Integrand size = 36, antiderivative size = 41 \[ \int \frac {\sqrt [3]{a} \sqrt [3]{-b} B-(-b)^{2/3} B x}{a+b x^3} \, dx=\frac {2 B \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{-b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a}} \] Output:
2/3*B*arctan(1/3*(a^(1/3)+2*(-b)^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/a^(1/3)
Leaf count is larger than twice the leaf count of optimal. \(129\) vs. \(2(41)=82\).
Time = 0.04 (sec) , antiderivative size = 129, normalized size of antiderivative = 3.15 \[ \int \frac {\sqrt [3]{a} \sqrt [3]{-b} B-(-b)^{2/3} B x}{a+b x^3} \, dx=\frac {\sqrt [3]{-b} B \left (2 \sqrt {3} \left (\sqrt [3]{-b}-\sqrt [3]{b}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+\left (\sqrt [3]{-b}+\sqrt [3]{b}\right ) \left (2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )\right )\right )}{6 \sqrt [3]{a} b^{2/3}} \] Input:
Integrate[(a^(1/3)*(-b)^(1/3)*B - (-b)^(2/3)*B*x)/(a + b*x^3),x]
Output:
((-b)^(1/3)*B*(2*Sqrt[3]*((-b)^(1/3) - b^(1/3))*ArcTan[(1 - (2*b^(1/3)*x)/ a^(1/3))/Sqrt[3]] + ((-b)^(1/3) + b^(1/3))*(2*Log[a^(1/3) + b^(1/3)*x] - L og[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])))/(6*a^(1/3)*b^(2/3))
Time = 0.31 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2019, 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.
\(\displaystyle \int \frac {\sqrt [3]{a} \sqrt [3]{-b} B-(-b)^{2/3} B x}{a+b x^3} \, dx\) |
\(\Big \downarrow \) 2019 |
\(\displaystyle \int \frac {1}{-\frac {a^{2/3} (-b)^{2/3}}{b B}+\frac {\sqrt [3]{a} x}{B}+\frac {\sqrt [3]{-b} x^2}{B}}dx\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle -\frac {2 B \int \frac {1}{-\left (\frac {2 \sqrt [3]{-b} x}{\sqrt [3]{a}}+1\right )^2-3}d\left (\frac {2 \sqrt [3]{-b} x}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{a}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {2 B \arctan \left (\frac {\frac {2 \sqrt [3]{-b} x}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a}}\) |
Input:
Int[(a^(1/3)*(-b)^(1/3)*B - (-b)^(2/3)*B*x)/(a + b*x^3),x]
Output:
(2*B*ArcTan[(1 + (2*(-b)^(1/3)*x)/a^(1/3))/Sqrt[3]])/(Sqrt[3]*a^(1/3))
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_) + (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[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(201\) vs. \(2(30)=60\).
Time = 0.24 (sec) , antiderivative size = 202, normalized size of antiderivative = 4.93
method | result | size |
default | \(-B \,b^{\frac {1}{3}} \left (-a^{\frac {1}{3}} \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (-b \right )^{\frac {1}{3}} \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )\right ) \left (-1\right )^{\frac {1}{3}}\) | \(202\) |
Input:
int((a^(1/3)*(-b)^(1/3)*B-(-b)^(2/3)*B*x)/(b*x^3+a),x,method=_RETURNVERBOS E)
Output:
-B*b^(1/3)*(-a^(1/3)*(1/3/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/6/b/(a/b)^(2/3 )*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3 ^(1/2)*(2/(a/b)^(1/3)*x-1)))+(-b)^(1/3)*(-1/3/b/(a/b)^(1/3)*ln(x+(a/b)^(1/ 3))+1/6/b/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3*3^(1/2)/b/(a/b )^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))))*(-1)^(1/3)
Time = 0.12 (sec) , antiderivative size = 114, normalized size of antiderivative = 2.78 \[ \int \frac {\sqrt [3]{a} \sqrt [3]{-b} B-(-b)^{2/3} B x}{a+b x^3} \, dx=\left [\sqrt {\frac {1}{3}} B \sqrt {-\frac {1}{a^{\frac {2}{3}}}} \log \left (\frac {2 \, b x^{3} + 3 \, a^{\frac {2}{3}} \left (-b\right )^{\frac {1}{3}} x - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}} x^{2} - a \left (-b\right )^{\frac {1}{3}} x - a^{\frac {4}{3}}\right )} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} - a}{b x^{3} + a}\right ), \frac {2 \, \sqrt {\frac {1}{3}} B \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (-b\right )^{\frac {1}{3}} x + a^{\frac {1}{3}}\right )}}{a^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}}}\right ] \] Input:
integrate((a^(1/3)*(-b)^(1/3)*B-(-b)^(2/3)*B*x)/(b*x^3+a),x, algorithm="fr icas")
Output:
[sqrt(1/3)*B*sqrt(-1/a^(2/3))*log((2*b*x^3 + 3*a^(2/3)*(-b)^(1/3)*x - 3*sq rt(1/3)*(2*a^(2/3)*(-b)^(2/3)*x^2 - a*(-b)^(1/3)*x - a^(4/3))*sqrt(-1/a^(2 /3)) - a)/(b*x^3 + a)), 2*sqrt(1/3)*B*arctan(sqrt(1/3)*(2*(-b)^(1/3)*x + a ^(1/3))/a^(1/3))/a^(1/3)]
Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 105, normalized size of antiderivative = 2.56 \[ \int \frac {\sqrt [3]{a} \sqrt [3]{-b} B-(-b)^{2/3} B x}{a+b x^3} \, dx=- \frac {B \left (- \frac {\sqrt {3} i \log {\left (- \frac {\sqrt [3]{a} \left (- b\right )^{\frac {2}{3}}}{2 b} - \frac {\sqrt {3} i \sqrt [3]{a} \left (- b\right )^{\frac {2}{3}}}{2 b} + x \right )}}{3} + \frac {\sqrt {3} i \log {\left (- \frac {\sqrt [3]{a} \left (- b\right )^{\frac {2}{3}}}{2 b} + \frac {\sqrt {3} i \sqrt [3]{a} \left (- b\right )^{\frac {2}{3}}}{2 b} + x \right )}}{3}\right )}{\sqrt [3]{a}} \] Input:
integrate((a**(1/3)*(-b)**(1/3)*B-(-b)**(2/3)*B*x)/(b*x**3+a),x)
Output:
-B*(-sqrt(3)*I*log(-a**(1/3)*(-b)**(2/3)/(2*b) - sqrt(3)*I*a**(1/3)*(-b)** (2/3)/(2*b) + x)/3 + sqrt(3)*I*log(-a**(1/3)*(-b)**(2/3)/(2*b) + sqrt(3)*I *a**(1/3)*(-b)**(2/3)/(2*b) + x)/3)/a**(1/3)
Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (30) = 60\).
Time = 0.14 (sec) , antiderivative size = 174, normalized size of antiderivative = 4.24 \[ \int \frac {\sqrt [3]{a} \sqrt [3]{-b} B-(-b)^{2/3} B x}{a+b x^3} \, dx=-\frac {\sqrt {3} {\left (B \left (-b\right )^{\frac {2}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} - B a^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (B \left (-b\right )^{\frac {2}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + B a^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}}\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (B \left (-b\right )^{\frac {2}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + B a^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}}\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate((a^(1/3)*(-b)^(1/3)*B-(-b)^(2/3)*B*x)/(b*x^3+a),x, algorithm="ma xima")
Output:
-1/3*sqrt(3)*(B*(-b)^(2/3)*(a/b)^(1/3) - B*a^(1/3)*(-b)^(1/3))*arctan(1/3* sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b*(a/b)^(2/3)) - 1/6*(B*(-b)^(2/ 3)*(a/b)^(1/3) + B*a^(1/3)*(-b)^(1/3))*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/ 3))/(b*(a/b)^(2/3)) + 1/3*(B*(-b)^(2/3)*(a/b)^(1/3) + B*a^(1/3)*(-b)^(1/3) )*log(x + (a/b)^(1/3))/(b*(a/b)^(2/3))
Time = 0.14 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.39 \[ \int \frac {\sqrt [3]{a} \sqrt [3]{-b} B-(-b)^{2/3} B x}{a+b x^3} \, dx=-\frac {2 \, \sqrt {3} B \left (-b\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} {\left (2 \, \left (-b\right )^{\frac {2}{3}} x + a^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}}\right )}}{3 \, \sqrt {a^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}}}}\right )}{3 \, \sqrt {a^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}}}} \] Input:
integrate((a^(1/3)*(-b)^(1/3)*B-(-b)^(2/3)*B*x)/(b*x^3+a),x, algorithm="gi ac")
Output:
-2/3*sqrt(3)*B*(-b)^(1/3)*arctan(-1/3*sqrt(3)*(2*(-b)^(2/3)*x + a^(1/3)*(- b)^(1/3))/sqrt(a^(2/3)*(-b)^(2/3)))/sqrt(a^(2/3)*(-b)^(2/3))
Time = 5.73 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt [3]{a} \sqrt [3]{-b} B-(-b)^{2/3} B x}{a+b x^3} \, dx=-\frac {2\,\sqrt {3}\,B\,\sqrt {-b}\,\mathrm {atanh}\left (\frac {\sqrt {3}\,\sqrt {-b}}{3\,\sqrt {b}}-\frac {2\,\sqrt {3}\,\sqrt {b}\,x}{3\,a^{1/3}\,{\left (-b\right )}^{1/6}}\right )}{3\,a^{1/3}\,\sqrt {b}} \] Input:
int(-(B*(-b)^(2/3)*x - B*a^(1/3)*(-b)^(1/3))/(a + b*x^3),x)
Output:
-(2*3^(1/2)*B*(-b)^(1/2)*atanh((3^(1/2)*(-b)^(1/2))/(3*b^(1/2)) - (2*3^(1/ 2)*b^(1/2)*x)/(3*a^(1/3)*(-b)^(1/6))))/(3*a^(1/3)*b^(1/2))
Time = 0.15 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt [3]{a} \sqrt [3]{-b} B-(-b)^{2/3} B x}{a+b x^3} \, dx=\frac {2 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b}{3 a^{\frac {1}{3}}} \] Input:
int((a^(1/3)*(-b)^(1/3)*B-(-b)^(2/3)*B*x)/(b*x^3+a),x)
Output:
(2*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b)/(3*a**(1/ 3))