Integrand size = 18, antiderivative size = 123 \[ \int \frac {1}{(1+x) \sqrt [3]{3+2 x+x^2}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} \sqrt [3]{3+2 x+x^2}}{\sqrt {3}}\right )}{2 \sqrt [3]{2}}+\frac {\log \left (-2+2^{2/3} \sqrt [3]{3+2 x+x^2}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (2+2^{2/3} \sqrt [3]{3+2 x+x^2}+\sqrt [3]{2} \left (3+2 x+x^2\right )^{2/3}\right )}{4 \sqrt [3]{2}} \]
1/4*3^(1/2)*arctan(1/3*3^(1/2)+1/3*2^(2/3)*(x^2+2*x+3)^(1/3)*3^(1/2))*2^(2 /3)+1/4*ln(-2+2^(2/3)*(x^2+2*x+3)^(1/3))*2^(2/3)-1/8*ln(2+2^(2/3)*(x^2+2*x +3)^(1/3)+2^(1/3)*(x^2+2*x+3)^(2/3))*2^(2/3)
Time = 0.16 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(1+x) \sqrt [3]{3+2 x+x^2}} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{3+2 x+x^2}}{\sqrt {3}}\right )+2 \log \left (-2+2^{2/3} \sqrt [3]{3+2 x+x^2}\right )-\log \left (2+2^{2/3} \sqrt [3]{3+2 x+x^2}+\sqrt [3]{2} \left (3+2 x+x^2\right )^{2/3}\right )}{4 \sqrt [3]{2}} \]
(2*Sqrt[3]*ArcTan[(1 + 2^(2/3)*(3 + 2*x + x^2)^(1/3))/Sqrt[3]] + 2*Log[-2 + 2^(2/3)*(3 + 2*x + x^2)^(1/3)] - Log[2 + 2^(2/3)*(3 + 2*x + x^2)^(1/3) + 2^(1/3)*(3 + 2*x + x^2)^(2/3)])/(4*2^(1/3))
Time = 0.22 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.59, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1118, 243, 67, 16, 1082, 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.
\(\displaystyle \int \frac {1}{(x+1) \sqrt [3]{x^2+2 x+3}} \, dx\) |
\(\Big \downarrow \) 1118 |
\(\displaystyle \int \frac {1}{(x+1) \sqrt [3]{(x+1)^2+2}}d(x+1)\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {1}{(x+1)^2 \sqrt [3]{x+3}}d(x+1)^2\) |
\(\Big \downarrow \) 67 |
\(\displaystyle \frac {1}{2} \left (-\frac {3 \int \frac {1}{-x+\sqrt [3]{2}-1}d\sqrt [3]{x+3}}{2 \sqrt [3]{2}}+\frac {3}{2} \int \frac {1}{(x+1)^4+\sqrt [3]{2} \sqrt [3]{x+3}+2^{2/3}}d\sqrt [3]{x+3}-\frac {\log \left ((x+1)^2\right )}{2 \sqrt [3]{2}}\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {1}{2} \left (\frac {3}{2} \int \frac {1}{(x+1)^4+\sqrt [3]{2} \sqrt [3]{x+3}+2^{2/3}}d\sqrt [3]{x+3}+\frac {3 \log \left (-x+\sqrt [3]{2}-1\right )}{2 \sqrt [3]{2}}-\frac {\log \left ((x+1)^2\right )}{2 \sqrt [3]{2}}\right )\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {1}{2} \left (-\frac {3 \int \frac {1}{-(x+1)^4-3}d\left (2^{2/3} \sqrt [3]{x+3}+1\right )}{\sqrt [3]{2}}+\frac {3 \log \left (-x+\sqrt [3]{2}-1\right )}{2 \sqrt [3]{2}}-\frac {\log \left ((x+1)^2\right )}{2 \sqrt [3]{2}}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{2} \left (\frac {\sqrt {3} \arctan \left (\frac {2^{2/3} \sqrt [3]{x+3}+1}{\sqrt {3}}\right )}{\sqrt [3]{2}}+\frac {3 \log \left (-x+\sqrt [3]{2}-1\right )}{2 \sqrt [3]{2}}-\frac {\log \left ((x+1)^2\right )}{2 \sqrt [3]{2}}\right )\) |
((Sqrt[3]*ArcTan[(1 + 2^(2/3)*(3 + x)^(1/3))/Sqrt[3]])/2^(1/3) + (3*Log[-1 + 2^(1/3) - x])/(2*2^(1/3)) - Log[(1 + x)^2]/(2*2^(1/3)))/2
3.19.7.3.1 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[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x ] + (Simp[3/(2*b) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Simp[3/(2*b*q) 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_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[1/e Subst[Int[x^m*(a - b^2/(4*c) + (c*x^2)/e^2)^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[2*c*d - b*e, 0]
Time = 6.52 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.68
method | result | size |
pseudoelliptic | \(\frac {2^{\frac {2}{3}} \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} \left (x^{2}+2 x +3\right )^{\frac {1}{3}}+1\right )}{3}\right )+2 \ln \left (\left (x^{2}+2 x +3\right )^{\frac {1}{3}}-2^{\frac {1}{3}}\right )-\ln \left (\left (x^{2}+2 x +3\right )^{\frac {2}{3}}+2^{\frac {1}{3}} \left (x^{2}+2 x +3\right )^{\frac {1}{3}}+2^{\frac {2}{3}}\right )\right )}{8}\) | \(84\) |
trager | \(\text {Expression too large to display}\) | \(805\) |
1/8*2^(2/3)*(2*3^(1/2)*arctan(1/3*3^(1/2)*(2^(2/3)*(x^2+2*x+3)^(1/3)+1))+2 *ln((x^2+2*x+3)^(1/3)-2^(1/3))-ln((x^2+2*x+3)^(2/3)+2^(1/3)*(x^2+2*x+3)^(1 /3)+2^(2/3)))
Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(1+x) \sqrt [3]{3+2 x+x^2}} \, dx=\frac {1}{4} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {1}{6}} {\left (2^{\frac {5}{6}} + 2 \, \sqrt {2} {\left (x^{2} + 2 \, x + 3\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{8} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (x^{2} + 2 \, x + 3\right )}^{\frac {1}{3}} + {\left (x^{2} + 2 \, x + 3\right )}^{\frac {2}{3}}\right ) + \frac {1}{4} \cdot 2^{\frac {2}{3}} \log \left (-2^{\frac {1}{3}} + {\left (x^{2} + 2 \, x + 3\right )}^{\frac {1}{3}}\right ) \]
1/4*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(1/6)*(2^(5/6) + 2*sqrt(2)*(x^2 + 2*x + 3)^(1/3))) - 1/8*2^(2/3)*log(2^(2/3) + 2^(1/3)*(x^2 + 2*x + 3)^(1/3 ) + (x^2 + 2*x + 3)^(2/3)) + 1/4*2^(2/3)*log(-2^(1/3) + (x^2 + 2*x + 3)^(1 /3))
\[ \int \frac {1}{(1+x) \sqrt [3]{3+2 x+x^2}} \, dx=\int \frac {1}{\left (x + 1\right ) \sqrt [3]{x^{2} + 2 x + 3}}\, dx \]
\[ \int \frac {1}{(1+x) \sqrt [3]{3+2 x+x^2}} \, dx=\int { \frac {1}{{\left (x^{2} + 2 \, x + 3\right )}^{\frac {1}{3}} {\left (x + 1\right )}} \,d x } \]
\[ \int \frac {1}{(1+x) \sqrt [3]{3+2 x+x^2}} \, dx=\int { \frac {1}{{\left (x^{2} + 2 \, x + 3\right )}^{\frac {1}{3}} {\left (x + 1\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{(1+x) \sqrt [3]{3+2 x+x^2}} \, dx=\int \frac {1}{\left (x+1\right )\,{\left (x^2+2\,x+3\right )}^{1/3}} \,d x \]