Integrand size = 10, antiderivative size = 114 \[ \int x^3 \text {csch}^{-1}\left (\sqrt {x}\right ) \, dx=-\frac {\sqrt {-1-x} \sqrt {x}}{4 \sqrt {-x}}-\frac {(-1-x)^{3/2} \sqrt {x}}{4 \sqrt {-x}}-\frac {3 (-1-x)^{5/2} \sqrt {x}}{20 \sqrt {-x}}-\frac {(-1-x)^{7/2} \sqrt {x}}{28 \sqrt {-x}}+\frac {1}{4} x^4 \text {csch}^{-1}\left (\sqrt {x}\right ) \] Output:
-1/4*(-1-x)^(1/2)*x^(1/2)/(-x)^(1/2)-1/4*(-1-x)^(3/2)*x^(1/2)/(-x)^(1/2)-3 /20*(-1-x)^(5/2)*x^(1/2)/(-x)^(1/2)-1/28*(-1-x)^(7/2)*x^(1/2)/(-x)^(1/2)+1 /4*x^4*arccsch(x^(1/2))
Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.41 \[ \int x^3 \text {csch}^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{140} \sqrt {1+\frac {1}{x}} \sqrt {x} \left (-16+8 x-6 x^2+5 x^3\right )+\frac {1}{4} x^4 \text {csch}^{-1}\left (\sqrt {x}\right ) \] Input:
Integrate[x^3*ArcCsch[Sqrt[x]],x]
Output:
(Sqrt[1 + x^(-1)]*Sqrt[x]*(-16 + 8*x - 6*x^2 + 5*x^3))/140 + (x^4*ArcCsch[ Sqrt[x]])/4
Time = 0.23 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.69, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6900, 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.
\(\displaystyle \int x^3 \text {csch}^{-1}\left (\sqrt {x}\right ) \, dx\) |
\(\Big \downarrow \) 6900 |
\(\displaystyle \frac {1}{4} x^4 \text {csch}^{-1}\left (\sqrt {x}\right )-\frac {\sqrt {x} \int \frac {x^3}{2 \sqrt {-x-1}}dx}{4 \sqrt {-x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} x^4 \text {csch}^{-1}\left (\sqrt {x}\right )-\frac {\sqrt {x} \int \frac {x^3}{\sqrt {-x-1}}dx}{8 \sqrt {-x}}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \frac {1}{4} x^4 \text {csch}^{-1}\left (\sqrt {x}\right )-\frac {\sqrt {x} \int \left (-(-x-1)^{5/2}-3 (-x-1)^{3/2}-3 \sqrt {-x-1}-\frac {1}{\sqrt {-x-1}}\right )dx}{8 \sqrt {-x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} x^4 \text {csch}^{-1}\left (\sqrt {x}\right )-\frac {\left (\frac {2}{7} (-x-1)^{7/2}+\frac {6}{5} (-x-1)^{5/2}+2 (-x-1)^{3/2}+2 \sqrt {-x-1}\right ) \sqrt {x}}{8 \sqrt {-x}}\) |
Input:
Int[x^3*ArcCsch[Sqrt[x]],x]
Output:
-1/8*((2*Sqrt[-1 - x] + 2*(-1 - x)^(3/2) + (6*(-1 - x)^(5/2))/5 + (2*(-1 - x)^(7/2))/7)*Sqrt[x])/Sqrt[-x] + (x^4*ArcCsch[Sqrt[x]])/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] :> 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_.) + 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 = 0.21 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.35
method | result | size |
parts | \(\frac {x^{4} \operatorname {arccsch}\left (\sqrt {x}\right )}{4}+\frac {\sqrt {\frac {x +1}{x}}\, \sqrt {x}\, \left (5 x^{3}-6 x^{2}+8 x -16\right )}{140}\) | \(40\) |
derivativedivides | \(\frac {x^{4} \operatorname {arccsch}\left (\sqrt {x}\right )}{4}+\frac {\left (x +1\right ) \left (5 x^{3}-6 x^{2}+8 x -16\right )}{140 \sqrt {\frac {x +1}{x}}\, \sqrt {x}}\) | \(43\) |
default | \(\frac {x^{4} \operatorname {arccsch}\left (\sqrt {x}\right )}{4}+\frac {\left (x +1\right ) \left (5 x^{3}-6 x^{2}+8 x -16\right )}{140 \sqrt {\frac {x +1}{x}}\, \sqrt {x}}\) | \(43\) |
Input:
int(x^3*arccsch(x^(1/2)),x,method=_RETURNVERBOSE)
Output:
1/4*x^4*arccsch(x^(1/2))+1/140*((x+1)/x)^(1/2)*x^(1/2)*(5*x^3-6*x^2+8*x-16 )
Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.48 \[ \int x^3 \text {csch}^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{4} \, x^{4} \log \left (\frac {x \sqrt {\frac {x + 1}{x}} + \sqrt {x}}{x}\right ) + \frac {1}{140} \, {\left (5 \, x^{3} - 6 \, x^{2} + 8 \, x - 16\right )} \sqrt {x} \sqrt {\frac {x + 1}{x}} \] Input:
integrate(x^3*arccsch(x^(1/2)),x, algorithm="fricas")
Output:
1/4*x^4*log((x*sqrt((x + 1)/x) + sqrt(x))/x) + 1/140*(5*x^3 - 6*x^2 + 8*x - 16)*sqrt(x)*sqrt((x + 1)/x)
\[ \int x^3 \text {csch}^{-1}\left (\sqrt {x}\right ) \, dx=\int x^{3} \operatorname {acsch}{\left (\sqrt {x} \right )}\, dx \] Input:
integrate(x**3*acsch(x**(1/2)),x)
Output:
Integral(x**3*acsch(sqrt(x)), x)
Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.51 \[ \int x^3 \text {csch}^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{28} \, x^{\frac {7}{2}} {\left (\frac {1}{x} + 1\right )}^{\frac {7}{2}} - \frac {3}{20} \, x^{\frac {5}{2}} {\left (\frac {1}{x} + 1\right )}^{\frac {5}{2}} + \frac {1}{4} \, x^{4} \operatorname {arcsch}\left (\sqrt {x}\right ) + \frac {1}{4} \, x^{\frac {3}{2}} {\left (\frac {1}{x} + 1\right )}^{\frac {3}{2}} - \frac {1}{4} \, \sqrt {x} \sqrt {\frac {1}{x} + 1} \] Input:
integrate(x^3*arccsch(x^(1/2)),x, algorithm="maxima")
Output:
1/28*x^(7/2)*(1/x + 1)^(7/2) - 3/20*x^(5/2)*(1/x + 1)^(5/2) + 1/4*x^4*arcc sch(sqrt(x)) + 1/4*x^(3/2)*(1/x + 1)^(3/2) - 1/4*sqrt(x)*sqrt(1/x + 1)
\[ \int x^3 \text {csch}^{-1}\left (\sqrt {x}\right ) \, dx=\int { x^{3} \operatorname {arcsch}\left (\sqrt {x}\right ) \,d x } \] Input:
integrate(x^3*arccsch(x^(1/2)),x, algorithm="giac")
Output:
integrate(x^3*arccsch(sqrt(x)), x)
Timed out. \[ \int x^3 \text {csch}^{-1}\left (\sqrt {x}\right ) \, dx=\int x^3\,\mathrm {asinh}\left (\frac {1}{\sqrt {x}}\right ) \,d x \] Input:
int(x^3*asinh(1/x^(1/2)),x)
Output:
int(x^3*asinh(1/x^(1/2)), x)
\[ \int x^3 \text {csch}^{-1}\left (\sqrt {x}\right ) \, dx=\int \mathit {acsch} \left (\sqrt {x}\right ) x^{3}d x \] Input:
int(x^3*acsch(x^(1/2)),x)
Output:
int(acsch(sqrt(x))*x**3,x)