Integrand size = 10, antiderivative size = 107 \[ \int x^2 \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=-\frac {1}{3} \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}+\frac {2}{9} \left (-1+\frac {1}{\sqrt {x}}\right )^{3/2} \left (1+\frac {1}{\sqrt {x}}\right )^{3/2} x^{3/2}-\frac {1}{15} \left (-1+\frac {1}{\sqrt {x}}\right )^{5/2} \left (1+\frac {1}{\sqrt {x}}\right )^{5/2} x^{5/2}+\frac {1}{3} x^3 \text {sech}^{-1}\left (\sqrt {x}\right ) \] Output:
-1/3*(-1+1/x^(1/2))^(1/2)*(1/x^(1/2)+1)^(1/2)*x^(1/2)+2/9*(-1+1/x^(1/2))^( 3/2)*(1/x^(1/2)+1)^(3/2)*x^(3/2)-1/15*(-1+1/x^(1/2))^(5/2)*(1/x^(1/2)+1)^( 5/2)*x^(5/2)+1/3*x^3*arcsech(x^(1/2))
Time = 0.02 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.67 \[ \int x^2 \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=-\frac {1}{45} \sqrt {\frac {1-\sqrt {x}}{1+\sqrt {x}}} \left (8+8 \sqrt {x}+4 x+4 x^{3/2}+3 x^2+3 x^{5/2}\right )+\frac {1}{3} x^3 \text {sech}^{-1}\left (\sqrt {x}\right ) \] Input:
Integrate[x^2*ArcSech[Sqrt[x]],x]
Output:
-1/45*(Sqrt[(1 - Sqrt[x])/(1 + Sqrt[x])]*(8 + 8*Sqrt[x] + 4*x + 4*x^(3/2) + 3*x^2 + 3*x^(5/2))) + (x^3*ArcSech[Sqrt[x]])/3
Time = 0.24 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.86, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6899, 27, 53, 2009}
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.
\(\displaystyle \int x^2 \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx\) |
\(\Big \downarrow \) 6899 |
\(\displaystyle \frac {\sqrt {1-x} \int \frac {x^2}{2 \sqrt {1-x}}dx}{3 \sqrt {\frac {1}{\sqrt {x}}-1} \sqrt {\frac {1}{\sqrt {x}}+1} \sqrt {x}}+\frac {1}{3} x^3 \text {sech}^{-1}\left (\sqrt {x}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {1-x} \int \frac {x^2}{\sqrt {1-x}}dx}{6 \sqrt {\frac {1}{\sqrt {x}}-1} \sqrt {\frac {1}{\sqrt {x}}+1} \sqrt {x}}+\frac {1}{3} x^3 \text {sech}^{-1}\left (\sqrt {x}\right )\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \frac {\sqrt {1-x} \int \left ((1-x)^{3/2}-2 \sqrt {1-x}+\frac {1}{\sqrt {1-x}}\right )dx}{6 \sqrt {\frac {1}{\sqrt {x}}-1} \sqrt {\frac {1}{\sqrt {x}}+1} \sqrt {x}}+\frac {1}{3} x^3 \text {sech}^{-1}\left (\sqrt {x}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} x^3 \text {sech}^{-1}\left (\sqrt {x}\right )+\frac {\left (-\frac {2}{5} (1-x)^{5/2}+\frac {4}{3} (1-x)^{3/2}-2 \sqrt {1-x}\right ) \sqrt {1-x}}{6 \sqrt {\frac {1}{\sqrt {x}}-1} \sqrt {\frac {1}{\sqrt {x}}+1} \sqrt {x}}\) |
Input:
Int[x^2*ArcSech[Sqrt[x]],x]
Output:
((-2*Sqrt[1 - x] + (4*(1 - x)^(3/2))/3 - (2*(1 - x)^(5/2))/5)*Sqrt[1 - x]) /(6*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*Sqrt[x]) + (x^3*ArcSech[Sqrt[ x]])/3
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] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[((a_.) + ArcSech[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Si mp[(c + d*x)^(m + 1)*((a + b*ArcSech[u])/(d*(m + 1))), x] + Simp[b*(Sqrt[1 - u^2]/(d*(m + 1)*u*Sqrt[-1 + 1/u]*Sqrt[1 + 1/u])) Int[SimplifyIntegrand[ (c + d*x)^(m + 1)*(D[u, x]/(u*Sqrt[1 - u^2])), x], x], x] /; FreeQ[{a, b, c , d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && !FunctionOfQ[(c + d*x)^(m + 1), u, x] && !FunctionOfExponentialQ[u, x]
Time = 0.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.46
method | result | size |
derivativedivides | \(\frac {x^{3} \operatorname {arcsech}\left (\sqrt {x}\right )}{3}-\frac {\sqrt {-\frac {\sqrt {x}-1}{\sqrt {x}}}\, \sqrt {x}\, \sqrt {\frac {\sqrt {x}+1}{\sqrt {x}}}\, \left (3 x^{2}+4 x +8\right )}{45}\) | \(49\) |
default | \(\frac {x^{3} \operatorname {arcsech}\left (\sqrt {x}\right )}{3}-\frac {\sqrt {-\frac {\sqrt {x}-1}{\sqrt {x}}}\, \sqrt {x}\, \sqrt {\frac {\sqrt {x}+1}{\sqrt {x}}}\, \left (3 x^{2}+4 x +8\right )}{45}\) | \(49\) |
parts | \(\frac {x^{3} \operatorname {arcsech}\left (\sqrt {x}\right )}{3}-\frac {\sqrt {-\frac {\sqrt {x}-1}{\sqrt {x}}}\, \sqrt {x}\, \sqrt {\frac {\sqrt {x}+1}{\sqrt {x}}}\, \left (3 x^{2}+4 x +8\right )}{45}\) | \(49\) |
Input:
int(x^2*arcsech(x^(1/2)),x,method=_RETURNVERBOSE)
Output:
1/3*x^3*arcsech(x^(1/2))-1/45*(-(x^(1/2)-1)/x^(1/2))^(1/2)*x^(1/2)*((x^(1/ 2)+1)/x^(1/2))^(1/2)*(3*x^2+4*x+8)
Time = 0.10 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.49 \[ \int x^2 \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{3} \, x^{3} \log \left (\frac {x \sqrt {-\frac {x - 1}{x}} + \sqrt {x}}{x}\right ) - \frac {1}{45} \, {\left (3 \, x^{2} + 4 \, x + 8\right )} \sqrt {x} \sqrt {-\frac {x - 1}{x}} \] Input:
integrate(x^2*arcsech(x^(1/2)),x, algorithm="fricas")
Output:
1/3*x^3*log((x*sqrt(-(x - 1)/x) + sqrt(x))/x) - 1/45*(3*x^2 + 4*x + 8)*sqr t(x)*sqrt(-(x - 1)/x)
\[ \int x^2 \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=\int x^{2} \operatorname {asech}{\left (\sqrt {x} \right )}\, dx \] Input:
integrate(x**2*asech(x**(1/2)),x)
Output:
Integral(x**2*asech(sqrt(x)), x)
Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.43 \[ \int x^2 \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=-\frac {1}{15} \, x^{\frac {5}{2}} {\left (\frac {1}{x} - 1\right )}^{\frac {5}{2}} + \frac {1}{3} \, x^{3} \operatorname {arsech}\left (\sqrt {x}\right ) + \frac {2}{9} \, x^{\frac {3}{2}} {\left (\frac {1}{x} - 1\right )}^{\frac {3}{2}} - \frac {1}{3} \, \sqrt {x} \sqrt {\frac {1}{x} - 1} \] Input:
integrate(x^2*arcsech(x^(1/2)),x, algorithm="maxima")
Output:
-1/15*x^(5/2)*(1/x - 1)^(5/2) + 1/3*x^3*arcsech(sqrt(x)) + 2/9*x^(3/2)*(1/ x - 1)^(3/2) - 1/3*sqrt(x)*sqrt(1/x - 1)
\[ \int x^2 \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=\int { x^{2} \operatorname {arsech}\left (\sqrt {x}\right ) \,d x } \] Input:
integrate(x^2*arcsech(x^(1/2)),x, algorithm="giac")
Output:
integrate(x^2*arcsech(sqrt(x)), x)
Timed out. \[ \int x^2 \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=\int x^2\,\mathrm {acosh}\left (\frac {1}{\sqrt {x}}\right ) \,d x \] Input:
int(x^2*acosh(1/x^(1/2)),x)
Output:
int(x^2*acosh(1/x^(1/2)), x)
\[ \int x^2 \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=\int \mathit {asech} \left (\sqrt {x}\right ) x^{2}d x \] Input:
int(x^2*asech(x^(1/2)),x)
Output:
int(asech(sqrt(x))*x**2,x)