Integrand size = 9, antiderivative size = 99 \[ \int e^{\coth ^{-1}(x)} x (1+x) \, dx=\sqrt {1+\frac {1}{x}} \sqrt {\frac {-1+x}{x}} x+\frac {1}{3} \left (1+\frac {1}{x}\right )^{3/2} \sqrt {\frac {-1+x}{x}} x^2+\frac {1}{3} \left (1+\frac {1}{x}\right )^{5/2} \sqrt {\frac {-1+x}{x}} x^3+\text {arctanh}\left (\sqrt {1+\frac {1}{x}} \sqrt {\frac {-1+x}{x}}\right ) \]
arctanh((1+1/x)^(1/2)*((-1+x)/x)^(1/2))+1/3*(1+1/x)^(3/2)*x^2*((-1+x)/x)^( 1/2)+1/3*(1+1/x)^(5/2)*x^3*((-1+x)/x)^(1/2)+x*(1+1/x)^(1/2)*((-1+x)/x)^(1/ 2)
Time = 0.04 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.41 \[ \int e^{\coth ^{-1}(x)} x (1+x) \, dx=\frac {1}{3} \sqrt {1-\frac {1}{x^2}} x \left (5+3 x+x^2\right )+\log \left (\left (1+\sqrt {1-\frac {1}{x^2}}\right ) x\right ) \]
Time = 0.25 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6729, 107, 105, 105, 103, 219}
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.
\(\displaystyle \int x (x+1) e^{\coth ^{-1}(x)} \, dx\) |
\(\Big \downarrow \) 6729 |
\(\displaystyle -\int \frac {\left (1+\frac {1}{x}\right )^{3/2} x^4}{\sqrt {1-\frac {1}{x}}}d\frac {1}{x}\) |
\(\Big \downarrow \) 107 |
\(\displaystyle \frac {1}{3} \sqrt {1-\frac {1}{x}} \left (\frac {1}{x}+1\right )^{5/2} x^3-\frac {2}{3} \int \frac {\left (1+\frac {1}{x}\right )^{3/2} x^3}{\sqrt {1-\frac {1}{x}}}d\frac {1}{x}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {1}{3} \sqrt {1-\frac {1}{x}} \left (\frac {1}{x}+1\right )^{5/2} x^3-\frac {2}{3} \left (\frac {3}{2} \int \frac {\sqrt {1+\frac {1}{x}} x^2}{\sqrt {1-\frac {1}{x}}}d\frac {1}{x}-\frac {1}{2} \sqrt {1-\frac {1}{x}} \left (\frac {1}{x}+1\right )^{3/2} x^2\right )\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {1}{3} \sqrt {1-\frac {1}{x}} \left (\frac {1}{x}+1\right )^{5/2} x^3-\frac {2}{3} \left (\frac {3}{2} \left (\int \frac {x}{\sqrt {1-\frac {1}{x}} \sqrt {1+\frac {1}{x}}}d\frac {1}{x}-\sqrt {1-\frac {1}{x}} \sqrt {\frac {1}{x}+1} x\right )-\frac {1}{2} \sqrt {1-\frac {1}{x}} \left (\frac {1}{x}+1\right )^{3/2} x^2\right )\) |
\(\Big \downarrow \) 103 |
\(\displaystyle \frac {1}{3} \sqrt {1-\frac {1}{x}} \left (\frac {1}{x}+1\right )^{5/2} x^3-\frac {2}{3} \left (\frac {3}{2} \left (-\int \frac {1}{1-\frac {1}{x^2}}d\left (\sqrt {1-\frac {1}{x}} \sqrt {1+\frac {1}{x}}\right )-\sqrt {1-\frac {1}{x}} \sqrt {\frac {1}{x}+1} x\right )-\frac {1}{2} \sqrt {1-\frac {1}{x}} \left (\frac {1}{x}+1\right )^{3/2} x^2\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{3} \sqrt {1-\frac {1}{x}} \left (\frac {1}{x}+1\right )^{5/2} x^3-\frac {2}{3} \left (\frac {3}{2} \left (-\text {arctanh}\left (\sqrt {1-\frac {1}{x}} \sqrt {\frac {1}{x}+1}\right )-\sqrt {1-\frac {1}{x}} \sqrt {\frac {1}{x}+1} x\right )-\frac {1}{2} \sqrt {1-\frac {1}{x}} \left (\frac {1}{x}+1\right )^{3/2} x^2\right )\) |
(Sqrt[1 - x^(-1)]*(1 + x^(-1))^(5/2)*x^3)/3 - (2*(-1/2*(Sqrt[1 - x^(-1)]*( 1 + x^(-1))^(3/2)*x^2) + (3*(-(Sqrt[1 - x^(-1)]*Sqrt[1 + x^(-1)]*x) - ArcT anh[Sqrt[1 - x^(-1)]*Sqrt[1 + x^(-1)]]))/2))/3
3.3.79.3.1 Defintions of rubi rules used
Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ ))), x_] :> Simp[b*f Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d *e - f*(b*c + a*d), 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(p _.), x_Symbol] :> Simp[(-d^p)*(e*x)^m*(1/x)^m 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}, x] && EqQ[a^2*c^2 - d^2, 0] && IntegerQ[p]
Time = 0.13 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.63
method | result | size |
trager | \(\frac {\left (1+x \right ) \left (x^{2}+3 x +5\right ) \sqrt {-\frac {1-x}{1+x}}}{3}+\ln \left (\sqrt {-\frac {1-x}{1+x}}\, x +\sqrt {-\frac {1-x}{1+x}}+x \right )\) | \(62\) |
risch | \(\frac {\left (x^{2}+3 x +5\right ) \left (x -1\right )}{3 \sqrt {\frac {x -1}{1+x}}}+\frac {\ln \left (x +\sqrt {x^{2}-1}\right ) \sqrt {\left (x -1\right ) \left (1+x \right )}}{\sqrt {\frac {x -1}{1+x}}\, \left (1+x \right )}\) | \(62\) |
default | \(\frac {\left (x -1\right ) \left (\left (\left (x -1\right ) \left (1+x \right )\right )^{\frac {3}{2}}+3 x \sqrt {x^{2}-1}+3 \ln \left (x +\sqrt {x^{2}-1}\right )+6 \sqrt {x^{2}-1}\right )}{3 \sqrt {\frac {x -1}{1+x}}\, \sqrt {\left (x -1\right ) \left (1+x \right )}}\) | \(67\) |
1/3*(1+x)*(x^2+3*x+5)*(-(1-x)/(1+x))^(1/2)+ln((-(1-x)/(1+x))^(1/2)*x+(-(1- x)/(1+x))^(1/2)+x)
Time = 0.25 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.58 \[ \int e^{\coth ^{-1}(x)} x (1+x) \, dx=\frac {1}{3} \, {\left (x^{3} + 4 \, x^{2} + 8 \, x + 5\right )} \sqrt {\frac {x - 1}{x + 1}} + \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) \]
1/3*(x^3 + 4*x^2 + 8*x + 5)*sqrt((x - 1)/(x + 1)) + log(sqrt((x - 1)/(x + 1)) + 1) - log(sqrt((x - 1)/(x + 1)) - 1)
\[ \int e^{\coth ^{-1}(x)} x (1+x) \, dx=\int \frac {x \left (x + 1\right )}{\sqrt {\frac {x - 1}{x + 1}}}\, dx \]
Time = 0.21 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.11 \[ \int e^{\coth ^{-1}(x)} x (1+x) \, dx=-\frac {2 \, {\left (3 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{2}} - 8 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {3}{2}} + 9 \, \sqrt {\frac {x - 1}{x + 1}}\right )}}{3 \, {\left (\frac {3 \, {\left (x - 1\right )}}{x + 1} - \frac {3 \, {\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac {{\left (x - 1\right )}^{3}}{{\left (x + 1\right )}^{3}} - 1\right )}} + \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) \]
-2/3*(3*((x - 1)/(x + 1))^(5/2) - 8*((x - 1)/(x + 1))^(3/2) + 9*sqrt((x - 1)/(x + 1)))/(3*(x - 1)/(x + 1) - 3*(x - 1)^2/(x + 1)^2 + (x - 1)^3/(x + 1 )^3 - 1) + log(sqrt((x - 1)/(x + 1)) + 1) - log(sqrt((x - 1)/(x + 1)) - 1)
Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.60 \[ \int e^{\coth ^{-1}(x)} x (1+x) \, dx=\frac {1}{3} \, \sqrt {x^{2} - 1} {\left (x {\left (\frac {x}{\mathrm {sgn}\left (x + 1\right )} + \frac {3}{\mathrm {sgn}\left (x + 1\right )}\right )} + \frac {5}{\mathrm {sgn}\left (x + 1\right )}\right )} - \frac {\log \left ({\left | -x + \sqrt {x^{2} - 1} \right |}\right )}{\mathrm {sgn}\left (x + 1\right )} \]
1/3*sqrt(x^2 - 1)*(x*(x/sgn(x + 1) + 3/sgn(x + 1)) + 5/sgn(x + 1)) - log(a bs(-x + sqrt(x^2 - 1)))/sgn(x + 1)
Time = 0.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.95 \[ \int e^{\coth ^{-1}(x)} x (1+x) \, dx=2\,\mathrm {atanh}\left (\sqrt {\frac {x-1}{x+1}}\right )-\frac {6\,\sqrt {\frac {x-1}{x+1}}-\frac {16\,{\left (\frac {x-1}{x+1}\right )}^{3/2}}{3}+2\,{\left (\frac {x-1}{x+1}\right )}^{5/2}}{\frac {3\,\left (x-1\right )}{x+1}-\frac {3\,{\left (x-1\right )}^2}{{\left (x+1\right )}^2}+\frac {{\left (x-1\right )}^3}{{\left (x+1\right )}^3}-1} \]