Integrand size = 10, antiderivative size = 54 \[ \int \frac {\csc ^{-1}\left (\sqrt {x}\right )}{x^3} \, dx=-\frac {\sqrt {-1+x}}{8 x^2}-\frac {3 \sqrt {-1+x}}{16 x}-\frac {\csc ^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {3}{16} \arctan \left (\sqrt {-1+x}\right ) \] Output:
-1/8*(-1+x)^(1/2)/x^2-3/16*(-1+x)^(1/2)/x-1/2*arccsc(x^(1/2))/x^2-3/16*arc tan((-1+x)^(1/2))
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.02 \[ \int \frac {\csc ^{-1}\left (\sqrt {x}\right )}{x^3} \, dx=\left (-\frac {1}{8 x^{3/2}}-\frac {3}{16 \sqrt {x}}\right ) \sqrt {\frac {-1+x}{x}}-\frac {\csc ^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {3}{16} \arcsin \left (\frac {1}{\sqrt {x}}\right ) \] Input:
Integrate[ArcCsc[Sqrt[x]]/x^3,x]
Output:
(-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
Time = 0.21 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5794, 27, 52, 52, 73, 216}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\csc ^{-1}\left (\sqrt {x}\right )}{x^3} \, dx\) |
\(\Big \downarrow \) 5794 |
\(\displaystyle -\frac {1}{2} \int \frac {1}{2 \sqrt {x-1} x^3}dx-\frac {\csc ^{-1}\left (\sqrt {x}\right )}{2 x^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{4} \int \frac {1}{\sqrt {x-1} x^3}dx-\frac {\csc ^{-1}\left (\sqrt {x}\right )}{2 x^2}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{4} \left (-\frac {3}{4} \int \frac {1}{\sqrt {x-1} x^2}dx-\frac {\sqrt {x-1}}{2 x^2}\right )-\frac {\csc ^{-1}\left (\sqrt {x}\right )}{2 x^2}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{4} \left (-\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {x-1} x}dx+\frac {\sqrt {x-1}}{x}\right )-\frac {\sqrt {x-1}}{2 x^2}\right )-\frac {\csc ^{-1}\left (\sqrt {x}\right )}{2 x^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{4} \left (-\frac {3}{4} \left (\int \frac {1}{x}d\sqrt {x-1}+\frac {\sqrt {x-1}}{x}\right )-\frac {\sqrt {x-1}}{2 x^2}\right )-\frac {\csc ^{-1}\left (\sqrt {x}\right )}{2 x^2}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{4} \left (-\frac {3}{4} \left (\arctan \left (\sqrt {x-1}\right )+\frac {\sqrt {x-1}}{x}\right )-\frac {\sqrt {x-1}}{2 x^2}\right )-\frac {\csc ^{-1}\left (\sqrt {x}\right )}{2 x^2}\) |
Input:
Int[ArcCsc[Sqrt[x]]/x^3,x]
Output:
-1/2*ArcCsc[Sqrt[x]]/x^2 + (-1/2*Sqrt[-1 + x]/x^2 - (3*(Sqrt[-1 + x]/x + A rcTan[Sqrt[-1 + x]]))/4)/4
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] - Simp[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] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_.) + ArcCsc[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Sim p[(c + d*x)^(m + 1)*((a + b*ArcCsc[u])/(d*(m + 1))), x] + Simp[b*(u/(d*(m + 1)*Sqrt[u^2])) Int[SimplifyIntegrand[(c + d*x)^(m + 1)*(D[u, x]/(u*Sqrt[ u^2 - 1])), x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && Invers eFunctionFreeQ[u, x] && !FunctionOfQ[(c + d*x)^(m + 1), u, x] && !Functio nOfExponentialQ[u, x]
Time = 0.16 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(-\frac {\operatorname {arccsc}\left (\sqrt {x}\right )}{2 x^{2}}+\frac {\sqrt {-1+x}\, \left (3 \arctan \left (\frac {1}{\sqrt {-1+x}}\right ) x^{2}-3 \sqrt {-1+x}\, x -2 \sqrt {-1+x}\right )}{16 \sqrt {\frac {-1+x}{x}}\, x^{\frac {5}{2}}}\) | \(57\) |
default | \(-\frac {\operatorname {arccsc}\left (\sqrt {x}\right )}{2 x^{2}}+\frac {\sqrt {-1+x}\, \left (3 \arctan \left (\frac {1}{\sqrt {-1+x}}\right ) x^{2}-3 \sqrt {-1+x}\, x -2 \sqrt {-1+x}\right )}{16 \sqrt {\frac {-1+x}{x}}\, x^{\frac {5}{2}}}\) | \(57\) |
parts | \(-\frac {\operatorname {arccsc}\left (\sqrt {x}\right )}{2 x^{2}}-\frac {\sqrt {\frac {-1+x}{x}}\, \left (3 \arctan \left (\sqrt {-1+x}\right ) x^{2}+3 \sqrt {-1+x}\, x +2 \sqrt {-1+x}\right )}{16 x^{\frac {3}{2}} \sqrt {-1+x}}\) | \(57\) |
Input:
int(arccsc(x^(1/2))/x^3,x,method=_RETURNVERBOSE)
Output:
-1/2*arccsc(x^(1/2))/x^2+1/16*(-1+x)^(1/2)*(3*arctan(1/(-1+x)^(1/2))*x^2-3 *(-1+x)^(1/2)*x-2*(-1+x)^(1/2))/((-1+x)/x)^(1/2)/x^(5/2)
Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.56 \[ \int \frac {\csc ^{-1}\left (\sqrt {x}\right )}{x^3} \, dx=\frac {{\left (3 \, x^{2} - 8\right )} \operatorname {arccsc}\left (\sqrt {x}\right ) - {\left (3 \, x + 2\right )} \sqrt {x - 1}}{16 \, x^{2}} \] Input:
integrate(arccsc(x^(1/2))/x^3,x, algorithm="fricas")
Output:
1/16*((3*x^2 - 8)*arccsc(sqrt(x)) - (3*x + 2)*sqrt(x - 1))/x^2
Result contains complex when optimal does not.
Time = 41.24 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.70 \[ \int \frac {\csc ^{-1}\left (\sqrt {x}\right )}{x^3} \, dx=- \frac {\begin {cases} \frac {3 i \operatorname {acosh}{\left (\frac {1}{\sqrt {x}} \right )}}{4} - \frac {3 i}{4 \sqrt {x} \sqrt {-1 + \frac {1}{x}}} + \frac {i}{4 x^{\frac {3}{2}} \sqrt {-1 + \frac {1}{x}}} + \frac {i}{2 x^{\frac {5}{2}} \sqrt {-1 + \frac {1}{x}}} & \text {for}\: \frac {1}{\left |{x}\right |} > 1 \\- \frac {3 \operatorname {asin}{\left (\frac {1}{\sqrt {x}} \right )}}{4} + \frac {3}{4 \sqrt {x} \sqrt {1 - \frac {1}{x}}} - \frac {1}{4 x^{\frac {3}{2}} \sqrt {1 - \frac {1}{x}}} - \frac {1}{2 x^{\frac {5}{2}} \sqrt {1 - \frac {1}{x}}} & \text {otherwise} \end {cases}}{4} - \frac {\operatorname {acsc}{\left (\sqrt {x} \right )}}{2 x^{2}} \] Input:
integrate(acsc(x**(1/2))/x**3,x)
Output:
-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)
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (38) = 76\).
Time = 0.14 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.48 \[ \int \frac {\csc ^{-1}\left (\sqrt {x}\right )}{x^3} \, dx=-\frac {3 \, x^{\frac {3}{2}} {\left (-\frac {1}{x} + 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {x} \sqrt {-\frac {1}{x} + 1}}{16 \, {\left (x^{2} {\left (\frac {1}{x} - 1\right )}^{2} - 2 \, x {\left (\frac {1}{x} - 1\right )} + 1\right )}} - \frac {\operatorname {arccsc}\left (\sqrt {x}\right )}{2 \, x^{2}} - \frac {3}{16} \, \arctan \left (\sqrt {x} \sqrt {-\frac {1}{x} + 1}\right ) \] Input:
integrate(arccsc(x^(1/2))/x^3,x, algorithm="maxima")
Output:
-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*arccsc(sqrt(x))/x^2 - 3/16*arctan(sqrt(x)* sqrt(-1/x + 1))
Time = 0.12 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.09 \[ \int \frac {\csc ^{-1}\left (\sqrt {x}\right )}{x^3} \, dx=-\frac {1}{2} \, {\left (\frac {1}{x} - 1\right )}^{2} \arcsin \left (\frac {1}{\sqrt {x}}\right ) - {\left (\frac {1}{x} - 1\right )} \arcsin \left (\frac {1}{\sqrt {x}}\right ) + \frac {{\left (-\frac {1}{x} + 1\right )}^{\frac {3}{2}}}{8 \, \sqrt {x}} - \frac {5 \, \sqrt {-\frac {1}{x} + 1}}{16 \, \sqrt {x}} - \frac {5}{16} \, \arcsin \left (\frac {1}{\sqrt {x}}\right ) \] Input:
integrate(arccsc(x^(1/2))/x^3,x, algorithm="giac")
Output:
-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 ))
Timed out. \[ \int \frac {\csc ^{-1}\left (\sqrt {x}\right )}{x^3} \, dx=\int \frac {\mathrm {asin}\left (\frac {1}{\sqrt {x}}\right )}{x^3} \,d x \] Input:
int(asin(1/x^(1/2))/x^3,x)
Output:
int(asin(1/x^(1/2))/x^3, x)
\[ \int \frac {\csc ^{-1}\left (\sqrt {x}\right )}{x^3} \, dx=\int \frac {\mathit {acsc} \left (\sqrt {x}\right )}{x^{3}}d x \] Input:
int(acsc(x^(1/2))/x^3,x)
Output:
int(acsc(sqrt(x))/x**3,x)