Integrand size = 11, antiderivative size = 83 \[ \int \frac {1}{1+\left (x^2\right )^{3/2}} \, dx=-\frac {x \arctan \left (\frac {1-2 \sqrt {x^2}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt {x^2}}-\frac {x \log \left (1+x^2-\sqrt {x^2}\right )}{6 \sqrt {x^2}}+\frac {x \log \left (1+\sqrt {x^2}\right )}{3 \sqrt {x^2}} \]
-1/6*x*ln(1+x^2-(x^2)^(1/2))/(x^2)^(1/2)+1/3*x*ln(1+(x^2)^(1/2))/(x^2)^(1/ 2)-1/3*x*arctan(1/3*(1-2*(x^2)^(1/2))*3^(1/2))*3^(1/2)/(x^2)^(1/2)
Time = 0.55 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.81 \[ \int \frac {1}{1+\left (x^2\right )^{3/2}} \, dx=\frac {x \left (\frac {\arctan \left (\frac {-1+2 \sqrt {x^2}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (1+x^2-\sqrt {x^2}\right )+\frac {1}{3} \log \left (1+\sqrt {x^2}\right )\right )}{\sqrt {x^2}} \]
(x*(ArcTan[(-1 + 2*Sqrt[x^2])/Sqrt[3]]/Sqrt[3] - Log[1 + x^2 - Sqrt[x^2]]/ 6 + Log[1 + Sqrt[x^2]]/3))/Sqrt[x^2]
Time = 0.23 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.87, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {786, 750, 16, 1142, 25, 1083, 217, 1103}
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.
\(\displaystyle \int \frac {1}{\left (x^2\right )^{3/2}+1} \, dx\) |
\(\Big \downarrow \) 786 |
\(\displaystyle \frac {x \int \frac {1}{\left (x^2\right )^{3/2}+1}d\sqrt {x^2}}{\sqrt {x^2}}\) |
\(\Big \downarrow \) 750 |
\(\displaystyle \frac {x \left (\frac {1}{3} \int \frac {2-\sqrt {x^2}}{x^2-\sqrt {x^2}+1}d\sqrt {x^2}+\frac {1}{3} \int \frac {1}{\sqrt {x^2}+1}d\sqrt {x^2}\right )}{\sqrt {x^2}}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {x \left (\frac {1}{3} \int \frac {2-\sqrt {x^2}}{x^2-\sqrt {x^2}+1}d\sqrt {x^2}+\frac {1}{3} \log \left (\sqrt {x^2}+1\right )\right )}{\sqrt {x^2}}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {x \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^2-\sqrt {x^2}+1}d\sqrt {x^2}-\frac {1}{2} \int -\frac {1-2 \sqrt {x^2}}{x^2-\sqrt {x^2}+1}d\sqrt {x^2}\right )+\frac {1}{3} \log \left (\sqrt {x^2}+1\right )\right )}{\sqrt {x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {x \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^2-\sqrt {x^2}+1}d\sqrt {x^2}+\frac {1}{2} \int \frac {1-2 \sqrt {x^2}}{x^2-\sqrt {x^2}+1}d\sqrt {x^2}\right )+\frac {1}{3} \log \left (\sqrt {x^2}+1\right )\right )}{\sqrt {x^2}}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {x \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {1-2 \sqrt {x^2}}{x^2-\sqrt {x^2}+1}d\sqrt {x^2}-3 \int \frac {1}{-x^2-3}d\left (2 \sqrt {x^2}-1\right )\right )+\frac {1}{3} \log \left (\sqrt {x^2}+1\right )\right )}{\sqrt {x^2}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {x \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {1-2 \sqrt {x^2}}{x^2-\sqrt {x^2}+1}d\sqrt {x^2}+\sqrt {3} \arctan \left (\frac {2 \sqrt {x^2}-1}{\sqrt {3}}\right )\right )+\frac {1}{3} \log \left (\sqrt {x^2}+1\right )\right )}{\sqrt {x^2}}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {x \left (\frac {1}{3} \left (\sqrt {3} \arctan \left (\frac {2 \sqrt {x^2}-1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (x^2-\sqrt {x^2}+1\right )\right )+\frac {1}{3} \log \left (\sqrt {x^2}+1\right )\right )}{\sqrt {x^2}}\) |
(x*((Sqrt[3]*ArcTan[(-1 + 2*Sqrt[x^2])/Sqrt[3]] - Log[1 + x^2 - Sqrt[x^2]] /2)/3 + Log[1 + Sqrt[x^2]]/3))/Sqrt[x^2]
3.30.29.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*((c_.)*(x_)^(q_.))^(n_))^(p_), x_Symbol] :> Simp[x/(c*x^q )^(1/q) Subst[Int[(a + b*x^(n*q))^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{ a, b, c, n, p, q}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Time = 3.78 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.30
method | result | size |
default | \(\frac {x^{3} \left (-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {x^{3}}{\left (x^{2}\right )^{\frac {3}{2}}}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {x^{3}}{\left (x^{2}\right )^{\frac {3}{2}}}\right )^{\frac {1}{3}}}\right )+2 \ln \left (x +\left (\frac {x^{3}}{\left (x^{2}\right )^{\frac {3}{2}}}\right )^{\frac {1}{3}}\right )-\ln \left (x^{2}-\left (\frac {x^{3}}{\left (x^{2}\right )^{\frac {3}{2}}}\right )^{\frac {1}{3}} x +\left (\frac {x^{3}}{\left (x^{2}\right )^{\frac {3}{2}}}\right )^{\frac {2}{3}}\right )\right )}{6 \left (x^{2}\right )^{\frac {3}{2}} \left (\frac {x^{3}}{\left (x^{2}\right )^{\frac {3}{2}}}\right )^{\frac {2}{3}}}\) | \(108\) |
meijerg | \(\frac {\frac {x {\left (\frac {\left (x^{2}\right )^{\frac {3}{2}}}{x^{3}}\right )}^{\frac {1}{3}} \ln \left (1+{\left (\left (x^{2}\right )^{\frac {3}{2}}\right )}^{\frac {1}{3}}\right )}{{\left (\left (x^{2}\right )^{\frac {3}{2}}\right )}^{\frac {1}{3}}}-\frac {x {\left (\frac {\left (x^{2}\right )^{\frac {3}{2}}}{x^{3}}\right )}^{\frac {1}{3}} \ln \left (1-{\left (\left (x^{2}\right )^{\frac {3}{2}}\right )}^{\frac {1}{3}}+{\left (\left (x^{2}\right )^{\frac {3}{2}}\right )}^{\frac {2}{3}}\right )}{2 {\left (\left (x^{2}\right )^{\frac {3}{2}}\right )}^{\frac {1}{3}}}+\frac {x {\left (\frac {\left (x^{2}\right )^{\frac {3}{2}}}{x^{3}}\right )}^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, {\left (\left (x^{2}\right )^{\frac {3}{2}}\right )}^{\frac {1}{3}}}{2-{\left (\left (x^{2}\right )^{\frac {3}{2}}\right )}^{\frac {1}{3}}}\right )}{{\left (\left (x^{2}\right )^{\frac {3}{2}}\right )}^{\frac {1}{3}}}}{3 {\left (\frac {\left (x^{2}\right )^{\frac {3}{2}}}{x^{3}}\right )}^{\frac {1}{3}}}\) | \(134\) |
1/6*x^3*(-2*3^(1/2)*arctan(1/3*3^(1/2)*(-2*x+(1/(x^2)^(3/2)*x^3)^(1/3))/(1 /(x^2)^(3/2)*x^3)^(1/3))+2*ln(x+(1/(x^2)^(3/2)*x^3)^(1/3))-ln(x^2-(1/(x^2) ^(3/2)*x^3)^(1/3)*x+(1/(x^2)^(3/2)*x^3)^(2/3)))/(x^2)^(3/2)/(1/(x^2)^(3/2) *x^3)^(2/3)
Time = 0.26 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.58 \[ \int \frac {1}{1+\left (x^2\right )^{3/2}} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \sqrt {x^{2}} - \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{6} \, \log \left (x^{2} - \sqrt {x^{2}} + 1\right ) + \frac {1}{3} \, \log \left (\sqrt {x^{2}} + 1\right ) \]
1/3*sqrt(3)*arctan(2/3*sqrt(3)*sqrt(x^2) - 1/3*sqrt(3)) - 1/6*log(x^2 - sq rt(x^2) + 1) + 1/3*log(sqrt(x^2) + 1)
\[ \int \frac {1}{1+\left (x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (\sqrt {x^{2}} + 1\right ) \left (x^{2} - \sqrt {x^{2}} + 1\right )}\, dx \]
Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.41 \[ \int \frac {1}{1+\left (x^2\right )^{3/2}} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{6} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{3} \, \log \left (x + 1\right ) \]
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.30 \[ \int \frac {1}{1+\left (x^2\right )^{3/2}} \, dx=-\frac {\sqrt {3} {\left (-i \, \sqrt {3} - 1\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {1}{\mathrm {sgn}\left (x\right )}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {1}{\mathrm {sgn}\left (x\right )}\right )^{\frac {1}{3}}}\right )}{6 \, \mathrm {sgn}\left (x\right )^{\frac {1}{3}}} - \frac {1}{9} i \, \pi \mathrm {sgn}\left (x\right ) - \frac {{\left (-i \, \sqrt {3} - 1\right )} \log \left (x^{2} + x \left (-\frac {1}{\mathrm {sgn}\left (x\right )}\right )^{\frac {1}{3}} + \left (-\frac {1}{\mathrm {sgn}\left (x\right )}\right )^{\frac {2}{3}}\right )}{12 \, \mathrm {sgn}\left (x\right )^{\frac {1}{3}}} - \frac {1}{3} \, \left (-\frac {1}{\mathrm {sgn}\left (x\right )}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {1}{\mathrm {sgn}\left (x\right )}\right )^{\frac {1}{3}} \right |}\right ) \]
-1/6*sqrt(3)*(-I*sqrt(3) - 1)*arctan(1/3*sqrt(3)*(2*x + (-1/sgn(x))^(1/3)) /(-1/sgn(x))^(1/3))/sgn(x)^(1/3) - 1/9*I*pi*sgn(x) - 1/12*(-I*sqrt(3) - 1) *log(x^2 + x*(-1/sgn(x))^(1/3) + (-1/sgn(x))^(2/3))/sgn(x)^(1/3) - 1/3*(-1 /sgn(x))^(1/3)*log(abs(x - (-1/sgn(x))^(1/3)))
Timed out. \[ \int \frac {1}{1+\left (x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (x^2\right )}^{3/2}+1} \,d x \]