Integrand size = 16, antiderivative size = 144 \[ \int \frac {1}{\left (1+\frac {2 x}{1+x^2}\right )^{3/2}} \, dx=\frac {3 (2+x)}{2 \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {1+x^2}{2 (1+x) \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {3 (1+x) \text {arcsinh}(x)}{\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {9 (1+x) \text {arctanh}\left (\frac {1-x}{\sqrt {2} \sqrt {1+x^2}}\right )}{2 \sqrt {2} \sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}} \]
3/2*(2+x)/(1+2*x/(x^2+1))^(1/2)+1/2*(-x^2-1)/(1+x)/(1+2*x/(x^2+1))^(1/2)-3 *(1+x)*arcsinh(x)/(x^2+1)^(1/2)/(1+2*x/(x^2+1))^(1/2)-9/4*(1+x)*arctanh(1/ 2*(1-x)*2^(1/2)/(x^2+1)^(1/2))*2^(1/2)/(x^2+1)^(1/2)/(1+2*x/(x^2+1))^(1/2)
Time = 0.27 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (1+\frac {2 x}{1+x^2}\right )^{3/2}} \, dx=\frac {(1+x) \left (\sqrt {1+x^2} \left (5+9 x+2 x^2\right )+9 \sqrt {2} (1+x)^2 \text {arctanh}\left (\frac {1+x-\sqrt {1+x^2}}{\sqrt {2}}\right )+6 (1+x)^2 \log \left (-x+\sqrt {1+x^2}\right )\right )}{2 \left (\frac {(1+x)^2}{1+x^2}\right )^{3/2} \left (1+x^2\right )^{3/2}} \]
((1 + x)*(Sqrt[1 + x^2]*(5 + 9*x + 2*x^2) + 9*Sqrt[2]*(1 + x)^2*ArcTanh[(1 + x - Sqrt[1 + x^2])/Sqrt[2]] + 6*(1 + x)^2*Log[-x + Sqrt[1 + x^2]]))/(2* ((1 + x)^2/(1 + x^2))^(3/2)*(1 + x^2)^(3/2))
Time = 0.29 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.72, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {7274, 1298, 27, 492, 590, 25, 719, 222, 488, 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.
| \(\displaystyle \int \frac {1}{\left (\frac {2 x}{x^2+1}+1\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 7274 |
| \(\displaystyle \frac {\sqrt {x^2+2 x+1} \int \frac {\left (x^2+1\right )^{3/2}}{\left (x^2+2 x+1\right )^{3/2}}dx}{\sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1}}\) |
| \(\Big \downarrow \) 1298 |
| \(\displaystyle \frac {8 (x+1) \int \frac {\left (x^2+1\right )^{3/2}}{8 (x+1)^3}dx}{\sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(x+1) \int \frac {\left (x^2+1\right )^{3/2}}{(x+1)^3}dx}{\sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1}}\) |
| \(\Big \downarrow \) 492 |
| \(\displaystyle \frac {(x+1) \left (\frac {3}{2} \int \frac {x \sqrt {x^2+1}}{(x+1)^2}dx-\frac {\left (x^2+1\right )^{3/2}}{2 (x+1)^2}\right )}{\sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1}}\) |
| \(\Big \downarrow \) 590 |
| \(\displaystyle \frac {(x+1) \left (\frac {3}{2} \left (\frac {(x+2) \sqrt {x^2+1}}{x+1}-\int -\frac {1-2 x}{(x+1) \sqrt {x^2+1}}dx\right )-\frac {\left (x^2+1\right )^{3/2}}{2 (x+1)^2}\right )}{\sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(x+1) \left (\frac {3}{2} \left (\int \frac {1-2 x}{(x+1) \sqrt {x^2+1}}dx+\frac {\sqrt {x^2+1} (x+2)}{x+1}\right )-\frac {\left (x^2+1\right )^{3/2}}{2 (x+1)^2}\right )}{\sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1}}\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \frac {(x+1) \left (\frac {3}{2} \left (-2 \int \frac {1}{\sqrt {x^2+1}}dx+3 \int \frac {1}{(x+1) \sqrt {x^2+1}}dx+\frac {\sqrt {x^2+1} (x+2)}{x+1}\right )-\frac {\left (x^2+1\right )^{3/2}}{2 (x+1)^2}\right )}{\sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {(x+1) \left (\frac {3}{2} \left (3 \int \frac {1}{(x+1) \sqrt {x^2+1}}dx-2 \text {arcsinh}(x)+\frac {\sqrt {x^2+1} (x+2)}{x+1}\right )-\frac {\left (x^2+1\right )^{3/2}}{2 (x+1)^2}\right )}{\sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1}}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {(x+1) \left (\frac {3}{2} \left (-3 \int \frac {1}{2-\frac {(1-x)^2}{x^2+1}}d\frac {1-x}{\sqrt {x^2+1}}-2 \text {arcsinh}(x)+\frac {\sqrt {x^2+1} (x+2)}{x+1}\right )-\frac {\left (x^2+1\right )^{3/2}}{2 (x+1)^2}\right )}{\sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {(x+1) \left (\frac {3}{2} \left (-2 \text {arcsinh}(x)-\frac {3 \text {arctanh}\left (\frac {1-x}{\sqrt {2} \sqrt {x^2+1}}\right )}{\sqrt {2}}+\frac {\sqrt {x^2+1} (x+2)}{x+1}\right )-\frac {\left (x^2+1\right )^{3/2}}{2 (x+1)^2}\right )}{\sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1}}\) |
((1 + x)*(-1/2*(1 + x^2)^(3/2)/(1 + x)^2 + (3*(((2 + x)*Sqrt[1 + x^2])/(1 + x) - 2*ArcSinh[x] - (3*ArcTanh[(1 - x)/(Sqrt[2]*Sqrt[1 + x^2])])/Sqrt[2] ))/2))/(Sqrt[1 + x^2]*Sqrt[1 + (2*x)/(1 + x^2)])
3.9.99.3.1 Defintions of rubi rules used
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_)^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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (c + d*x)^(n + 1)*((a + b*x^2)^p/(d*(n + 1))), x] - Simp[2*b*(p/(d*(n + 1)) ) Int[x*(c + d*x)^(n + 1)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && GtQ[p, 0] && (IntegerQ[p] || LtQ[n, -1]) && NeQ[n, -1] && !IL tQ[n + 2*p + 1, 0] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
Int[(x_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c + d*x)^(n + 1))*(a + b*x^2)^p*((c*(2*p + 1) - d*(n + 1)*x)/(d^2*( n + 1)*(n + 2*p + 2))), x] + Simp[2*(p/(d^2*(n + 1)*(n + 2*p + 2))) Int[( c + d*x)^(n + 1)*(a + b*x^2)^(p - 1)*(a*d*(n + 1) + b*c*(2*p + 1)*x), x], x ] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0] && LtQ[n, -1] && !ILtQ[n + 2*p + 1, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q_.), x_ Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x) ^(2*FracPart[p])) Int[(b + 2*c*x)^(2*p)*(d + f*x^2)^q, x], x] /; FreeQ[{a , b, c, d, f, p, q}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
Int[(u_.)*((a_.) + (b_.)*(v_)^(n_)*(x_)^(m_.))^(p_), x_Symbol] :> Simp[(a + b*x^m*v^n)^FracPart[p]/(v^(n*FracPart[p])*(b*x^m + a/v^n)^FracPart[p]) I nt[u*v^(n*p)*(b*x^m + a/v^n)^p, x], x] /; FreeQ[{a, b, m, p}, x] && !Integ erQ[p] && ILtQ[n, 0] && BinomialQ[v, x]
Time = 0.29 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.76
| method | result | size |
| risch | \(\frac {2 x^{4}+9 x^{3}+7 x^{2}+9 x +5}{2 \left (x +1\right ) \left (x^{2}+1\right ) \sqrt {\frac {\left (x +1\right )^{2}}{x^{2}+1}}}+\frac {\left (-3 \,\operatorname {arcsinh}\left (x \right )-\frac {9 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (-2 x +2\right ) \sqrt {2}}{4 \sqrt {\left (x +1\right )^{2}-2 x}}\right )}{4}\right ) \left (x +1\right )}{\sqrt {\frac {\left (x +1\right )^{2}}{x^{2}+1}}\, \sqrt {x^{2}+1}}\) | \(109\) |
| trager | \(\frac {\left (x^{2}+1\right ) \left (2 x^{2}+9 x +5\right ) \sqrt {-\frac {-x^{2}-2 x -1}{x^{2}+1}}}{2 \left (x +1\right )^{3}}+3 \ln \left (-\frac {\sqrt {-\frac {-x^{2}-2 x -1}{x^{2}+1}}\, x^{2}-x^{2}+\sqrt {-\frac {-x^{2}-2 x -1}{x^{2}+1}}-x}{x +1}\right )-\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {2 \sqrt {-\frac {-x^{2}-2 x -1}{x^{2}+1}}\, x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}+2 \sqrt {-\frac {-x^{2}-2 x -1}{x^{2}+1}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{\left (x +1\right )^{2}}\right )}{4}\) | \(192\) |
| default | \(-\frac {\left (x +1\right ) \left (-\left (x^{2}+1\right )^{\frac {5}{2}} x +\left (x^{2}+1\right )^{\frac {3}{2}} x^{3}+\left (x^{2}+1\right )^{\frac {5}{2}}-\left (x^{2}+1\right )^{\frac {3}{2}} x^{2}-18 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (x -1\right ) \sqrt {2}}{2 \sqrt {x^{2}+1}}\right ) x^{2}-5 x \left (x^{2}+1\right )^{\frac {3}{2}}+6 \sqrt {x^{2}+1}\, x^{3}+24 \,\operatorname {arcsinh}\left (x \right ) x^{2}-36 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (x -1\right ) \sqrt {2}}{2 \sqrt {x^{2}+1}}\right ) x -3 \left (x^{2}+1\right )^{\frac {3}{2}}-6 \sqrt {x^{2}+1}\, x^{2}+48 \,\operatorname {arcsinh}\left (x \right ) x -18 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (x -1\right ) \sqrt {2}}{2 \sqrt {x^{2}+1}}\right )-30 x \sqrt {x^{2}+1}+24 \,\operatorname {arcsinh}\left (x \right )-18 \sqrt {x^{2}+1}\right )}{8 \left (\frac {x^{2}+2 x +1}{x^{2}+1}\right )^{\frac {3}{2}} \left (x^{2}+1\right )^{\frac {3}{2}}}\) | \(217\) |
1/2*(2*x^4+9*x^3+7*x^2+9*x+5)/(x+1)/(x^2+1)/((x+1)^2/(x^2+1))^(1/2)+(-3*ar csinh(x)-9/4*2^(1/2)*arctanh(1/4*(-2*x+2)*2^(1/2)/((x+1)^2-2*x)^(1/2)))/(( x+1)^2/(x^2+1))^(1/2)/(x^2+1)^(1/2)*(x+1)
Time = 0.29 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.42 \[ \int \frac {1}{\left (1+\frac {2 x}{1+x^2}\right )^{3/2}} \, dx=\frac {10 \, x^{3} + 9 \, \sqrt {2} {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} \log \left (-\frac {x^{2} + \sqrt {2} {\left (x^{2} - 1\right )} + {\left (2 \, x^{2} + \sqrt {2} {\left (x^{2} + 1\right )} + 2\right )} \sqrt {\frac {x^{2} + 2 \, x + 1}{x^{2} + 1}} - 1}{x^{2} + 2 \, x + 1}\right ) + 30 \, x^{2} + 12 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} \log \left (-\frac {x^{2} - {\left (x^{2} + 1\right )} \sqrt {\frac {x^{2} + 2 \, x + 1}{x^{2} + 1}} + x}{x + 1}\right ) + 2 \, {\left (2 \, x^{4} + 9 \, x^{3} + 7 \, x^{2} + 9 \, x + 5\right )} \sqrt {\frac {x^{2} + 2 \, x + 1}{x^{2} + 1}} + 30 \, x + 10}{4 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} \]
1/4*(10*x^3 + 9*sqrt(2)*(x^3 + 3*x^2 + 3*x + 1)*log(-(x^2 + sqrt(2)*(x^2 - 1) + (2*x^2 + sqrt(2)*(x^2 + 1) + 2)*sqrt((x^2 + 2*x + 1)/(x^2 + 1)) - 1) /(x^2 + 2*x + 1)) + 30*x^2 + 12*(x^3 + 3*x^2 + 3*x + 1)*log(-(x^2 - (x^2 + 1)*sqrt((x^2 + 2*x + 1)/(x^2 + 1)) + x)/(x + 1)) + 2*(2*x^4 + 9*x^3 + 7*x ^2 + 9*x + 5)*sqrt((x^2 + 2*x + 1)/(x^2 + 1)) + 30*x + 10)/(x^3 + 3*x^2 + 3*x + 1)
\[ \int \frac {1}{\left (1+\frac {2 x}{1+x^2}\right )^{3/2}} \, dx=\int \frac {1}{\left (\frac {2 x}{x^{2} + 1} + 1\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{\left (1+\frac {2 x}{1+x^2}\right )^{3/2}} \, dx=\int { \frac {1}{{\left (\frac {2 \, x}{x^{2} + 1} + 1\right )}^{\frac {3}{2}}} \,d x } \]
Time = 0.40 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\left (1+\frac {2 x}{1+x^2}\right )^{3/2}} \, dx=\frac {9 \, \sqrt {2} \log \left (\frac {{\left | -2 \, x - 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 1} - 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 1} - 2 \right |}}\right )}{4 \, \mathrm {sgn}\left (x + 1\right )} + \frac {3 \, \log \left (-x + \sqrt {x^{2} + 1}\right )}{\mathrm {sgn}\left (x + 1\right )} + \frac {\sqrt {x^{2} + 1}}{\mathrm {sgn}\left (x + 1\right )} + \frac {7 \, {\left (x - \sqrt {x^{2} + 1}\right )}^{3} + 5 \, {\left (x - \sqrt {x^{2} + 1}\right )}^{2} - 13 \, x + 13 \, \sqrt {x^{2} + 1} + 5}{{\left ({\left (x - \sqrt {x^{2} + 1}\right )}^{2} + 2 \, x - 2 \, \sqrt {x^{2} + 1} - 1\right )}^{2} \mathrm {sgn}\left (x + 1\right )} \]
9/4*sqrt(2)*log(abs(-2*x - 2*sqrt(2) + 2*sqrt(x^2 + 1) - 2)/abs(-2*x + 2*s qrt(2) + 2*sqrt(x^2 + 1) - 2))/sgn(x + 1) + 3*log(-x + sqrt(x^2 + 1))/sgn( x + 1) + sqrt(x^2 + 1)/sgn(x + 1) + (7*(x - sqrt(x^2 + 1))^3 + 5*(x - sqrt (x^2 + 1))^2 - 13*x + 13*sqrt(x^2 + 1) + 5)/(((x - sqrt(x^2 + 1))^2 + 2*x - 2*sqrt(x^2 + 1) - 1)^2*sgn(x + 1))
Timed out. \[ \int \frac {1}{\left (1+\frac {2 x}{1+x^2}\right )^{3/2}} \, dx=\int \frac {1}{{\left (\frac {2\,x}{x^2+1}+1\right )}^{3/2}} \,d x \]