\(\int \frac {c-2 d x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx\) [128]

Optimal result

Integrand size = 30, antiderivative size = 46 \[ \int \frac {c-2 d x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx=-\frac {2 \text {arctanh}\left (\frac {(c-2 d x)^2}{3 \sqrt {c} \sqrt {c^3-8 d^3 x^3}}\right )}{3 \sqrt {c} d} \] Output:

-2/3*arctanh(1/3*(-2*d*x+c)^2/c^(1/2)/(-8*d^3*x^3+c^3)^(1/2))/c^(1/2)/d
 

Mathematica [A] (verified)

Time = 1.57 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.96 \[ \int \frac {c-2 d x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx=-\frac {2 \text {arctanh}\left (\frac {3 \sqrt {c} \sqrt {c^3-8 d^3 x^3}}{(c-2 d x)^2}\right )}{3 \sqrt {c} d} \] Input:

Integrate[(c - 2*d*x)/((c + d*x)*Sqrt[c^3 - 8*d^3*x^3]),x]
 

Output:

(-2*ArcTanh[(3*Sqrt[c]*Sqrt[c^3 - 8*d^3*x^3])/(c - 2*d*x)^2])/(3*Sqrt[c]*d 
)
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2563, 219}

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.

\(\displaystyle \int \frac {c-2 d x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx\)

\(\Big \downarrow \) 2563

\(\displaystyle -\frac {2 c \int \frac {1}{9-\frac {(c-2 d x)^4}{c \left (c^3-8 d^3 x^3\right )}}d\frac {(c-2 d x)^2}{c^2 \sqrt {c^3-8 d^3 x^3}}}{d}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {2 \text {arctanh}\left (\frac {(c-2 d x)^2}{3 \sqrt {c} \sqrt {c^3-8 d^3 x^3}}\right )}{3 \sqrt {c} d}\)

Input:

Int[(c - 2*d*x)/((c + d*x)*Sqrt[c^3 - 8*d^3*x^3]),x]
 

Output:

(-2*ArcTanh[(c - 2*d*x)^2/(3*Sqrt[c]*Sqrt[c^3 - 8*d^3*x^3])])/(3*Sqrt[c]*d 
)
 

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 2563

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ 
Symbol] :> Simp[-2*(e/d)   Subst[Int[1/(9 - a*x^2), x], x, (1 + f*(x/e))^2/ 
Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] & 
& EqQ[b*c^3 + 8*a*d^3, 0] && EqQ[2*d*e + c*f, 0]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.63 (sec) , antiderivative size = 503, normalized size of antiderivative = 10.93

method	result	size
default	\(-\frac {4 \left (\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}\right ) \sqrt {\frac {x -\frac {c}{2 d}}{\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}}}\, \sqrt {\frac {x -\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {c}{2 d}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\, \sqrt {\frac {x -\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {c}{2 d}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -\frac {c}{2 d}}{\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}}}, \sqrt {\frac {\frac {c}{2 d}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {c}{2 d}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\right )}{\sqrt {-8 d^{3} x^{3}+c^{3}}}+\frac {4 \left (\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}\right ) \sqrt {\frac {x -\frac {c}{2 d}}{\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}}}\, \sqrt {\frac {x -\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {c}{2 d}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\, \sqrt {\frac {x -\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {c}{2 d}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -\frac {c}{2 d}}{\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}}}, \frac {2 \left (\frac {c}{2 d}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}\right ) d}{3 c}, \sqrt {\frac {\frac {c}{2 d}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {c}{2 d}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\right )}{\sqrt {-8 d^{3} x^{3}+c^{3}}}\)	\(503\)
elliptic	\(-\frac {4 \left (\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}\right ) \sqrt {\frac {x -\frac {c}{2 d}}{\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}}}\, \sqrt {\frac {x -\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {c}{2 d}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\, \sqrt {\frac {x -\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {c}{2 d}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -\frac {c}{2 d}}{\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}}}, \sqrt {\frac {\frac {c}{2 d}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {c}{2 d}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\right )}{\sqrt {-8 d^{3} x^{3}+c^{3}}}+\frac {4 \left (\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}\right ) \sqrt {\frac {x -\frac {c}{2 d}}{\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}}}\, \sqrt {\frac {x -\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {c}{2 d}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\, \sqrt {\frac {x -\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {c}{2 d}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -\frac {c}{2 d}}{\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}-\frac {c}{2 d}}}, \frac {2 \left (\frac {c}{2 d}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}\right ) d}{3 c}, \sqrt {\frac {\frac {c}{2 d}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) c}{2 d}}{\frac {c}{2 d}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) c}{2 d}}}\right )}{\sqrt {-8 d^{3} x^{3}+c^{3}}}\)	\(503\)

Input:

int((-2*d*x+c)/(d*x+c)/(-8*d^3*x^3+c^3)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-4*(1/2*(-1/2-1/2*I*3^(1/2))*c/d-1/2*c/d)*((x-1/2*c/d)/(1/2*(-1/2-1/2*I*3^ 
(1/2))*c/d-1/2*c/d))^(1/2)*((x-1/2*(-1/2+1/2*I*3^(1/2))*c/d)/(1/2*c/d-1/2* 
(-1/2+1/2*I*3^(1/2))*c/d))^(1/2)*((x-1/2*(-1/2-1/2*I*3^(1/2))*c/d)/(1/2*c/ 
d-1/2*(-1/2-1/2*I*3^(1/2))*c/d))^(1/2)/(-8*d^3*x^3+c^3)^(1/2)*EllipticF((( 
x-1/2*c/d)/(1/2*(-1/2-1/2*I*3^(1/2))*c/d-1/2*c/d))^(1/2),((1/2*c/d-1/2*(-1 
/2-1/2*I*3^(1/2))*c/d)/(1/2*c/d-1/2*(-1/2+1/2*I*3^(1/2))*c/d))^(1/2))+4*(1 
/2*(-1/2-1/2*I*3^(1/2))*c/d-1/2*c/d)*((x-1/2*c/d)/(1/2*(-1/2-1/2*I*3^(1/2) 
)*c/d-1/2*c/d))^(1/2)*((x-1/2*(-1/2+1/2*I*3^(1/2))*c/d)/(1/2*c/d-1/2*(-1/2 
+1/2*I*3^(1/2))*c/d))^(1/2)*((x-1/2*(-1/2-1/2*I*3^(1/2))*c/d)/(1/2*c/d-1/2 
*(-1/2-1/2*I*3^(1/2))*c/d))^(1/2)/(-8*d^3*x^3+c^3)^(1/2)*EllipticPi(((x-1/ 
2*c/d)/(1/2*(-1/2-1/2*I*3^(1/2))*c/d-1/2*c/d))^(1/2),2/3*(1/2*c/d-1/2*(-1/ 
2-1/2*I*3^(1/2))*c/d)/c*d,((1/2*c/d-1/2*(-1/2-1/2*I*3^(1/2))*c/d)/(1/2*c/d 
-1/2*(-1/2+1/2*I*3^(1/2))*c/d))^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (38) = 76\).

Time = 0.17 (sec) , antiderivative size = 294, normalized size of antiderivative = 6.39 \[ \int \frac {c-2 d x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx=\left [\frac {\log \left (\frac {8 \, d^{6} x^{6} - 240 \, c d^{5} x^{5} + 408 \, c^{2} d^{4} x^{4} + 88 \, c^{3} d^{3} x^{3} + 156 \, c^{4} d^{2} x^{2} + 12 \, c^{5} d x + 17 \, c^{6} - 3 \, {\left (8 \, d^{4} x^{4} - 52 \, c d^{3} x^{3} + 12 \, c^{2} d^{2} x^{2} - 4 \, c^{3} d x + 5 \, c^{4}\right )} \sqrt {-8 \, d^{3} x^{3} + c^{3}} \sqrt {c}}{d^{6} x^{6} + 6 \, c d^{5} x^{5} + 15 \, c^{2} d^{4} x^{4} + 20 \, c^{3} d^{3} x^{3} + 15 \, c^{4} d^{2} x^{2} + 6 \, c^{5} d x + c^{6}}\right )}{6 \, \sqrt {c} d}, -\frac {\sqrt {-c} \arctan \left (\frac {{\left (4 \, d^{3} x^{3} - 24 \, c d^{2} x^{2} - 6 \, c^{2} d x - 5 \, c^{3}\right )} \sqrt {-8 \, d^{3} x^{3} + c^{3}} \sqrt {-c}}{3 \, {\left (16 \, c d^{4} x^{4} - 8 \, c^{2} d^{3} x^{3} - 2 \, c^{4} d x + c^{5}\right )}}\right )}{3 \, c d}\right ] \] Input:

integrate((-2*d*x+c)/(d*x+c)/(-8*d^3*x^3+c^3)^(1/2),x, algorithm="fricas")
 

Output:

[1/6*log((8*d^6*x^6 - 240*c*d^5*x^5 + 408*c^2*d^4*x^4 + 88*c^3*d^3*x^3 + 1 
56*c^4*d^2*x^2 + 12*c^5*d*x + 17*c^6 - 3*(8*d^4*x^4 - 52*c*d^3*x^3 + 12*c^ 
2*d^2*x^2 - 4*c^3*d*x + 5*c^4)*sqrt(-8*d^3*x^3 + c^3)*sqrt(c))/(d^6*x^6 + 
6*c*d^5*x^5 + 15*c^2*d^4*x^4 + 20*c^3*d^3*x^3 + 15*c^4*d^2*x^2 + 6*c^5*d*x 
 + c^6))/(sqrt(c)*d), -1/3*sqrt(-c)*arctan(1/3*(4*d^3*x^3 - 24*c*d^2*x^2 - 
 6*c^2*d*x - 5*c^3)*sqrt(-8*d^3*x^3 + c^3)*sqrt(-c)/(16*c*d^4*x^4 - 8*c^2* 
d^3*x^3 - 2*c^4*d*x + c^5))/(c*d)]
 

Sympy [F]

\[ \int \frac {c-2 d x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx=- \int \left (- \frac {c}{c \sqrt {c^{3} - 8 d^{3} x^{3}} + d x \sqrt {c^{3} - 8 d^{3} x^{3}}}\right )\, dx - \int \frac {2 d x}{c \sqrt {c^{3} - 8 d^{3} x^{3}} + d x \sqrt {c^{3} - 8 d^{3} x^{3}}}\, dx \] Input:

integrate((-2*d*x+c)/(d*x+c)/(-8*d**3*x**3+c**3)**(1/2),x)
                                                                                    
                                                                                    
 

Output:

-Integral(-c/(c*sqrt(c**3 - 8*d**3*x**3) + d*x*sqrt(c**3 - 8*d**3*x**3)), 
x) - Integral(2*d*x/(c*sqrt(c**3 - 8*d**3*x**3) + d*x*sqrt(c**3 - 8*d**3*x 
**3)), x)
 

Maxima [F]

\[ \int \frac {c-2 d x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx=\int { -\frac {2 \, d x - c}{\sqrt {-8 \, d^{3} x^{3} + c^{3}} {\left (d x + c\right )}} \,d x } \] Input:

integrate((-2*d*x+c)/(d*x+c)/(-8*d^3*x^3+c^3)^(1/2),x, algorithm="maxima")
 

Output:

-integrate((2*d*x - c)/(sqrt(-8*d^3*x^3 + c^3)*(d*x + c)), x)
 

Giac [F]

\[ \int \frac {c-2 d x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx=\int { -\frac {2 \, d x - c}{\sqrt {-8 \, d^{3} x^{3} + c^{3}} {\left (d x + c\right )}} \,d x } \] Input:

integrate((-2*d*x+c)/(d*x+c)/(-8*d^3*x^3+c^3)^(1/2),x, algorithm="giac")
 

Output:

integrate(-(2*d*x - c)/(sqrt(-8*d^3*x^3 + c^3)*(d*x + c)), x)
 

Mupad [B] (verification not implemented)

Time = 23.10 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.46 \[ \int \frac {c-2 d x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx=\frac {\ln \left (\frac {\left (\sqrt {c^3-8\,d^3\,x^3}-c^{3/2}\right )\,{\left (\sqrt {c^3-8\,d^3\,x^3}+c^{3/2}+4\,\sqrt {c}\,d\,x\right )}^3}{x^3\,{\left (c+d\,x\right )}^3}\right )}{3\,\sqrt {c}\,d} \] Input:

int((c - 2*d*x)/((c^3 - 8*d^3*x^3)^(1/2)*(c + d*x)),x)
 

Output:

log((((c^3 - 8*d^3*x^3)^(1/2) - c^(3/2))*((c^3 - 8*d^3*x^3)^(1/2) + c^(3/2 
) + 4*c^(1/2)*d*x)^3)/(x^3*(c + d*x)^3))/(3*c^(1/2)*d)
 

Reduce [F]

\[ \int \frac {c-2 d x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx=\int \frac {\sqrt {-8 d^{3} x^{3}+c^{3}}}{4 d^{3} x^{3}+6 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \] Input:

int((-2*d*x+c)/(d*x+c)/(-8*d^3*x^3+c^3)^(1/2),x)
 

Output:

int(sqrt(c**3 - 8*d**3*x**3)/(c**3 + 3*c**2*d*x + 6*c*d**2*x**2 + 4*d**3*x 
**3),x)