Integrand size = 10, antiderivative size = 125 \[ \int \frac {1}{1+a+b x^3} \, dx=-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{1+a}}}{\sqrt {3}}\right )}{\sqrt {3} (1+a)^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{1+a}+\sqrt [3]{b} x\right )}{3 (1+a)^{2/3} \sqrt [3]{b}}-\frac {\log \left ((1+a)^{2/3}-\sqrt [3]{1+a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 (1+a)^{2/3} \sqrt [3]{b}} \]
1/3*ln((1+a)^(1/3)+b^(1/3)*x)/(1+a)^(2/3)/b^(1/3)-1/6*ln((1+a)^(2/3)-(1+a) ^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(1+a)^(2/3)/b^(1/3)-1/3*arctan(1/3*(1-2*b^(1 /3)*x/(1+a)^(1/3))*3^(1/2))/(1+a)^(2/3)/b^(1/3)*3^(1/2)
Time = 0.02 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.81 \[ \int \frac {1}{1+a+b x^3} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {-1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{1+a}}}{\sqrt {3}}\right )+2 \log \left (\sqrt [3]{1+a}+\sqrt [3]{b} x\right )-\log \left ((1+a)^{2/3}-\sqrt [3]{1+a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 (1+a)^{2/3} \sqrt [3]{b}} \]
(2*Sqrt[3]*ArcTan[(-1 + (2*b^(1/3)*x)/(1 + a)^(1/3))/Sqrt[3]] + 2*Log[(1 + a)^(1/3) + b^(1/3)*x] - Log[(1 + a)^(2/3) - (1 + a)^(1/3)*b^(1/3)*x + b^( 2/3)*x^2])/(6*(1 + a)^(2/3)*b^(1/3))
Time = 0.31 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.98, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {750, 16, 1142, 25, 27, 1082, 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.
| \(\displaystyle \int \frac {1}{a+b x^3+1} \, dx\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {\int \frac {2 \sqrt [3]{a+1}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a+1} \sqrt [3]{b} x+(a+1)^{2/3}}dx}{3 (a+1)^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a+1}}dx}{3 (a+1)^{2/3}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\int \frac {2 \sqrt [3]{a+1}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a+1} \sqrt [3]{b} x+(a+1)^{2/3}}dx}{3 (a+1)^{2/3}}+\frac {\log \left (\sqrt [3]{a+1}+\sqrt [3]{b} x\right )}{3 (a+1)^{2/3} \sqrt [3]{b}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {3}{2} \sqrt [3]{a+1} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a+1} \sqrt [3]{b} x+(a+1)^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a+1}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a+1} \sqrt [3]{b} x+(a+1)^{2/3}}dx}{2 \sqrt [3]{b}}}{3 (a+1)^{2/3}}+\frac {\log \left (\sqrt [3]{a+1}+\sqrt [3]{b} x\right )}{3 (a+1)^{2/3} \sqrt [3]{b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3}{2} \sqrt [3]{a+1} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a+1} \sqrt [3]{b} x+(a+1)^{2/3}}dx+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a+1}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a+1} \sqrt [3]{b} x+(a+1)^{2/3}}dx}{2 \sqrt [3]{b}}}{3 (a+1)^{2/3}}+\frac {\log \left (\sqrt [3]{a+1}+\sqrt [3]{b} x\right )}{3 (a+1)^{2/3} \sqrt [3]{b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3}{2} \sqrt [3]{a+1} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a+1} \sqrt [3]{b} x+(a+1)^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{a+1}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a+1} \sqrt [3]{b} x+(a+1)^{2/3}}dx}{3 (a+1)^{2/3}}+\frac {\log \left (\sqrt [3]{a+1}+\sqrt [3]{b} x\right )}{3 (a+1)^{2/3} \sqrt [3]{b}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {1}{2} \int \frac {\sqrt [3]{a+1}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a+1} \sqrt [3]{b} x+(a+1)^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+1}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+1}}\right )}{\sqrt [3]{b}}}{3 (a+1)^{2/3}}+\frac {\log \left (\sqrt [3]{a+1}+\sqrt [3]{b} x\right )}{3 (a+1)^{2/3} \sqrt [3]{b}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {1}{2} \int \frac {\sqrt [3]{a+1}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a+1} \sqrt [3]{b} x+(a+1)^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+1}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 (a+1)^{2/3}}+\frac {\log \left (\sqrt [3]{a+1}+\sqrt [3]{b} x\right )}{3 (a+1)^{2/3} \sqrt [3]{b}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+1}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}-\frac {\log \left (-\sqrt [3]{a+1} \sqrt [3]{b} x+(a+1)^{2/3}+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}}{3 (a+1)^{2/3}}+\frac {\log \left (\sqrt [3]{a+1}+\sqrt [3]{b} x\right )}{3 (a+1)^{2/3} \sqrt [3]{b}}\) |
Log[(1 + a)^(1/3) + b^(1/3)*x]/(3*(1 + a)^(2/3)*b^(1/3)) + (-((Sqrt[3]*Arc Tan[(1 - (2*b^(1/3)*x)/(1 + a)^(1/3))/Sqrt[3]])/b^(1/3)) - Log[(1 + a)^(2/ 3) - (1 + a)^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(2*b^(1/3)))/(3*(1 + a)^(2/3))
3.4.63.3.1 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_)*(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[(-(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_.)*(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_))/((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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.69 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.22
| method | result | size |
| risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a +1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 b}\) | \(28\) |
| default | \(\frac {\ln \left (x +\left (\frac {1+a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {1+a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {1+a}{b}\right )^{\frac {1}{3}} x +\left (\frac {1+a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {1+a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1+a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {1+a}{b}\right )^{\frac {2}{3}}}\) | \(105\) |
Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (90) = 180\).
Time = 0.28 (sec) , antiderivative size = 446, normalized size of antiderivative = 3.57 \[ \int \frac {1}{1+a+b x^3} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} {\left (a + 1\right )} b \sqrt {-\frac {\left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, {\left (a + 1\right )} b x^{3} - 3 \, \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}} {\left (a + 1\right )} x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (a + 1\right )} b x^{2} + \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}} x - \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}} {\left (a + 1\right )}\right )} \sqrt {-\frac {\left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}}}{b}} - 2 \, a - 1}{b x^{3} + a + 1}\right ) - \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}} \log \left ({\left (a + 1\right )} b x^{2} - \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}} x + \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}} {\left (a + 1\right )}\right ) + 2 \, \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}} \log \left ({\left (a + 1\right )} b x + \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}}\right )}{6 \, {\left (a^{2} + 2 \, a + 1\right )} b}, \frac {6 \, \sqrt {\frac {1}{3}} {\left (a + 1\right )} b \sqrt {\frac {\left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}} x - \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}} {\left (a + 1\right )}\right )} \sqrt {\frac {\left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}}}{b}}}{a^{2} + 2 \, a + 1}\right ) - \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}} \log \left ({\left (a + 1\right )} b x^{2} - \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}} x + \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}} {\left (a + 1\right )}\right ) + 2 \, \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}} \log \left ({\left (a + 1\right )} b x + \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}}\right )}{6 \, {\left (a^{2} + 2 \, a + 1\right )} b}\right ] \]
[1/6*(3*sqrt(1/3)*(a + 1)*b*sqrt(-((a^2 + 2*a + 1)*b)^(1/3)/b)*log((2*(a + 1)*b*x^3 - 3*((a^2 + 2*a + 1)*b)^(1/3)*(a + 1)*x - a^2 + 3*sqrt(1/3)*(2*( a + 1)*b*x^2 + ((a^2 + 2*a + 1)*b)^(2/3)*x - ((a^2 + 2*a + 1)*b)^(1/3)*(a + 1))*sqrt(-((a^2 + 2*a + 1)*b)^(1/3)/b) - 2*a - 1)/(b*x^3 + a + 1)) - ((a ^2 + 2*a + 1)*b)^(2/3)*log((a + 1)*b*x^2 - ((a^2 + 2*a + 1)*b)^(2/3)*x + ( (a^2 + 2*a + 1)*b)^(1/3)*(a + 1)) + 2*((a^2 + 2*a + 1)*b)^(2/3)*log((a + 1 )*b*x + ((a^2 + 2*a + 1)*b)^(2/3)))/((a^2 + 2*a + 1)*b), 1/6*(6*sqrt(1/3)* (a + 1)*b*sqrt(((a^2 + 2*a + 1)*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*((a^2 + 2* a + 1)*b)^(2/3)*x - ((a^2 + 2*a + 1)*b)^(1/3)*(a + 1))*sqrt(((a^2 + 2*a + 1)*b)^(1/3)/b)/(a^2 + 2*a + 1)) - ((a^2 + 2*a + 1)*b)^(2/3)*log((a + 1)*b* x^2 - ((a^2 + 2*a + 1)*b)^(2/3)*x + ((a^2 + 2*a + 1)*b)^(1/3)*(a + 1)) + 2 *((a^2 + 2*a + 1)*b)^(2/3)*log((a + 1)*b*x + ((a^2 + 2*a + 1)*b)^(2/3)))/( (a^2 + 2*a + 1)*b)]
Time = 0.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.26 \[ \int \frac {1}{1+a+b x^3} \, dx=\operatorname {RootSum} {\left (t^{3} \cdot \left (27 a^{2} b + 54 a b + 27 b\right ) - 1, \left ( t \mapsto t \log {\left (3 t a + 3 t + x \right )} \right )\right )} \]
Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.91 \[ \int \frac {1}{1+a+b x^3} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a + 1}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a + 1}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b \left (\frac {a + 1}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (x^{2} - x \left (\frac {a + 1}{b}\right )^{\frac {1}{3}} + \left (\frac {a + 1}{b}\right )^{\frac {2}{3}}\right )}{6 \, b \left (\frac {a + 1}{b}\right )^{\frac {2}{3}}} + \frac {\log \left (x + \left (\frac {a + 1}{b}\right )^{\frac {1}{3}}\right )}{3 \, b \left (\frac {a + 1}{b}\right )^{\frac {2}{3}}} \]
1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - ((a + 1)/b)^(1/3))/((a + 1)/b)^(1/3) )/(b*((a + 1)/b)^(2/3)) - 1/6*log(x^2 - x*((a + 1)/b)^(1/3) + ((a + 1)/b)^ (2/3))/(b*((a + 1)/b)^(2/3)) + 1/3*log(x + ((a + 1)/b)^(1/3))/(b*((a + 1)/ b)^(2/3))
Time = 0.27 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.14 \[ \int \frac {1}{1+a+b x^3} \, dx=\frac {{\left (-a b^{2} - b^{2}\right )}^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a + 1}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a + 1}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} a b + \sqrt {3} b} + \frac {{\left (-a b^{2} - b^{2}\right )}^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a + 1}{b}\right )^{\frac {1}{3}} + \left (-\frac {a + 1}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (a b + b\right )}} - \frac {\left (-\frac {a + 1}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a + 1}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (a + 1\right )}} \]
(-a*b^2 - b^2)^(1/3)*arctan(1/3*sqrt(3)*(2*x + (-(a + 1)/b)^(1/3))/(-(a + 1)/b)^(1/3))/(sqrt(3)*a*b + sqrt(3)*b) + 1/6*(-a*b^2 - b^2)^(1/3)*log(x^2 + x*(-(a + 1)/b)^(1/3) + (-(a + 1)/b)^(2/3))/(a*b + b) - 1/3*(-(a + 1)/b)^ (1/3)*log(abs(x - (-(a + 1)/b)^(1/3)))/(a + 1)
Time = 5.56 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.10 \[ \int \frac {1}{1+a+b x^3} \, dx=\frac {\ln \left (a+b^{1/3}\,x\,{\left (a+1\right )}^{2/3}+1\right )}{3\,b^{1/3}\,{\left (a+1\right )}^{2/3}}+\frac {\ln \left (3\,b^2\,x+\frac {\left (9\,a\,b^2+9\,b^2\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,b^{1/3}\,{\left (a+1\right )}^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,b^{1/3}\,{\left (a+1\right )}^{2/3}}-\frac {\ln \left (3\,b^2\,x-\frac {\left (9\,a\,b^2+9\,b^2\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,b^{1/3}\,{\left (a+1\right )}^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,b^{1/3}\,{\left (a+1\right )}^{2/3}} \]
log(a + b^(1/3)*x*(a + 1)^(2/3) + 1)/(3*b^(1/3)*(a + 1)^(2/3)) + (log(3*b^ 2*x + ((9*a*b^2 + 9*b^2)*(3^(1/2)*1i - 1))/(6*b^(1/3)*(a + 1)^(2/3)))*(3^( 1/2)*1i - 1))/(6*b^(1/3)*(a + 1)^(2/3)) - (log(3*b^2*x - ((9*a*b^2 + 9*b^2 )*(3^(1/2)*1i + 1))/(6*b^(1/3)*(a + 1)^(2/3)))*(3^(1/2)*1i + 1))/(6*b^(1/3 )*(a + 1)^(2/3))