sqrt {x}}\right ) \]

[In]

integrate(arccsc(x^(1/2))/x^3,x, algorithm="giac")

[Out]

-1/2*(1/x - 1)^2*arcsin(1/sqrt(x)) - (1/x - 1)*arcsin(1/sqrt(x)) + 1/8*(-1/x + 1)^(3/2)/sqrt(x) - 5/16*sqrt(-1
/x + 1)/sqrt(x) - 5/16*arcsin(1/sqrt(x))

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc ^{-1}\left (\sqrt {x}\right )}{x^3} \, dx=\int \frac {\mathrm {asin}\left (\frac {1}{\sqrt {x}}\right )}{x^3} \,d x \]

[In]

int(asin(1/x^(1/2))/x^3,x)

[Out]

int(asin(1/x^(1/2))/x^3, x)