Integrand size = 10, antiderivative size = 63 \[ \int \frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{x^2} \, dx=\frac {\sqrt {-1-x}}{2 \sqrt {-x} \sqrt {x}}-\frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{x}-\frac {\sqrt {x} \arctan \left (\sqrt {-1-x}\right )}{2 \sqrt {-x}} \] Output:
1/2*(-1-x)^(1/2)/(-x)^(1/2)/x^(1/2)-arccsch(x^(1/2))/x-1/2*x^(1/2)*arctan( (-1-x)^(1/2))/(-x)^(1/2)
Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.67 \[ \int \frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{x^2} \, dx=\frac {\sqrt {\frac {1+x}{x}}}{2 \sqrt {x}}-\frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{x}-\frac {1}{2} \text {arcsinh}\left (\frac {1}{\sqrt {x}}\right ) \] Input:
Integrate[ArcCsch[Sqrt[x]]/x^2,x]
Output:
Sqrt[(1 + x)/x]/(2*Sqrt[x]) - ArcCsch[Sqrt[x]]/x - ArcSinh[1/Sqrt[x]]/2
Time = 0.21 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.86, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6900, 27, 52, 73, 217}
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.
\(\displaystyle \int \frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 6900 |
\(\displaystyle \frac {\sqrt {x} \int \frac {1}{2 \sqrt {-x-1} x^2}dx}{\sqrt {-x}}-\frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {x} \int \frac {1}{\sqrt {-x-1} x^2}dx}{2 \sqrt {-x}}-\frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{x}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {\sqrt {x} \left (\frac {\sqrt {-x-1}}{x}-\frac {1}{2} \int \frac {1}{\sqrt {-x-1} x}dx\right )}{2 \sqrt {-x}}-\frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{x}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\sqrt {x} \left (\int \frac {1}{x}d\sqrt {-x-1}+\frac {\sqrt {-x-1}}{x}\right )}{2 \sqrt {-x}}-\frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{x}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\sqrt {x} \left (\frac {\sqrt {-x-1}}{x}-\arctan \left (\sqrt {-x-1}\right )\right )}{2 \sqrt {-x}}-\frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{x}\) |
Input:
Int[ArcCsch[Sqrt[x]]/x^2,x]
Output:
-(ArcCsch[Sqrt[x]]/x) + (Sqrt[x]*(Sqrt[-1 - x]/x - ArcTan[Sqrt[-1 - x]]))/ (2*Sqrt[-x])
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[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_.) + ArcCsch[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Si mp[(c + d*x)^(m + 1)*((a + b*ArcCsch[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*Sq rt[-1 - u^2])), x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && In verseFunctionFreeQ[u, x] && !FunctionOfQ[(c + d*x)^(m + 1), u, x] && !Fun ctionOfExponentialQ[u, x]
Time = 0.16 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.73
method | result | size |
derivativedivides | \(-\frac {\operatorname {arccsch}\left (\sqrt {x}\right )}{x}-\frac {\sqrt {x +1}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {x +1}}\right ) x -\sqrt {x +1}\right )}{2 \sqrt {\frac {x +1}{x}}\, x^{\frac {3}{2}}}\) | \(46\) |
default | \(-\frac {\operatorname {arccsch}\left (\sqrt {x}\right )}{x}-\frac {\sqrt {x +1}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {x +1}}\right ) x -\sqrt {x +1}\right )}{2 \sqrt {\frac {x +1}{x}}\, x^{\frac {3}{2}}}\) | \(46\) |
parts | \(-\frac {\operatorname {arccsch}\left (\sqrt {x}\right )}{x}-\frac {\sqrt {\frac {x +1}{x}}\, \sqrt {x}\, \left (\ln \left (\sqrt {x +1}+1\right ) x -\ln \left (\sqrt {x +1}-1\right ) x -2 \sqrt {x +1}\right )}{4 \sqrt {x +1}\, \left (\sqrt {x +1}+1\right ) \left (\sqrt {x +1}-1\right )}\) | \(77\) |
Input:
int(arccsch(x^(1/2))/x^2,x,method=_RETURNVERBOSE)
Output:
-arccsch(x^(1/2))/x-1/2*(x+1)^(1/2)*(arctanh(1/(x+1)^(1/2))*x-(x+1)^(1/2)) /((x+1)/x)^(1/2)/x^(3/2)
Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.70 \[ \int \frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{x^2} \, dx=-\frac {{\left (x + 2\right )} \log \left (\frac {x \sqrt {\frac {x + 1}{x}} + \sqrt {x}}{x}\right ) - \sqrt {x} \sqrt {\frac {x + 1}{x}}}{2 \, x} \] Input:
integrate(arccsch(x^(1/2))/x^2,x, algorithm="fricas")
Output:
-1/2*((x + 2)*log((x*sqrt((x + 1)/x) + sqrt(x))/x) - sqrt(x)*sqrt((x + 1)/ x))/x
\[ \int \frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{x^2} \, dx=\int \frac {\operatorname {acsch}{\left (\sqrt {x} \right )}}{x^{2}}\, dx \] Input:
integrate(acsch(x**(1/2))/x**2,x)
Output:
Integral(acsch(sqrt(x))/x**2, x)
Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.03 \[ \int \frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{x^2} \, dx=\frac {\sqrt {x} \sqrt {\frac {1}{x} + 1}}{2 \, {\left (x {\left (\frac {1}{x} + 1\right )} - 1\right )}} - \frac {\operatorname {arcsch}\left (\sqrt {x}\right )}{x} - \frac {1}{4} \, \log \left (\sqrt {x} \sqrt {\frac {1}{x} + 1} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {x} \sqrt {\frac {1}{x} + 1} - 1\right ) \] Input:
integrate(arccsch(x^(1/2))/x^2,x, algorithm="maxima")
Output:
1/2*sqrt(x)*sqrt(1/x + 1)/(x*(1/x + 1) - 1) - arccsch(sqrt(x))/x - 1/4*log (sqrt(x)*sqrt(1/x + 1) + 1) + 1/4*log(sqrt(x)*sqrt(1/x + 1) - 1)
\[ \int \frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{x^2} \, dx=\int { \frac {\operatorname {arcsch}\left (\sqrt {x}\right )}{x^{2}} \,d x } \] Input:
integrate(arccsch(x^(1/2))/x^2,x, algorithm="giac")
Output:
integrate(arccsch(sqrt(x))/x^2, x)
Time = 3.63 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.52 \[ \int \frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{x^2} \, dx=\frac {\sqrt {\frac {1}{x}+1}}{2\,\sqrt {x}}-\frac {2\,\mathrm {asinh}\left (\frac {1}{\sqrt {x}}\right )\,\left (\frac {1}{2\,\sqrt {x}}+\frac {\sqrt {x}}{4}\right )}{\sqrt {x}} \] Input:
int(asinh(1/x^(1/2))/x^2,x)
Output:
(1/x + 1)^(1/2)/(2*x^(1/2)) - (2*asinh(1/x^(1/2))*(1/(2*x^(1/2)) + x^(1/2) /4))/x^(1/2)
\[ \int \frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{x^2} \, dx=\int \frac {\mathit {acsch} \left (\sqrt {x}\right )}{x^{2}}d x \] Input:
int(acsch(x^(1/2))/x^2,x)
Output:
int(acsch(sqrt(x))/x**2,x)