Integrand size = 29, antiderivative size = 32 \[ \int \frac {1-\sqrt [3]{2} x}{\left (2^{2/3}+x\right ) \sqrt {1+x^3}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {3} \left (1+\sqrt [3]{2} x\right )}{\sqrt {1+x^3}}\right )}{\sqrt {3}} \]
Time = 1.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {1-\sqrt [3]{2} x}{\left (2^{2/3}+x\right ) \sqrt {1+x^3}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {1+x^3}}{\sqrt {3} \left (1+\sqrt [3]{2} x\right )}\right )}{\sqrt {3}} \]
Time = 0.23 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2562, 216}
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.
| \(\displaystyle \int \frac {1-\sqrt [3]{2} x}{\left (x+2^{2/3}\right ) \sqrt {x^3+1}} \, dx\) |
| \(\Big \downarrow \) 2562 |
| \(\displaystyle 2 \int \frac {1}{\frac {3 \left (\sqrt [3]{2} x+1\right )^2}{x^3+1}+1}d\frac {\sqrt [3]{2} x+1}{\sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2 \arctan \left (\frac {\sqrt {3} \left (\sqrt [3]{2} x+1\right )}{\sqrt {x^3+1}}\right )}{\sqrt {3}}\) |
3.1.84.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ Symbol] :> Simp[2*(e/d) Subst[Int[1/(1 + 3*a*x^2), x], x, (1 + 2*d*(x/c)) /Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b*c^3 - 4*a*d^3, 0] && EqQ[2*d*e + c*f, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.96 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.47
| method | result | size |
| trager | \(-\frac {2^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \,2^{\frac {1}{3}}\right ) \ln \left (\frac {12 \sqrt {x^{3}+1}\, x +3 \,2^{\frac {2}{3}} x^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \,2^{\frac {1}{3}}\right )-\operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \,2^{\frac {1}{3}}\right ) x^{3}+6 \sqrt {x^{3}+1}\, 2^{\frac {2}{3}}+6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \,2^{\frac {1}{3}}\right ) 2^{\frac {1}{3}} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \,2^{\frac {1}{3}}\right )}{\left (2^{\frac {1}{3}} x +2\right )^{3}}\right )}{6}\) | \(111\) |
| default | \(-\frac {2 \,2^{\frac {1}{3}} \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {6 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \Pi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{2^{\frac {2}{3}}-1}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}\, \left (2^{\frac {2}{3}}-1\right )}\) | \(258\) |
| elliptic | \(-\frac {2 \,2^{\frac {1}{3}} \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {6 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \Pi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{2^{\frac {2}{3}}-1}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}\, \left (2^{\frac {2}{3}}-1\right )}\) | \(258\) |
-1/6*2^(1/3)*RootOf(_Z^2+6*2^(1/3))*ln((12*(x^3+1)^(1/2)*x+3*2^(2/3)*x^2*R ootOf(_Z^2+6*2^(1/3))-RootOf(_Z^2+6*2^(1/3))*x^3+6*(x^3+1)^(1/2)*2^(2/3)+6 *RootOf(_Z^2+6*2^(1/3))*2^(1/3)*x+2*RootOf(_Z^2+6*2^(1/3)))/(2^(1/3)*x+2)^ 3)
Exception generated. \[ \int \frac {1-\sqrt [3]{2} x}{\left (2^{2/3}+x\right ) \sqrt {1+x^3}} \, dx=\text {Exception raised: TypeError} \]
\[ \int \frac {1-\sqrt [3]{2} x}{\left (2^{2/3}+x\right ) \sqrt {1+x^3}} \, dx=- \int \frac {\sqrt [3]{2} x}{x \sqrt {x^{3} + 1} + 2^{\frac {2}{3}} \sqrt {x^{3} + 1}}\, dx - \int \left (- \frac {1}{x \sqrt {x^{3} + 1} + 2^{\frac {2}{3}} \sqrt {x^{3} + 1}}\right )\, dx \]
-Integral(2**(1/3)*x/(x*sqrt(x**3 + 1) + 2**(2/3)*sqrt(x**3 + 1)), x) - In tegral(-1/(x*sqrt(x**3 + 1) + 2**(2/3)*sqrt(x**3 + 1)), x)
\[ \int \frac {1-\sqrt [3]{2} x}{\left (2^{2/3}+x\right ) \sqrt {1+x^3}} \, dx=\int { -\frac {2^{\frac {1}{3}} x - 1}{\sqrt {x^{3} + 1} {\left (x + 2^{\frac {2}{3}}\right )}} \,d x } \]
\[ \int \frac {1-\sqrt [3]{2} x}{\left (2^{2/3}+x\right ) \sqrt {1+x^3}} \, dx=\int { -\frac {2^{\frac {1}{3}} x - 1}{\sqrt {x^{3} + 1} {\left (x + 2^{\frac {2}{3}}\right )}} \,d x } \]
Time = 1.77 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.09 \[ \int \frac {1-\sqrt [3]{2} x}{\left (2^{2/3}+x\right ) \sqrt {1+x^3}} \, dx=\frac {\sqrt {3}\,\ln \left (\frac {\left (\sqrt {3}\,1{}\mathrm {i}+\sqrt {x^3+1}+2^{1/3}\,\sqrt {3}\,x\,1{}\mathrm {i}\right )\,{\left (\sqrt {3}\,1{}\mathrm {i}-\sqrt {x^3+1}+2^{1/3}\,\sqrt {3}\,x\,1{}\mathrm {i}\right )}^3}{{\left (x+2^{2/3}\right )}^6}\right )\,1{}\mathrm {i}}{3} \]
(3^(1/2)*log(((3^(1/2)*1i + (x^3 + 1)^(1/2) + 2^(1/3)*3^(1/2)*x*1i)*(3^(1/ 2)*1i - (x^3 + 1)^(1/2) + 2^(1/3)*3^(1/2)*x*1i)^3)/(x + 2^(2/3))^6)*1i)/3
\[ \int \frac {1-\sqrt [3]{2} x}{\left (2^{2/3}+x\right ) \sqrt {1+x^3}} \, dx=-2^{\frac {1}{3}} \left (\int \frac {x}{\sqrt {x^{3}+1}\, 2^{\frac {2}{3}}+\sqrt {x^{3}+1}\, x}d x \right )+\int \frac {1}{\sqrt {x^{3}+1}\, 2^{\frac {2}{3}}+\sqrt {x^{3}+1}\, x}d x \]