\(\int \frac {\csc ^{-1}(\sqrt {x})}{x^2} \, dx\) [7]

Optimal result

Integrand size = 10, antiderivative size = 38 \[ \int \frac {\csc ^{-1}\left (\sqrt {x}\right )}{x^2} \, dx=-\frac {\sqrt {-1+x}}{2 x}-\frac {\csc ^{-1}\left (\sqrt {x}\right )}{x}-\frac {1}{2} \arctan \left (\sqrt {-1+x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5379, 12, 44, 65, 209} \[ \int \frac {\csc ^{-1}\left (\sqrt {x}\right )}{x^2} \, dx=-\frac {1}{2} \arctan \left (\sqrt {x-1}\right )-\frac {\sqrt {x-1}}{2 x}-\frac {\csc ^{-1}\left (\sqrt {x}\right )}{x} \]

[In]

Int[ArcCsc[Sqrt[x]]/x^2,x]

[Out]

-1/2*Sqrt[-1 + x]/x - ArcCsc[Sqrt[x]]/x - ArcTan[Sqrt[-1 + x]]/2

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 )}{x}-\int \frac {1}{2 \sqrt {-1+x} x^2} \, dx \\ & = -\frac {\csc ^{-1}\left (\sqrt {x}\right )}{x}-\frac {1}{2} \int \frac {1}{\sqrt {-1+x} x^2} \, dx \\ & = -\frac {\sqrt {-1+x}}{2 x}-\frac {\csc ^{-1}\left (\sqrt {x}\right )}{x}-\frac {1}{4} \int \frac {1}{\sqrt {-1+x} x} \, dx \\ & = -\frac {\sqrt {-1+x}}{2 x}-\frac {\csc ^{-1}\left (\sqrt {x}\right )}{x}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x}\right ) \\ & = -\frac {\sqrt {-1+x}}{2 x}-\frac {\csc ^{-1}\left (\sqrt {x}\right )}{x}-\frac {1}{2} \arctan \left (\sqrt {-1+x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.84 \[ \int \frac {\csc ^{-1}\left (\sqrt {x}\right )}{x^2} \, dx=-\frac {\sqrt {-1+x}+2 \csc ^{-1}\left (\sqrt {x}\right )-x \arcsin \left (\frac {1}{\sqrt {x}}\right )}{2 x} \]

[In]

Integrate[ArcCsc[Sqrt[x]]/x^2,x]

[Out]

-1/2*(Sqrt[-1 + x] + 2*ArcCsc[Sqrt[x]] - x*ArcSin[1/Sqrt[x]])/x

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.16

method	result	size
parts	\(-\frac {\operatorname {arccsc}\left (\sqrt {x}\right )}{x}-\frac {\sqrt {\frac {x -1}{x}}\, \left (\arctan \left (\sqrt {x -1}\right ) x +\sqrt {x -1}\right )}{2 \sqrt {x}\, \sqrt {x -1}}\)	\(44\)
derivativedivides	\(-\frac {\operatorname {arccsc}\left (\sqrt {x}\right )}{x}+\frac {\sqrt {x -1}\, \left (\arctan \left (\frac {1}{\sqrt {x -1}}\right ) x -\sqrt {x -1}\right )}{2 \sqrt {\frac {x -1}{x}}\, x^{\frac {3}{2}}}\)	\(46\)
default	\(-\frac {\operatorname {arccsc}\left (\sqrt {x}\right )}{x}+\frac {\sqrt {x -1}\, \left (\arctan \left (\frac {1}{\sqrt {x -1}}\right ) x -\sqrt {x -1}\right )}{2 \sqrt {\frac {x -1}{x}}\, x^{\frac {3}{2}}}\)	\(46\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.55 \[ \int \frac {\csc ^{-1}\left (\sqrt {x}\right )}{x^2} \, dx=\frac {{\left (x - 2\right )} \operatorname {arccsc}\left (\sqrt {x}\right ) - \sqrt {x - 1}}{2 \, x} \]

[In]

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

[Out]

1/2*((x - 2)*arccsc(sqrt(x)) - sqrt(x - 1))/x

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 12.86 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.00 \[ \int \frac {\csc ^{-1}\left (\sqrt {x}\right )}{x^2} \, dx=- \frac {\begin {cases} i \operatorname {acosh}{\left (\frac {1}{\sqrt {x}} \right )} - \frac {i}{\sqrt {x} \sqrt {-1 + \frac {1}{x}}} + \frac {i}{x^{\frac {3}{2}} \sqrt {-1 + \frac {1}{x}}} & \text {for}\: \frac {1}{\left |{x}\right |} > 1 \\- \operatorname {asin}{\left (\frac {1}{\sqrt {x}} \right )} + \frac {\sqrt {1 - \frac {1}{x}}}{\sqrt {x}} & \text {otherwise} \end {cases}}{2} - \frac {\operatorname {acsc}{\left (\sqrt {x} \right )}}{x} \]

[In]

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

[Out]

-Piecewise((I*acosh(1/sqrt(x)) - I/(sqrt(x)*sqrt(-1 + 1/x)) + I/(x**(3/2)*sqrt(-1 + 1/x)), 1/Abs(x) > 1), (-as
in(1/sqrt(x)) + sqrt(1 - 1/x)/sqrt(x), True))/2 - acsc(sqrt(x))/x

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.34 \[ \int \frac {\csc ^{-1}\left (\sqrt {x}\right )}{x^2} \, dx=\frac {\sqrt {x} \sqrt {-\frac {1}{x} + 1}}{2 \, {\left (x {\left (\frac {1}{x} - 1\right )} - 1\right )}} - \frac {\operatorname {arccsc}\left (\sqrt {x}\right )}{x} - \frac {1}{2} \, \arctan \left (\sqrt {x} \sqrt {-\frac {1}{x} + 1}\right ) \]

[In]

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

[Out]

1/2*sqrt(x)*sqrt(-1/x + 1)/(x*(1/x - 1) - 1) - arccsc(sqrt(x))/x - 1/2*arctan(sqrt(x)*sqrt(-1/x + 1))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.84 \[ \int \frac {\csc ^{-1}\left (\sqrt {x}\right )}{x^2} \, dx=-{\left (\frac {1}{x} - 1\right )} \arcsin \left (\frac {1}{\sqrt {x}}\right ) - \frac {\sqrt {-\frac {1}{x} + 1}}{2 \, \sqrt {x}} - \frac {1}{2} \, \arcsin \left (\frac {1}{\sqrt {x}}\right ) \]

[In]

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

[Out]

-(1/x - 1)*arcsin(1/sqrt(x)) - 1/2*sqrt(-1/x + 1)/sqrt(x) - 1/2*arcsin(1/sqrt(x))

Mupad [B] (verification not implemented)

Time = 0.85 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.74 \[ \int \frac {\csc ^{-1}\left (\sqrt {x}\right )}{x^2} \, dx=-\frac {\sqrt {1-\frac {1}{x}}}{2\,\sqrt {x}}-\frac {\mathrm {asin}\left (\frac {1}{\sqrt {x}}\right )\,\left (\frac {2}{x}-1\right )}{2} \]

[In]

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

[Out]

- (1 - 1/x)^(1/2)/(2*x^(1/2)) - (asin(1/x^(1/2))*(2/x - 1))/2