3.1.82 \(\int \frac {c-2 d x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx\) [82]

Optimal. Leaf size=46 \[ -\frac {2 \tanh ^{-1}\left (\frac {(c-2 d x)^2}{3 \sqrt {c} \sqrt {c^3-8 d^3 x^3}}\right )}{3 \sqrt {c} d} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2163, 212} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {(c-2 d x)^2}{3 \sqrt {c} \sqrt {c^3-8 d^3 x^3}}\right )}{3 \sqrt {c} d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2163

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Dist[-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]

Rubi steps

\begin {align*} \int \frac {c-2 d x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx &=-\frac {(2 c) \text {Subst}\left (\int \frac {1}{9-c^3 x^2} \, dx,x,\frac {\left (1-\frac {2 d x}{c}\right )^2}{\sqrt {c^3-8 d^3 x^3}}\right )}{d}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {(c-2 d x)^2}{3 \sqrt {c} \sqrt {c^3-8 d^3 x^3}}\right )}{3 \sqrt {c} d}\\ \end {align*}

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Mathematica [A]
time = 1.38, size = 44, normalized size = 0.96 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {3 \sqrt {c} \sqrt {c^3-8 d^3 x^3}}{(c-2 d x)^2}\right )}{3 \sqrt {c} d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.38, size = 503, normalized size = 10.93

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (38) = 76\).
time = 0.44, size = 294, normalized size = 6.39 \begin {gather*} \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 ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[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 + 156*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)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 3.13, size = 67, normalized size = 1.46 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

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