Integrand size = 13, antiderivative size = 132 \[ \int e^{\coth ^{-1}(x)} x (1+x)^{3/2} \, dx=\frac {46 \sqrt {1-\frac {1}{x}} (1+x)^{3/2}}{21 \left (1+\frac {1}{x}\right )^{3/2}}+\frac {92 \sqrt {1-\frac {1}{x}} (1+x)^{3/2}}{21 \left (1+\frac {1}{x}\right )^{3/2} x}+\frac {8 \sqrt {1-\frac {1}{x}} x (1+x)^{3/2}}{7 \left (1+\frac {1}{x}\right )^{3/2}}+\frac {2 \sqrt {1-\frac {1}{x}} x^2 (1+x)^{3/2}}{7 \left (1+\frac {1}{x}\right )^{3/2}} \] Output:
46/21*(1-1/x)^(1/2)*(1+x)^(3/2)/(1+1/x)^(3/2)+92/21*(1-1/x)^(1/2)*(1+x)^(3 /2)/(1+1/x)^(3/2)/x+8/7*(1-1/x)^(1/2)*x*(1+x)^(3/2)/(1+1/x)^(3/2)+2/7*(1-1 /x)^(1/2)*x^2*(1+x)^(3/2)/(1+1/x)^(3/2)
Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.35 \[ \int e^{\coth ^{-1}(x)} x (1+x)^{3/2} \, dx=\frac {2 \sqrt {\frac {-1+x}{x}} \sqrt {1+x} \left (46+23 x+12 x^2+3 x^3\right )}{21 \sqrt {1+\frac {1}{x}}} \] Input:
Integrate[E^ArcCoth[x]*x*(1 + x)^(3/2),x]
Output:
(2*Sqrt[(-1 + x)/x]*Sqrt[1 + x]*(46 + 23*x + 12*x^2 + 3*x^3))/(21*Sqrt[1 + x^(-1)])
Time = 0.44 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.91, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {6730, 100, 27, 87, 55, 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.
\(\displaystyle \int x (x+1)^{3/2} e^{\coth ^{-1}(x)} \, dx\) |
\(\Big \downarrow \) 6730 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{3/2} (x+1)^{3/2} \int \frac {\left (1+\frac {1}{x}\right )^2}{\sqrt {1-\frac {1}{x}} \left (\frac {1}{x}\right )^{9/2}}d\frac {1}{x}}{\left (\frac {1}{x}+1\right )^{3/2}}\) |
\(\Big \downarrow \) 100 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{3/2} (x+1)^{3/2} \left (\frac {2}{7} \int \frac {20+\frac {7}{x}}{2 \sqrt {1-\frac {1}{x}} \left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}-\frac {2 \sqrt {1-\frac {1}{x}}}{7 \left (\frac {1}{x}\right )^{7/2}}\right )}{\left (\frac {1}{x}+1\right )^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{3/2} (x+1)^{3/2} \left (\frac {1}{7} \int \frac {20+\frac {7}{x}}{\sqrt {1-\frac {1}{x}} \left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}-\frac {2 \sqrt {1-\frac {1}{x}}}{7 \left (\frac {1}{x}\right )^{7/2}}\right )}{\left (\frac {1}{x}+1\right )^{3/2}}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{3/2} (x+1)^{3/2} \left (\frac {1}{7} \left (23 \int \frac {1}{\sqrt {1-\frac {1}{x}} \left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}-\frac {8 \sqrt {1-\frac {1}{x}}}{\left (\frac {1}{x}\right )^{5/2}}\right )-\frac {2 \sqrt {1-\frac {1}{x}}}{7 \left (\frac {1}{x}\right )^{7/2}}\right )}{\left (\frac {1}{x}+1\right )^{3/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{3/2} (x+1)^{3/2} \left (\frac {1}{7} \left (23 \left (\frac {2}{3} \int \frac {1}{\sqrt {1-\frac {1}{x}} \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}-\frac {2 \sqrt {1-\frac {1}{x}}}{3 \left (\frac {1}{x}\right )^{3/2}}\right )-\frac {8 \sqrt {1-\frac {1}{x}}}{\left (\frac {1}{x}\right )^{5/2}}\right )-\frac {2 \sqrt {1-\frac {1}{x}}}{7 \left (\frac {1}{x}\right )^{7/2}}\right )}{\left (\frac {1}{x}+1\right )^{3/2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {\left (\frac {1}{7} \left (23 \left (-\frac {4 \sqrt {1-\frac {1}{x}}}{3 \sqrt {\frac {1}{x}}}-\frac {2 \sqrt {1-\frac {1}{x}}}{3 \left (\frac {1}{x}\right )^{3/2}}\right )-\frac {8 \sqrt {1-\frac {1}{x}}}{\left (\frac {1}{x}\right )^{5/2}}\right )-\frac {2 \sqrt {1-\frac {1}{x}}}{7 \left (\frac {1}{x}\right )^{7/2}}\right ) \left (\frac {1}{x}\right )^{3/2} (x+1)^{3/2}}{\left (\frac {1}{x}+1\right )^{3/2}}\) |
Input:
Int[E^ArcCoth[x]*x*(1 + x)^(3/2),x]
Output:
-((((23*((-2*Sqrt[1 - x^(-1)])/(3*(x^(-1))^(3/2)) - (4*Sqrt[1 - x^(-1)])/( 3*Sqrt[x^(-1)])) - (8*Sqrt[1 - x^(-1)])/(x^(-1))^(5/2))/7 - (2*Sqrt[1 - x^ (-1)])/(7*(x^(-1))^(7/2)))*(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_))^(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*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 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_.))*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(p _), x_Symbol] :> Simp[(-(e*x)^m)*(1/x)^(m + p)*((c + d*x)^p/(1 + c/(d*x))^p ) Subst[Int[((1 + c*(x/d))^p*((1 + x/a)^(n/2)/x^(m + p + 2)))/(1 - x/a)^( n/2), x], x, 1/x], x] /; FreeQ[{a, c, d, e, m, n, p}, x] && EqQ[a^2*c^2 - d ^2, 0] && !IntegerQ[p]
Time = 0.14 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.28
method | result | size |
gosper | \(\frac {2 \left (x -1\right ) \left (3 x^{3}+12 x^{2}+23 x +46\right )}{21 \sqrt {1+x}\, \sqrt {\frac {x -1}{1+x}}}\) | \(37\) |
default | \(\frac {2 \left (x -1\right ) \left (3 x^{3}+12 x^{2}+23 x +46\right )}{21 \sqrt {1+x}\, \sqrt {\frac {x -1}{1+x}}}\) | \(37\) |
risch | \(\frac {2 \left (x -1\right ) \left (3 x^{3}+12 x^{2}+23 x +46\right )}{21 \sqrt {1+x}\, \sqrt {\frac {x -1}{1+x}}}\) | \(37\) |
orering | \(\frac {2 \left (x -1\right ) \left (3 x^{3}+12 x^{2}+23 x +46\right )}{21 \sqrt {1+x}\, \sqrt {\frac {x -1}{1+x}}}\) | \(37\) |
Input:
int(1/((x-1)/(1+x))^(1/2)*x*(1+x)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/21*(x-1)*(3*x^3+12*x^2+23*x+46)/(1+x)^(1/2)/((x-1)/(1+x))^(1/2)
Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.25 \[ \int e^{\coth ^{-1}(x)} x (1+x)^{3/2} \, dx=\frac {2}{21} \, {\left (3 \, x^{3} + 12 \, x^{2} + 23 \, x + 46\right )} \sqrt {x + 1} \sqrt {\frac {x - 1}{x + 1}} \] Input:
integrate(1/((x-1)/(1+x))^(1/2)*x*(1+x)^(3/2),x, algorithm="fricas")
Output:
2/21*(3*x^3 + 12*x^2 + 23*x + 46)*sqrt(x + 1)*sqrt((x - 1)/(x + 1))
\[ \int e^{\coth ^{-1}(x)} x (1+x)^{3/2} \, dx=\int \frac {x \left (x + 1\right )^{\frac {3}{2}}}{\sqrt {\frac {x - 1}{x + 1}}}\, dx \] Input:
integrate(1/((x-1)/(1+x))**(1/2)*x*(1+x)**(3/2),x)
Output:
Integral(x*(x + 1)**(3/2)/sqrt((x - 1)/(x + 1)), x)
Time = 0.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.20 \[ \int e^{\coth ^{-1}(x)} x (1+x)^{3/2} \, dx=\frac {2 \, {\left (3 \, x^{4} + 9 \, x^{3} + 11 \, x^{2} + 23 \, x - 46\right )}}{21 \, \sqrt {x - 1}} \] Input:
integrate(1/((x-1)/(1+x))^(1/2)*x*(1+x)^(3/2),x, algorithm="maxima")
Output:
2/21*(3*x^4 + 9*x^3 + 11*x^2 + 23*x - 46)/sqrt(x - 1)
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.36 \[ \int e^{\coth ^{-1}(x)} x (1+x)^{3/2} \, dx=-\frac {64}{21} i \, \sqrt {2} \mathrm {sgn}\left (x + 1\right ) + \frac {2 \, {\left (3 \, {\left (x - 1\right )}^{\frac {7}{2}} + 21 \, {\left (x - 1\right )}^{\frac {5}{2}} + 56 \, {\left (x - 1\right )}^{\frac {3}{2}} + 84 \, \sqrt {x - 1}\right )}}{21 \, \mathrm {sgn}\left (x + 1\right )} \] Input:
integrate(1/((x-1)/(1+x))^(1/2)*x*(1+x)^(3/2),x, algorithm="giac")
Output:
-64/21*I*sqrt(2)*sgn(x + 1) + 2/21*(3*(x - 1)^(7/2) + 21*(x - 1)^(5/2) + 5 6*(x - 1)^(3/2) + 84*sqrt(x - 1))/sgn(x + 1)
Time = 13.75 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.36 \[ \int e^{\coth ^{-1}(x)} x (1+x)^{3/2} \, dx=\sqrt {\frac {x-1}{x+1}}\,\left (\frac {46\,x\,\sqrt {x+1}}{21}+\frac {92\,\sqrt {x+1}}{21}+\frac {8\,x^2\,\sqrt {x+1}}{7}+\frac {2\,x^3\,\sqrt {x+1}}{7}\right ) \] Input:
int((x*(x + 1)^(3/2))/((x - 1)/(x + 1))^(1/2),x)
Output:
((x - 1)/(x + 1))^(1/2)*((46*x*(x + 1)^(1/2))/21 + (92*(x + 1)^(1/2))/21 + (8*x^2*(x + 1)^(1/2))/7 + (2*x^3*(x + 1)^(1/2))/7)
Time = 0.15 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.16 \[ \int e^{\coth ^{-1}(x)} x (1+x)^{3/2} \, dx=\frac {2 \sqrt {x -1}\, \left (3 x^{3}+12 x^{2}+23 x +46\right )}{21} \] Input:
int(1/((x-1)/(1+x))^(1/2)*x*(1+x)^(3/2),x)
Output:
(2*sqrt(x - 1)*(3*x**3 + 12*x**2 + 23*x + 46))/21