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

Optimal. Leaf size=95 \[ \frac{3 \log \left (d (2 c+d x)-d \sqrt [3]{2 c^3+d^3 x^3}\right )}{2 d}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 (2 c+d x)}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt{3}}\right )}{d}-\frac{\log (c+d x)}{d} \]

[Out]

-((Sqrt[3]*ArcTan[(1 + (2*(2*c + d*x))/(2*c^3 + d^3*x^3)^(1/3))/Sqrt[3]])/d) - L
og[c + d*x]/d + (3*Log[d*(2*c + d*x) - d*(2*c^3 + d^3*x^3)^(1/3)])/(2*d)

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Rubi [A]  time = 0.219754, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.032 \[ \frac{3 \log \left (d (2 c+d x)-d \sqrt [3]{2 c^3+d^3 x^3}\right )}{2 d}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 (2 c+d x)}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt{3}}\right )}{d}-\frac{\log (c+d x)}{d} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[3]*ArcTan[(1 + (2*(2*c + d*x))/(2*c^3 + d^3*x^3)^(1/3))/Sqrt[3]])/d) - L
og[c + d*x]/d + (3*Log[d*(2*c + d*x) - d*(2*c^3 + d^3*x^3)^(1/3)])/(2*d)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.164976, size = 0, normalized size = 0. \[ \int \frac{c-d x}{(c+d x) \sqrt [3]{2 c^3+d^3 x^3}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

Integrate[(c - d*x)/((c + d*x)*(2*c^3 + d^3*x^3)^(1/3)), x]

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{\frac{-dx+c}{dx+c}{\frac{1}{\sqrt [3]{{d}^{3}{x}^{3}+2\,{c}^{3}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{d x - c}{{\left (d^{3} x^{3} + 2 \, c^{3}\right )}^{\frac{1}{3}}{\left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{d x - c}{{\left (d^{3} x^{3} + 2 \, c^{3}\right )}^{\frac{1}{3}}{\left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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