Integrand size = 15, antiderivative size = 67 \[ \int \frac {\sqrt [3]{1+\sqrt {x}}}{x} \, dx=6 \sqrt [3]{1+\sqrt {x}}-2 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1+\sqrt {x}}}{\sqrt {3}}\right )+3 \log \left (1-\sqrt [3]{1+\sqrt {x}}\right )-\frac {\log (x)}{2} \]
-1/2*ln(x)+3*ln(1-(1+x^(1/2))^(1/3))-2*arctan(1/3*(1+2*(1+x^(1/2))^(1/3))* 3^(1/2))*3^(1/2)+6*(1+x^(1/2))^(1/3)
Time = 0.04 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.28 \[ \int \frac {\sqrt [3]{1+\sqrt {x}}}{x} \, dx=6 \sqrt [3]{1+\sqrt {x}}-2 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1+\sqrt {x}}}{\sqrt {3}}\right )+2 \log \left (-1+\sqrt [3]{1+\sqrt {x}}\right )-\log \left (1+\sqrt [3]{1+\sqrt {x}}+\left (1+\sqrt {x}\right )^{2/3}\right ) \]
6*(1 + Sqrt[x])^(1/3) - 2*Sqrt[3]*ArcTan[(1 + 2*(1 + Sqrt[x])^(1/3))/Sqrt[ 3]] + 2*Log[-1 + (1 + Sqrt[x])^(1/3)] - Log[1 + (1 + Sqrt[x])^(1/3) + (1 + Sqrt[x])^(2/3)]
Time = 0.18 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {798, 60, 69, 16, 1083, 217}
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.
\(\displaystyle \int \frac {\sqrt [3]{\sqrt {x}+1}}{x} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \frac {\sqrt [3]{\sqrt {x}+1}}{\sqrt {x}}d\sqrt {x}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle 2 \left (\int \frac {1}{\left (\sqrt {x}+1\right )^{2/3} \sqrt {x}}d\sqrt {x}+3 \sqrt [3]{\sqrt {x}+1}\right )\) |
\(\Big \downarrow \) 69 |
\(\displaystyle 2 \left (-\frac {3}{2} \int \frac {1}{1-\sqrt [3]{\sqrt {x}+1}}d\sqrt [3]{\sqrt {x}+1}-\frac {3}{2} \int \frac {1}{x+\sqrt [3]{\sqrt {x}+1}+1}d\sqrt [3]{\sqrt {x}+1}+3 \sqrt [3]{\sqrt {x}+1}-\frac {1}{2} \log \left (\sqrt {x}\right )\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle 2 \left (-\frac {3}{2} \int \frac {1}{x+\sqrt [3]{\sqrt {x}+1}+1}d\sqrt [3]{\sqrt {x}+1}+3 \sqrt [3]{\sqrt {x}+1}+\frac {3}{2} \log \left (1-\sqrt [3]{\sqrt {x}+1}\right )-\frac {\log \left (\sqrt {x}\right )}{2}\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle 2 \left (3 \int \frac {1}{-x-3}d\left (2 \sqrt [3]{\sqrt {x}+1}+1\right )+3 \sqrt [3]{\sqrt {x}+1}+\frac {3}{2} \log \left (1-\sqrt [3]{\sqrt {x}+1}\right )-\frac {\log \left (\sqrt {x}\right )}{2}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle 2 \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{\sqrt {x}+1}+1}{\sqrt {3}}\right )+3 \sqrt [3]{\sqrt {x}+1}+\frac {3}{2} \log \left (1-\sqrt [3]{\sqrt {x}+1}\right )-\frac {\log \left (\sqrt {x}\right )}{2}\right )\) |
2*(3*(1 + Sqrt[x])^(1/3) - Sqrt[3]*ArcTan[(1 + 2*(1 + Sqrt[x])^(1/3))/Sqrt [3]] + (3*Log[1 - (1 + Sqrt[x])^(1/3)])/2 - Log[Sqrt[x]]/2)
3.10.51.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Simp[3/(2*b*q) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1 /3)], x] - Simp[3/(2*b*q^2) Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
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]
Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 1.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.72
method | result | size |
meijerg | \(-\frac {2 \left (-\Gamma \left (\frac {2}{3}\right ) \sqrt {x}\, {}_{3}^{}{\moversetsp {}{\mundersetsp {}{F_{2}^{}}}}\left (\frac {2}{3},1,1;2,2;-\sqrt {x}\right )-3 \left (3+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+\frac {\ln \left (x \right )}{2}\right ) \Gamma \left (\frac {2}{3}\right )\right )}{3 \Gamma \left (\frac {2}{3}\right )}\) | \(48\) |
derivativedivides | \(6 \left (1+\sqrt {x}\right )^{\frac {1}{3}}-\ln \left (\left (1+\sqrt {x}\right )^{\frac {2}{3}}+\left (1+\sqrt {x}\right )^{\frac {1}{3}}+1\right )-2 \arctan \left (\frac {\left (1+2 \left (1+\sqrt {x}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}+2 \ln \left (\left (1+\sqrt {x}\right )^{\frac {1}{3}}-1\right )\) | \(64\) |
default | \(6 \left (1+\sqrt {x}\right )^{\frac {1}{3}}-\ln \left (\left (1+\sqrt {x}\right )^{\frac {2}{3}}+\left (1+\sqrt {x}\right )^{\frac {1}{3}}+1\right )-2 \arctan \left (\frac {\left (1+2 \left (1+\sqrt {x}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}+2 \ln \left (\left (1+\sqrt {x}\right )^{\frac {1}{3}}-1\right )\) | \(64\) |
-2/3/GAMMA(2/3)*(-GAMMA(2/3)*x^(1/2)*hypergeom([2/3,1,1],[2,2],-x^(1/2))-3 *(3+1/6*Pi*3^(1/2)-3/2*ln(3)+1/2*ln(x))*GAMMA(2/3))
Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt [3]{1+\sqrt {x}}}{x} \, dx=-2 \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + 6 \, {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} - \log \left ({\left (\sqrt {x} + 1\right )}^{\frac {2}{3}} + {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} + 1\right ) + 2 \, \log \left ({\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
-2*sqrt(3)*arctan(2/3*sqrt(3)*(sqrt(x) + 1)^(1/3) + 1/3*sqrt(3)) + 6*(sqrt (x) + 1)^(1/3) - log((sqrt(x) + 1)^(2/3) + (sqrt(x) + 1)^(1/3) + 1) + 2*lo g((sqrt(x) + 1)^(1/3) - 1)
Result contains complex when optimal does not.
Time = 0.64 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.58 \[ \int \frac {\sqrt [3]{1+\sqrt {x}}}{x} \, dx=- \frac {2 \sqrt [6]{x} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{\sqrt {x}}} \right )}}{\Gamma \left (\frac {2}{3}\right )} \]
Time = 0.26 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt [3]{1+\sqrt {x}}}{x} \, dx=-2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + 6 \, {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} - \log \left ({\left (\sqrt {x} + 1\right )}^{\frac {2}{3}} + {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} + 1\right ) + 2 \, \log \left ({\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
-2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(sqrt(x) + 1)^(1/3) + 1)) + 6*(sqrt(x) + 1)^(1/3) - log((sqrt(x) + 1)^(2/3) + (sqrt(x) + 1)^(1/3) + 1) + 2*log((sqr t(x) + 1)^(1/3) - 1)
Time = 0.33 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt [3]{1+\sqrt {x}}}{x} \, dx=-2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + 6 \, {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} - \log \left ({\left (\sqrt {x} + 1\right )}^{\frac {2}{3}} + {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} + 1\right ) + 2 \, \log \left ({\left | {\left (\sqrt {x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
-2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(sqrt(x) + 1)^(1/3) + 1)) + 6*(sqrt(x) + 1)^(1/3) - log((sqrt(x) + 1)^(2/3) + (sqrt(x) + 1)^(1/3) + 1) + 2*log(abs( (sqrt(x) + 1)^(1/3) - 1))
Time = 21.54 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt [3]{1+\sqrt {x}}}{x} \, dx=2\,\ln \left ({\left (\sqrt {x}+1\right )}^{1/3}-1\right )+6\,{\left (\sqrt {x}+1\right )}^{1/3}+\ln \left ({\left (\sqrt {x}+1\right )}^{1/3}+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )-\ln \left ({\left (\sqrt {x}+1\right )}^{1/3}+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right ) \]