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

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

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Rubi [A]  time = 0.202288, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{2 \tanh ^{-1}\left (\frac{(c-2 d x)^2}{3 \sqrt{c} \sqrt{c^3-8 d^3 x^3}}\right )}{3 \sqrt{c} d} \]

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)

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Rubi in Sympy [A]  time = 144.56, size = 549, normalized size = 11.93 \[ - \frac{2 \sqrt [4]{3} \sqrt{\frac{c^{2} + 2 c d x + 4 d^{2} x^{2}}{\left (c \left (1 + \sqrt{3}\right ) - 2 d x\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (c - 2 d x\right ) F\left (\operatorname{asin}{\left (- \frac{- c \left (-1 + \sqrt{3}\right ) - 2 d x}{c \left (1 + \sqrt{3}\right ) - 2 d x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{d \sqrt{\frac{c \left (c - 2 d x\right )}{\left (c \left (1 + \sqrt{3}\right ) - 2 d x\right )^{2}}} \left (\sqrt{3} + 3\right ) \sqrt{c^{3} - 8 d^{3} x^{3}}} - \frac{3^{\frac{3}{4}} \sqrt{\frac{c^{2} \left (1 + \frac{2 d x}{c} + \frac{4 d^{2} x^{2}}{c^{2}}\right )}{\left (c \left (1 + \sqrt{3}\right ) - 2 d x\right )^{2}}} \sqrt{3 + 2 \sqrt{3}} \sqrt{- \sqrt{3} + 2} \left (c - 2 d x\right ) \operatorname{atanh}{\left (\frac{\sqrt{- \frac{\left (c \left (-1 + \sqrt{3}\right ) + 2 d x\right )^{2}}{\left (c \left (1 + \sqrt{3}\right ) - 2 d x\right )^{2}} + 1}}{\sqrt{3 + 2 \sqrt{3}} \sqrt{\frac{\left (c \left (-1 + \sqrt{3}\right ) + 2 d x\right )^{2}}{\left (c \left (1 + \sqrt{3}\right ) - 2 d x\right )^{2}} - 4 \sqrt{3} + 7}} \right )}}{3 d \sqrt{\frac{c \left (c - 2 d x\right )}{\left (c \left (1 + \sqrt{3}\right ) - 2 d x\right )^{2}}} \sqrt{c^{3} - 8 d^{3} x^{3}}} + \frac{12 \sqrt [4]{3} \sqrt{\frac{c^{2} \left (1 + \frac{2 d x}{c} + \frac{4 d^{2} x^{2}}{c^{2}}\right )}{\left (c \left (1 + \sqrt{3}\right ) - 2 d x\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (c - 2 d x\right ) \Pi \left (4 \sqrt{3} + 7; \operatorname{asin}{\left (\frac{c \left (-1 + \sqrt{3}\right ) + 2 d x}{c \left (1 + \sqrt{3}\right ) - 2 d x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{d \sqrt{\frac{c \left (c - 2 d x\right )}{\left (c \left (1 + \sqrt{3}\right ) - 2 d x\right )^{2}}} \sqrt{- 4 \sqrt{3} + 7} \left (- \sqrt{3} + 3\right ) \left (\sqrt{3} + 3\right ) \sqrt{c^{3} - 8 d^{3} x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-2*d*x+c)/(d*x+c)/(-8*d**3*x**3+c**3)**(1/2),x)

[Out]

-2*3**(1/4)*sqrt((c**2 + 2*c*d*x + 4*d**2*x**2)/(c*(1 + sqrt(3)) - 2*d*x)**2)*sq
rt(sqrt(3) + 2)*(c - 2*d*x)*elliptic_f(asin(-(-c*(-1 + sqrt(3)) - 2*d*x)/(c*(1 +
sqrt(3)) - 2*d*x)), -7 - 4*sqrt(3))/(d*sqrt(c*(c - 2*d*x)/(c*(1 + sqrt(3)) - 2*
d*x)**2)*(sqrt(3) + 3)*sqrt(c**3 - 8*d**3*x**3)) - 3**(3/4)*sqrt(c**2*(1 + 2*d*x
/c + 4*d**2*x**2/c**2)/(c*(1 + sqrt(3)) - 2*d*x)**2)*sqrt(3 + 2*sqrt(3))*sqrt(-s
qrt(3) + 2)*(c - 2*d*x)*atanh(sqrt(-(c*(-1 + sqrt(3)) + 2*d*x)**2/(c*(1 + sqrt(3
)) - 2*d*x)**2 + 1)/(sqrt(3 + 2*sqrt(3))*sqrt((c*(-1 + sqrt(3)) + 2*d*x)**2/(c*(
1 + sqrt(3)) - 2*d*x)**2 - 4*sqrt(3) + 7)))/(3*d*sqrt(c*(c - 2*d*x)/(c*(1 + sqrt
(3)) - 2*d*x)**2)*sqrt(c**3 - 8*d**3*x**3)) + 12*3**(1/4)*sqrt(c**2*(1 + 2*d*x/c
+ 4*d**2*x**2/c**2)/(c*(1 + sqrt(3)) - 2*d*x)**2)*sqrt(-sqrt(3) + 2)*(c - 2*d*x
)*elliptic_pi(4*sqrt(3) + 7, asin((c*(-1 + sqrt(3)) + 2*d*x)/(c*(1 + sqrt(3)) -
2*d*x)), -7 - 4*sqrt(3))/(d*sqrt(c*(c - 2*d*x)/(c*(1 + sqrt(3)) - 2*d*x)**2)*sqr
t(-4*sqrt(3) + 7)*(-sqrt(3) + 3)*(sqrt(3) + 3)*sqrt(c**3 - 8*d**3*x**3))

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Mathematica [C]  time = 1.03501, size = 295, normalized size = 6.41 \[ -\frac{2 \sqrt{\frac{c-2 d x}{\left (1+\sqrt [3]{-1}\right ) c}} \left (\left (\sqrt [3]{-1}-2\right ) \left (\sqrt [3]{-1} c+2 d x\right ) \sqrt{\frac{\sqrt [3]{-1} \left (c+2 \sqrt [3]{-1} d x\right )}{\left (1+\sqrt [3]{-1}\right ) c}} F\left (\sin ^{-1}\left (\sqrt{\frac{c-2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}}\right )|\sqrt [3]{-1}\right )+\sqrt [3]{-1} \sqrt{3} \left (1+\sqrt [3]{-1}\right ) c \sqrt{\frac{c-2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \sqrt{\frac{c^2+2 c d x+4 d^2 x^2}{c^2}} \Pi \left (\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{c-2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}}\right )|\sqrt [3]{-1}\right )\right )}{\left (\sqrt [3]{-1}-2\right ) d \sqrt{\frac{c-2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \sqrt{c^3-8 d^3 x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-2*Sqrt[(c - 2*d*x)/((1 + (-1)^(1/3))*c)]*((-2 + (-1)^(1/3))*((-1)^(1/3)*c + 2*
d*x)*Sqrt[((-1)^(1/3)*(c + 2*(-1)^(1/3)*d*x))/((1 + (-1)^(1/3))*c)]*EllipticF[Ar
cSin[Sqrt[(c - 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]], (-1)^(1/3)] + (-1)^(1/3
)*Sqrt[3]*(1 + (-1)^(1/3))*c*Sqrt[(c - 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]*S
qrt[(c^2 + 2*c*d*x + 4*d^2*x^2)/c^2]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), Arc
Sin[Sqrt[(c - 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]], (-1)^(1/3)]))/((-2 + (-1
)^(1/3))*d*Sqrt[(c - 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]*Sqrt[c^3 - 8*d^3*x^
3])

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Maple [C]  time = 0.2, size = 650, normalized size = 14.1 \[ \text{result too large to display} \]

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)

[Out]

-4*(1/2*(-1/2+1/2*I*3^(1/2))*c/d-1/2*(-1/2-1/2*I*3^(1/2))*c/d)*((x-1/2*(-1/2-1/2
*I*3^(1/2))*c/d)/(1/2*(-1/2+1/2*I*3^(1/2))*c/d-1/2*(-1/2-1/2*I*3^(1/2))*c/d))^(1
/2)*((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*(-1/2-1/2*I*3^(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*(-1/2-1/2*I*3^(1/2))*c/d)/(1/2*(-1/
2+1/2*I*3^(1/2))*c/d-1/2*(-1/2-1/2*I*3^(1/2))*c/d))^(1/2),((1/2*(-1/2-1/2*I*3^(1
/2))*c/d-1/2*(-1/2+1/2*I*3^(1/2))*c/d)/(1/2*(-1/2-1/2*I*3^(1/2))*c/d-1/2*c/d))^(
1/2))+6*c/d*(1/2*(-1/2+1/2*I*3^(1/2))*c/d-1/2*(-1/2-1/2*I*3^(1/2))*c/d)*((x-1/2*
(-1/2-1/2*I*3^(1/2))*c/d)/(1/2*(-1/2+1/2*I*3^(1/2))*c/d-1/2*(-1/2-1/2*I*3^(1/2))
*c/d))^(1/2)*((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*(-1/2-1/2*I*3^(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)/(1/2*(-1/2-1/2*I*3^(1/2))*c/d+c/d)*EllipticP
i(((x-1/2*(-1/2-1/2*I*3^(1/2))*c/d)/(1/2*(-1/2+1/2*I*3^(1/2))*c/d-1/2*(-1/2-1/2*
I*3^(1/2))*c/d))^(1/2),(1/2*(-1/2-1/2*I*3^(1/2))*c/d-1/2*(-1/2+1/2*I*3^(1/2))*c/
d)/(1/2*(-1/2-1/2*I*3^(1/2))*c/d+c/d),((1/2*(-1/2-1/2*I*3^(1/2))*c/d-1/2*(-1/2+1
/2*I*3^(1/2))*c/d)/(1/2*(-1/2-1/2*I*3^(1/2))*c/d-1/2*c/d))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{2 \, d x - c}{\sqrt{-8 \, d^{3} x^{3} + c^{3}}{\left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*d*x - c)/(sqrt(-8*d^3*x^3 + c^3)*(d*x + c)),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 [A]  time = 0.388651, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (-\frac{3 \,{\left (8 \, c d^{4} x^{4} - 52 \, c^{2} d^{3} x^{3} + 12 \, c^{3} d^{2} x^{2} - 4 \, c^{4} d x + 5 \, c^{5}\right )} \sqrt{-8 \, d^{3} x^{3} + c^{3}} -{\left (8 \, d^{6} x^{6} - 240 \, c d^{5} x^{5} + 408 \, c^{2} d^{4} x^{4} + 88 \, c^{3} d^{3} x^{3} + 156 \, c^{4} d^{2} x^{2} + 12 \, c^{5} d x + 17 \, c^{6}\right )} \sqrt{c}}{d^{6} x^{6} + 6 \, c d^{5} x^{5} + 15 \, c^{2} d^{4} x^{4} + 20 \, c^{3} d^{3} x^{3} + 15 \, c^{4} d^{2} x^{2} + 6 \, c^{5} d x + c^{6}}\right )}{6 \, \sqrt{c} d}, -\frac{\arctan \left (\frac{{\left (4 \, d^{3} x^{3} - 24 \, c d^{2} x^{2} - 6 \, c^{2} d x - 5 \, c^{3}\right )} \sqrt{-c}}{3 \, \sqrt{-8 \, d^{3} x^{3} + c^{3}}{\left (2 \, c d x - c^{2}\right )}}\right )}{3 \, \sqrt{-c} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*d*x - c)/(sqrt(-8*d^3*x^3 + c^3)*(d*x + c)),x, algorithm="fricas")

[Out]

[1/6*log(-(3*(8*c*d^4*x^4 - 52*c^2*d^3*x^3 + 12*c^3*d^2*x^2 - 4*c^4*d*x + 5*c^5)
*sqrt(-8*d^3*x^3 + c^3) - (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)*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*arctan(1/3*(4*d^3*x^3 - 24*c*d^2*x^2 - 6*c^2*d*x - 5*c^3)*sqrt(-c)/(sq
rt(-8*d^3*x^3 + c^3)*(2*c*d*x - c^2)))/(sqrt(-c)*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{c}{c \sqrt{c^{3} - 8 d^{3} x^{3}} + d x \sqrt{c^{3} - 8 d^{3} x^{3}}}\right )\, dx - \int \frac{2 d x}{c \sqrt{c^{3} - 8 d^{3} x^{3}} + d x \sqrt{c^{3} - 8 d^{3} x^{3}}}\, dx \]

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) - I
ntegral(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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{2 \, d x - c}{\sqrt{-8 \, d^{3} x^{3} + c^{3}}{\left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*d*x - c)/(sqrt(-8*d^3*x^3 + c^3)*(d*x + c)),x, algorithm="giac")

[Out]

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