Integrand size = 20, antiderivative size = 85 \[ \int \frac {a-b x^3}{\left (a+b x^3\right )^{4/3}} \, dx=\frac {2 x}{\sqrt [3]{a+b x^3}}-\frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}+\frac {\log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{b}} \] Output:
2*x/(b*x^3+a)^(1/3)-1/3*arctan(1/3*(1+2*b^(1/3)*x/(b*x^3+a)^(1/3))*3^(1/2) )*3^(1/2)/b^(1/3)+1/2*ln(-b^(1/3)*x+(b*x^3+a)^(1/3))/b^(1/3)
Time = 0.33 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.67 \[ \int \frac {a-b x^3}{\left (a+b x^3\right )^{4/3}} \, dx=\frac {2 x}{\sqrt [3]{a+b x^3}}-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )}{\sqrt {3} \sqrt [3]{b}}+\frac {\log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{3 \sqrt [3]{b}}-\frac {\log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{6 \sqrt [3]{b}} \] Input:
Integrate[(a - b*x^3)/(a + b*x^3)^(4/3),x]
Output:
(2*x)/(a + b*x^3)^(1/3) - ArcTan[(Sqrt[3]*b^(1/3)*x)/(b^(1/3)*x + 2*(a + b *x^3)^(1/3))]/(Sqrt[3]*b^(1/3)) + Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)]/(3 *b^(1/3)) - Log[b^(2/3)*x^2 + b^(1/3)*x*(a + b*x^3)^(1/3) + (a + b*x^3)^(2 /3)]/(6*b^(1/3))
Time = 0.29 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {910, 769}
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.
\(\displaystyle \int \frac {a-b x^3}{\left (a+b x^3\right )^{4/3}} \, dx\) |
\(\Big \downarrow \) 910 |
\(\displaystyle \frac {2 x}{\sqrt [3]{a+b x^3}}-\int \frac {1}{\sqrt [3]{b x^3+a}}dx\) |
\(\Big \downarrow \) 769 |
\(\displaystyle -\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}+\frac {2 x}{\sqrt [3]{a+b x^3}}+\frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}\) |
Input:
Int[(a - b*x^3)/(a + b*x^3)^(4/3),x]
Output:
(2*x)/(a + b*x^3)^(1/3) - ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqr t[3]]/(Sqrt[3]*b^(1/3)) + Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)]/(2*b^(1/3) )
Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]* (x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^ 3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[(-(b*c - a*d))*x*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] - Simp[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)) Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/ n + p, 0])
Time = 0.90 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.34
method | result | size |
pseudoelliptic | \(\frac {2 x}{\left (b \,x^{3}+a \right )^{\frac {1}{3}}}+\frac {\ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )}{3 b^{\frac {1}{3}}}-\frac {\ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{6 b^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}+x \right )}{3 x}\right )}{3 b^{\frac {1}{3}}}\) | \(114\) |
Input:
int((-b*x^3+a)/(b*x^3+a)^(4/3),x,method=_RETURNVERBOSE)
Output:
2*x/(b*x^3+a)^(1/3)+1/3/b^(1/3)*ln((-b^(1/3)*x+(b*x^3+a)^(1/3))/x)-1/6/b^( 1/3)*ln((b^(2/3)*x^2+b^(1/3)*(b*x^3+a)^(1/3)*x+(b*x^3+a)^(2/3))/x^2)+1/3*3 ^(1/2)/b^(1/3)*arctan(1/3*3^(1/2)*(2*(b*x^3+a)^(1/3)/b^(1/3)+x)/x)
Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (66) = 132\).
Time = 0.09 (sec) , antiderivative size = 372, normalized size of antiderivative = 4.38 \[ \int \frac {a-b x^3}{\left (a+b x^3\right )^{4/3}} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} {\left (b^{2} x^{3} + a b\right )} \sqrt {-\frac {1}{b^{\frac {2}{3}}}} \log \left (3 \, b x^{3} - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {2}{3}} x^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (b^{\frac {4}{3}} x^{3} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x^{2} - 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b^{\frac {2}{3}} x\right )} \sqrt {-\frac {1}{b^{\frac {2}{3}}}} + 2 \, a\right ) + 12 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b x + 2 \, {\left (b x^{3} + a\right )} b^{\frac {2}{3}} \log \left (-\frac {b^{\frac {1}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - {\left (b x^{3} + a\right )} b^{\frac {2}{3}} \log \left (\frac {b^{\frac {2}{3}} x^{2} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{6 \, {\left (b^{2} x^{3} + a b\right )}}, \frac {12 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b x + 2 \, {\left (b x^{3} + a\right )} b^{\frac {2}{3}} \log \left (-\frac {b^{\frac {1}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - {\left (b x^{3} + a\right )} b^{\frac {2}{3}} \log \left (\frac {b^{\frac {2}{3}} x^{2} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) + \frac {6 \, \sqrt {\frac {1}{3}} {\left (b^{2} x^{3} + a b\right )} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (b^{\frac {1}{3}} x + 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )}}{b^{\frac {1}{3}} x}\right )}{b^{\frac {1}{3}}}}{6 \, {\left (b^{2} x^{3} + a b\right )}}\right ] \] Input:
integrate((-b*x^3+a)/(b*x^3+a)^(4/3),x, algorithm="fricas")
Output:
[1/6*(3*sqrt(1/3)*(b^2*x^3 + a*b)*sqrt(-1/b^(2/3))*log(3*b*x^3 - 3*(b*x^3 + a)^(1/3)*b^(2/3)*x^2 - 3*sqrt(1/3)*(b^(4/3)*x^3 + (b*x^3 + a)^(1/3)*b*x^ 2 - 2*(b*x^3 + a)^(2/3)*b^(2/3)*x)*sqrt(-1/b^(2/3)) + 2*a) + 12*(b*x^3 + a )^(2/3)*b*x + 2*(b*x^3 + a)*b^(2/3)*log(-(b^(1/3)*x - (b*x^3 + a)^(1/3))/x ) - (b*x^3 + a)*b^(2/3)*log((b^(2/3)*x^2 + (b*x^3 + a)^(1/3)*b^(1/3)*x + ( b*x^3 + a)^(2/3))/x^2))/(b^2*x^3 + a*b), 1/6*(12*(b*x^3 + a)^(2/3)*b*x + 2 *(b*x^3 + a)*b^(2/3)*log(-(b^(1/3)*x - (b*x^3 + a)^(1/3))/x) - (b*x^3 + a) *b^(2/3)*log((b^(2/3)*x^2 + (b*x^3 + a)^(1/3)*b^(1/3)*x + (b*x^3 + a)^(2/3 ))/x^2) + 6*sqrt(1/3)*(b^2*x^3 + a*b)*arctan(sqrt(1/3)*(b^(1/3)*x + 2*(b*x ^3 + a)^(1/3))/(b^(1/3)*x))/b^(1/3))/(b^2*x^3 + a*b)]
Result contains complex when optimal does not.
Time = 3.23 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.82 \[ \int \frac {a-b x^3}{\left (a+b x^3\right )^{4/3}} \, dx=\frac {x \Gamma \left (\frac {1}{3}\right )}{3 \sqrt [3]{a} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {4}{3}\right )} - \frac {b x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {4}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {4}{3}} \Gamma \left (\frac {7}{3}\right )} \] Input:
integrate((-b*x**3+a)/(b*x**3+a)**(4/3),x)
Output:
x*gamma(1/3)/(3*a**(1/3)*(1 + b*x**3/a)**(1/3)*gamma(4/3)) - b*x**4*gamma( 4/3)*hyper((4/3, 4/3), (7/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(4/3)*gamma (7/3))
Time = 0.11 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.53 \[ \int \frac {a-b x^3}{\left (a+b x^3\right )^{4/3}} \, dx=\frac {1}{6} \, b {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {4}{3}}} + \frac {6 \, x}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} b} - \frac {\log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{b^{\frac {4}{3}}} + \frac {2 \, \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {4}{3}}}\right )} + \frac {x}{{\left (b x^{3} + a\right )}^{\frac {1}{3}}} \] Input:
integrate((-b*x^3+a)/(b*x^3+a)^(4/3),x, algorithm="maxima")
Output:
1/6*b*(2*sqrt(3)*arctan(1/3*sqrt(3)*(b^(1/3) + 2*(b*x^3 + a)^(1/3)/x)/b^(1 /3))/b^(4/3) + 6*x/((b*x^3 + a)^(1/3)*b) - log(b^(2/3) + (b*x^3 + a)^(1/3) *b^(1/3)/x + (b*x^3 + a)^(2/3)/x^2)/b^(4/3) + 2*log(-b^(1/3) + (b*x^3 + a) ^(1/3)/x)/b^(4/3)) + x/(b*x^3 + a)^(1/3)
\[ \int \frac {a-b x^3}{\left (a+b x^3\right )^{4/3}} \, dx=\int { -\frac {b x^{3} - a}{{\left (b x^{3} + a\right )}^{\frac {4}{3}}} \,d x } \] Input:
integrate((-b*x^3+a)/(b*x^3+a)^(4/3),x, algorithm="giac")
Output:
integrate(-(b*x^3 - a)/(b*x^3 + a)^(4/3), x)
Timed out. \[ \int \frac {a-b x^3}{\left (a+b x^3\right )^{4/3}} \, dx=\int \frac {a-b\,x^3}{{\left (b\,x^3+a\right )}^{4/3}} \,d x \] Input:
int((a - b*x^3)/(a + b*x^3)^(4/3),x)
Output:
int((a - b*x^3)/(a + b*x^3)^(4/3), x)
\[ \int \frac {a-b x^3}{\left (a+b x^3\right )^{4/3}} \, dx=-\left (\int \frac {x^{3}}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} a +\left (b \,x^{3}+a \right )^{\frac {1}{3}} b \,x^{3}}d x \right ) b +\left (\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} a +\left (b \,x^{3}+a \right )^{\frac {1}{3}} b \,x^{3}}d x \right ) a \] Input:
int((-b*x^3+a)/(b*x^3+a)^(4/3),x)
Output:
- int(x**3/((a + b*x**3)**(1/3)*a + (a + b*x**3)**(1/3)*b*x**3),x)*b + in t(1/((a + b*x**3)**(1/3)*a + (a + b*x**3)**(1/3)*b*x**3),x)*a