Integrand size = 9, antiderivative size = 115 \[ \int \frac {1}{a+d x^3} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{d}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{d} x\right )}{3 a^{2/3} \sqrt [3]{d}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{d}} \] Output:
-1/3*arctan(1/3*(a^(1/3)-2*d^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/a^(2/3)/d^( 1/3)+1/3*ln(a^(1/3)+d^(1/3)*x)/a^(2/3)/d^(1/3)-1/6*ln(a^(2/3)-a^(1/3)*d^(1 /3)*x+d^(2/3)*x^2)/a^(2/3)/d^(1/3)
Time = 0.02 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.77 \[ \int \frac {1}{a+d x^3} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{d} x\right )+\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{d}} \] Input:
Integrate[(a + d*x^3)^(-1),x]
Output:
-1/6*(2*Sqrt[3]*ArcTan[(1 - (2*d^(1/3)*x)/a^(1/3))/Sqrt[3]] - 2*Log[a^(1/3 ) + d^(1/3)*x] + Log[a^(2/3) - a^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(a^(2/3)* d^(1/3))
Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.97, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, 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.
\(\displaystyle \int \frac {1}{a+d x^3} \, dx\) |
\(\Big \downarrow \) 750 |
\(\displaystyle \frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{d} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{d} x+\sqrt [3]{a}}dx}{3 a^{2/3}}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{d} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{d} x\right )}{3 a^{2/3} \sqrt [3]{d}}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{d} x+a^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{d} \left (\sqrt [3]{a}-2 \sqrt [3]{d} x\right )}{d^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{d} x+a^{2/3}}dx}{2 \sqrt [3]{d}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{d} x\right )}{3 a^{2/3} \sqrt [3]{d}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{d} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [3]{d} \left (\sqrt [3]{a}-2 \sqrt [3]{d} x\right )}{d^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{d} x+a^{2/3}}dx}{2 \sqrt [3]{d}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{d} x\right )}{3 a^{2/3} \sqrt [3]{d}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{d} x+a^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{d} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{d} x\right )}{3 a^{2/3} \sqrt [3]{d}}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{d} x+a^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{d}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{d} x\right )}{3 a^{2/3} \sqrt [3]{d}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{d} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{d} x\right )}{3 a^{2/3} \sqrt [3]{d}}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{d} x+d^{2/3} x^2\right )}{2 \sqrt [3]{d}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{d} x\right )}{3 a^{2/3} \sqrt [3]{d}}\) |
Input:
Int[(a + d*x^3)^(-1),x]
Output:
Log[a^(1/3) + d^(1/3)*x]/(3*a^(2/3)*d^(1/3)) + (-((Sqrt[3]*ArcTan[(1 - (2* d^(1/3)*x)/a^(1/3))/Sqrt[3]])/d^(1/3)) - Log[a^(2/3) - a^(1/3)*d^(1/3)*x + d^(2/3)*x^2]/(2*d^(1/3)))/(3*a^(2/3))
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 = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.23
method | result | size |
risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 d}\) | \(27\) |
default | \(\frac {\ln \left (x +\left (\frac {a}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {a}{d}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{d}\right )^{\frac {1}{3}} x +\left (\frac {a}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {a}{d}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {a}{d}\right )^{\frac {2}{3}}}\) | \(91\) |
Input:
int(1/(d*x^3+a),x,method=_RETURNVERBOSE)
Output:
1/3/d*sum(1/_R^2*ln(x-_R),_R=RootOf(_Z^3*d+a))
Time = 0.09 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.60 \[ \int \frac {1}{a+d x^3} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} a d \sqrt {-\frac {\left (a^{2} d\right )^{\frac {1}{3}}}{d}} \log \left (\frac {2 \, a d x^{3} - 3 \, \left (a^{2} d\right )^{\frac {1}{3}} a x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a d x^{2} + \left (a^{2} d\right )^{\frac {2}{3}} x - \left (a^{2} d\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} d\right )^{\frac {1}{3}}}{d}}}{d x^{3} + a}\right ) - \left (a^{2} d\right )^{\frac {2}{3}} \log \left (a d x^{2} - \left (a^{2} d\right )^{\frac {2}{3}} x + \left (a^{2} d\right )^{\frac {1}{3}} a\right ) + 2 \, \left (a^{2} d\right )^{\frac {2}{3}} \log \left (a d x + \left (a^{2} d\right )^{\frac {2}{3}}\right )}{6 \, a^{2} d}, \frac {6 \, \sqrt {\frac {1}{3}} a d \sqrt {\frac {\left (a^{2} d\right )^{\frac {1}{3}}}{d}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} d\right )^{\frac {2}{3}} x - \left (a^{2} d\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} d\right )^{\frac {1}{3}}}{d}}}{a^{2}}\right ) - \left (a^{2} d\right )^{\frac {2}{3}} \log \left (a d x^{2} - \left (a^{2} d\right )^{\frac {2}{3}} x + \left (a^{2} d\right )^{\frac {1}{3}} a\right ) + 2 \, \left (a^{2} d\right )^{\frac {2}{3}} \log \left (a d x + \left (a^{2} d\right )^{\frac {2}{3}}\right )}{6 \, a^{2} d}\right ] \] Input:
integrate(1/(d*x^3+a),x, algorithm="fricas")
Output:
[1/6*(3*sqrt(1/3)*a*d*sqrt(-(a^2*d)^(1/3)/d)*log((2*a*d*x^3 - 3*(a^2*d)^(1 /3)*a*x - a^2 + 3*sqrt(1/3)*(2*a*d*x^2 + (a^2*d)^(2/3)*x - (a^2*d)^(1/3)*a )*sqrt(-(a^2*d)^(1/3)/d))/(d*x^3 + a)) - (a^2*d)^(2/3)*log(a*d*x^2 - (a^2* d)^(2/3)*x + (a^2*d)^(1/3)*a) + 2*(a^2*d)^(2/3)*log(a*d*x + (a^2*d)^(2/3)) )/(a^2*d), 1/6*(6*sqrt(1/3)*a*d*sqrt((a^2*d)^(1/3)/d)*arctan(sqrt(1/3)*(2* (a^2*d)^(2/3)*x - (a^2*d)^(1/3)*a)*sqrt((a^2*d)^(1/3)/d)/a^2) - (a^2*d)^(2 /3)*log(a*d*x^2 - (a^2*d)^(2/3)*x + (a^2*d)^(1/3)*a) + 2*(a^2*d)^(2/3)*log (a*d*x + (a^2*d)^(2/3)))/(a^2*d)]
Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.17 \[ \int \frac {1}{a+d x^3} \, dx=\operatorname {RootSum} {\left (27 t^{3} a^{2} d - 1, \left ( t \mapsto t \log {\left (3 t a + x \right )} \right )\right )} \] Input:
integrate(1/(d*x**3+a),x)
Output:
RootSum(27*_t**3*a**2*d - 1, Lambda(_t, _t*log(3*_t*a + x)))
Time = 0.12 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.85 \[ \int \frac {1}{a+d x^3} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{d}\right )^{\frac {1}{3}}}\right )}{3 \, d \left (\frac {a}{d}\right )^{\frac {2}{3}}} - \frac {\log \left (x^{2} - x \left (\frac {a}{d}\right )^{\frac {1}{3}} + \left (\frac {a}{d}\right )^{\frac {2}{3}}\right )}{6 \, d \left (\frac {a}{d}\right )^{\frac {2}{3}}} + \frac {\log \left (x + \left (\frac {a}{d}\right )^{\frac {1}{3}}\right )}{3 \, d \left (\frac {a}{d}\right )^{\frac {2}{3}}} \] Input:
integrate(1/(d*x^3+a),x, algorithm="maxima")
Output:
1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - (a/d)^(1/3))/(a/d)^(1/3))/(d*(a/d)^( 2/3)) - 1/6*log(x^2 - x*(a/d)^(1/3) + (a/d)^(2/3))/(d*(a/d)^(2/3)) + 1/3*l og(x + (a/d)^(1/3))/(d*(a/d)^(2/3))
Time = 0.14 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int \frac {1}{a+d x^3} \, dx=-\frac {\left (-\frac {a}{d}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a} + \frac {\sqrt {3} \left (-a d^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{d}\right )^{\frac {1}{3}}}\right )}{3 \, a d} + \frac {\left (-a d^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{d}\right )^{\frac {1}{3}} + \left (-\frac {a}{d}\right )^{\frac {2}{3}}\right )}{6 \, a d} \] Input:
integrate(1/(d*x^3+a),x, algorithm="giac")
Output:
-1/3*(-a/d)^(1/3)*log(abs(x - (-a/d)^(1/3)))/a + 1/3*sqrt(3)*(-a*d^2)^(1/3 )*arctan(1/3*sqrt(3)*(2*x + (-a/d)^(1/3))/(-a/d)^(1/3))/(a*d) + 1/6*(-a*d^ 2)^(1/3)*log(x^2 + x*(-a/d)^(1/3) + (-a/d)^(2/3))/(a*d)
Time = 0.13 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.86 \[ \int \frac {1}{a+d x^3} \, dx=\frac {\ln \left (d^{1/3}\,x+a^{1/3}\right )}{3\,a^{2/3}\,d^{1/3}}+\frac {\ln \left (3\,d^2\,x+\frac {3\,a^{1/3}\,d^{5/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{2/3}\,d^{1/3}}-\frac {\ln \left (3\,d^2\,x-\frac {3\,a^{1/3}\,d^{5/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{2/3}\,d^{1/3}} \] Input:
int(1/(a + d*x^3),x)
Output:
log(d^(1/3)*x + a^(1/3))/(3*a^(2/3)*d^(1/3)) + (log(3*d^2*x + (3*a^(1/3)*d ^(5/3)*(3^(1/2)*1i - 1))/2)*(3^(1/2)*1i - 1))/(6*a^(2/3)*d^(1/3)) - (log(3 *d^2*x - (3*a^(1/3)*d^(5/3)*(3^(1/2)*1i + 1))/2)*(3^(1/2)*1i + 1))/(6*a^(2 /3)*d^(1/3))
Time = 0.15 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.58 \[ \int \frac {1}{a+d x^3} \, dx=\frac {-2 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 d^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right )-\mathrm {log}\left (a^{\frac {2}{3}}-d^{\frac {1}{3}} a^{\frac {1}{3}} x +d^{\frac {2}{3}} x^{2}\right )+2 \,\mathrm {log}\left (a^{\frac {1}{3}}+d^{\frac {1}{3}} x \right )}{6 a^{\frac {2}{3}} d^{\frac {1}{3}}} \] Input:
int(1/(d*x^3+a),x)
Output:
(a**(1/3)*( - 2*sqrt(3)*atan((a**(1/3) - 2*d**(1/3)*x)/(a**(1/3)*sqrt(3))) - log(a**(2/3) - d**(1/3)*a**(1/3)*x + d**(2/3)*x**2) + 2*log(a**(1/3) + d**(1/3)*x)))/(6*d**(1/3)*a)