Integrand size = 12, antiderivative size = 98 \[ \int e^{\coth ^{-1}(x)} (1+x)^{3/2} \, dx=\frac {28 \sqrt {1-\frac {1}{x}} (1+x)^{3/2}}{15 \left (1+\frac {1}{x}\right )^{3/2}}+\frac {86 \sqrt {1-\frac {1}{x}} (1+x)^{3/2}}{15 \left (1+\frac {1}{x}\right )^{3/2} x}+\frac {2 \sqrt {1-\frac {1}{x}} x (1+x)^{3/2}}{5 \left (1+\frac {1}{x}\right )^{3/2}} \] Output:
28/15*(1-1/x)^(1/2)*(1+x)^(3/2)/(1+1/x)^(3/2)+86/15*(1-1/x)^(1/2)*(1+x)^(3 /2)/(1+1/x)^(3/2)/x+2/5*(1-1/x)^(1/2)*x*(1+x)^(3/2)/(1+1/x)^(3/2)
Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.42 \[ \int e^{\coth ^{-1}(x)} (1+x)^{3/2} \, dx=\frac {2 \sqrt {\frac {-1+x}{x}} \sqrt {1+x} \left (43+14 x+3 x^2\right )}{15 \sqrt {1+\frac {1}{x}}} \] Input:
Integrate[E^ArcCoth[x]*(1 + x)^(3/2),x]
Output:
(2*Sqrt[(-1 + x)/x]*Sqrt[1 + x]*(43 + 14*x + 3*x^2))/(15*Sqrt[1 + x^(-1)])
Time = 0.39 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6727, 100, 27, 87, 48}
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.
\(\displaystyle \int (x+1)^{3/2} e^{\coth ^{-1}(x)} \, dx\) |
\(\Big \downarrow \) 6727 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{3/2} (x+1)^{3/2} \int \frac {\left (1+\frac {1}{x}\right )^2}{\sqrt {1-\frac {1}{x}} \left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}}{\left (\frac {1}{x}+1\right )^{3/2}}\) |
\(\Big \downarrow \) 100 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{3/2} (x+1)^{3/2} \left (\frac {2}{5} \int \frac {14+\frac {5}{x}}{2 \sqrt {1-\frac {1}{x}} \left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}-\frac {2 \sqrt {1-\frac {1}{x}}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )}{\left (\frac {1}{x}+1\right )^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{3/2} (x+1)^{3/2} \left (\frac {1}{5} \int \frac {14+\frac {5}{x}}{\sqrt {1-\frac {1}{x}} \left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}-\frac {2 \sqrt {1-\frac {1}{x}}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )}{\left (\frac {1}{x}+1\right )^{3/2}}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{3/2} (x+1)^{3/2} \left (\frac {1}{5} \left (\frac {43}{3} \int \frac {1}{\sqrt {1-\frac {1}{x}} \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}-\frac {28 \sqrt {1-\frac {1}{x}}}{3 \left (\frac {1}{x}\right )^{3/2}}\right )-\frac {2 \sqrt {1-\frac {1}{x}}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )}{\left (\frac {1}{x}+1\right )^{3/2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {\left (\frac {1}{5} \left (-\frac {86 \sqrt {1-\frac {1}{x}}}{3 \sqrt {\frac {1}{x}}}-\frac {28 \sqrt {1-\frac {1}{x}}}{3 \left (\frac {1}{x}\right )^{3/2}}\right )-\frac {2 \sqrt {1-\frac {1}{x}}}{5 \left (\frac {1}{x}\right )^{5/2}}\right ) \left (\frac {1}{x}\right )^{3/2} (x+1)^{3/2}}{\left (\frac {1}{x}+1\right )^{3/2}}\) |
Input:
Int[E^ArcCoth[x]*(1 + x)^(3/2),x]
Output:
-(((((-28*Sqrt[1 - x^(-1)])/(3*(x^(-1))^(3/2)) - (86*Sqrt[1 - x^(-1)])/(3* Sqrt[x^(-1)]))/5 - (2*Sqrt[1 - x^(-1)])/(5*(x^(-1))^(5/2)))*(x^(-1))^(3/2) *(1 + x)^(3/2))/(1 + x^(-1))^(3/2))
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] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_), x_Symbol] :> Si mp[(-(1/x)^p)*((c + d*x)^p/(1 + c/(d*x))^p) Subst[Int[((1 + c*(x/d))^p*(( 1 + x/a)^(n/2)/x^(p + 2)))/(1 - x/a)^(n/2), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && !IntegerQ[p]
Time = 0.14 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.33
method | result | size |
gosper | \(\frac {2 \left (x -1\right ) \left (3 x^{2}+14 x +43\right )}{15 \sqrt {1+x}\, \sqrt {\frac {x -1}{1+x}}}\) | \(32\) |
default | \(\frac {2 \left (x -1\right ) \left (3 x^{2}+14 x +43\right )}{15 \sqrt {1+x}\, \sqrt {\frac {x -1}{1+x}}}\) | \(32\) |
risch | \(\frac {2 \left (x -1\right ) \left (3 x^{2}+14 x +43\right )}{15 \sqrt {1+x}\, \sqrt {\frac {x -1}{1+x}}}\) | \(32\) |
orering | \(\frac {2 \left (x -1\right ) \left (3 x^{2}+14 x +43\right )}{15 \sqrt {1+x}\, \sqrt {\frac {x -1}{1+x}}}\) | \(32\) |
Input:
int(1/((x-1)/(1+x))^(1/2)*(1+x)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/15*(x-1)*(3*x^2+14*x+43)/(1+x)^(1/2)/((x-1)/(1+x))^(1/2)
Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.29 \[ \int e^{\coth ^{-1}(x)} (1+x)^{3/2} \, dx=\frac {2}{15} \, {\left (3 \, x^{2} + 14 \, x + 43\right )} \sqrt {x + 1} \sqrt {\frac {x - 1}{x + 1}} \] Input:
integrate(1/((x-1)/(1+x))^(1/2)*(1+x)^(3/2),x, algorithm="fricas")
Output:
2/15*(3*x^2 + 14*x + 43)*sqrt(x + 1)*sqrt((x - 1)/(x + 1))
Time = 39.53 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.71 \[ \int e^{\coth ^{-1}(x)} (1+x)^{3/2} \, dx=2 \left (\begin {cases} 4 \sqrt {2} \left (\frac {\sqrt {2} \left (x - 1\right )^{\frac {5}{2}}}{40} + \frac {\sqrt {2} \left (x - 1\right )^{\frac {3}{2}}}{6} + \frac {\sqrt {2} \sqrt {x - 1}}{2}\right ) & \text {for}\: \sqrt {x + 1} > - \sqrt {2} \wedge \sqrt {x + 1} < \sqrt {2} \end {cases}\right ) \] Input:
integrate(1/((x-1)/(1+x))**(1/2)*(1+x)**(3/2),x)
Output:
2*Piecewise((4*sqrt(2)*(sqrt(2)*(x - 1)**(5/2)/40 + sqrt(2)*(x - 1)**(3/2) /6 + sqrt(2)*sqrt(x - 1)/2), (sqrt(x + 1) < sqrt(2)) & (sqrt(x + 1) > -sqr t(2))))
Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.22 \[ \int e^{\coth ^{-1}(x)} (1+x)^{3/2} \, dx=\frac {2 \, {\left (3 \, x^{3} + 11 \, x^{2} + 29 \, x - 43\right )}}{15 \, \sqrt {x - 1}} \] Input:
integrate(1/((x-1)/(1+x))^(1/2)*(1+x)^(3/2),x, algorithm="maxima")
Output:
2/15*(3*x^3 + 11*x^2 + 29*x - 43)/sqrt(x - 1)
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.41 \[ \int e^{\coth ^{-1}(x)} (1+x)^{3/2} \, dx=-\frac {64}{15} i \, \sqrt {2} \mathrm {sgn}\left (x + 1\right ) + \frac {2 \, {\left (3 \, {\left (x - 1\right )}^{\frac {5}{2}} + 20 \, {\left (x - 1\right )}^{\frac {3}{2}} + 60 \, \sqrt {x - 1}\right )}}{15 \, \mathrm {sgn}\left (x + 1\right )} \] Input:
integrate(1/((x-1)/(1+x))^(1/2)*(1+x)^(3/2),x, algorithm="giac")
Output:
-64/15*I*sqrt(2)*sgn(x + 1) + 2/15*(3*(x - 1)^(5/2) + 20*(x - 1)^(3/2) + 6 0*sqrt(x - 1))/sgn(x + 1)
Time = 13.62 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.39 \[ \int e^{\coth ^{-1}(x)} (1+x)^{3/2} \, dx=\sqrt {\frac {x-1}{x+1}}\,\left (\frac {28\,x\,\sqrt {x+1}}{15}+\frac {86\,\sqrt {x+1}}{15}+\frac {2\,x^2\,\sqrt {x+1}}{5}\right ) \] Input:
int((x + 1)^(3/2)/((x - 1)/(x + 1))^(1/2),x)
Output:
((x - 1)/(x + 1))^(1/2)*((28*x*(x + 1)^(1/2))/15 + (86*(x + 1)^(1/2))/15 + (2*x^2*(x + 1)^(1/2))/5)
Time = 0.17 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.16 \[ \int e^{\coth ^{-1}(x)} (1+x)^{3/2} \, dx=\frac {2 \sqrt {x -1}\, \left (3 x^{2}+14 x +43\right )}{15} \] Input:
int(1/((x-1)/(1+x))^(1/2)*(1+x)^(3/2),x)
Output:
(2*sqrt(x - 1)*(3*x**2 + 14*x + 43))/15