3.19 \(\int \frac{1}{(c+d x) \sqrt [3]{2 c^3+d^3 x^3}} \, dx\)

Optimal. Leaf size=186 \[ -\frac{\log \left (\sqrt [3]{2 c^3+d^3 x^3}-d x\right )}{4 c d}+\frac{3 \log \left (d (2 c+d x)-d \sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c d}+\frac{\tan ^{-1}\left (\frac{\frac{2 d x}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt{3}}\right )}{2 \sqrt{3} c d}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 (2 c+d x)}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt{3}}\right )}{2 c d}-\frac{\log (c+d x)}{2 c d} \]

[Out]

ArcTan[(1 + (2*d*x)/(2*c^3 + d^3*x^3)^(1/3))/Sqrt[3]]/(2*Sqrt[3]*c*d) - (Sqrt[3]
*ArcTan[(1 + (2*(2*c + d*x))/(2*c^3 + d^3*x^3)^(1/3))/Sqrt[3]])/(2*c*d) - Log[c
+ d*x]/(2*c*d) - Log[-(d*x) + (2*c^3 + d^3*x^3)^(1/3)]/(4*c*d) + (3*Log[d*(2*c +
d*x) - d*(2*c^3 + d^3*x^3)^(1/3)])/(4*c*d)

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Rubi [A]  time = 0.354867, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ -\frac{\log \left (\sqrt [3]{2 c^3+d^3 x^3}-d x\right )}{4 c d}+\frac{3 \log \left (d (2 c+d x)-d \sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c d}+\frac{\tan ^{-1}\left (\frac{\frac{2 d x}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt{3}}\right )}{2 \sqrt{3} c d}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 (2 c+d x)}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt{3}}\right )}{2 c d}-\frac{\log (c+d x)}{2 c d} \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[(1 + (2*d*x)/(2*c^3 + d^3*x^3)^(1/3))/Sqrt[3]]/(2*Sqrt[3]*c*d) - (Sqrt[3]
*ArcTan[(1 + (2*(2*c + d*x))/(2*c^3 + d^3*x^3)^(1/3))/Sqrt[3]])/(2*c*d) - Log[c
+ d*x]/(2*c*d) - Log[-(d*x) + (2*c^3 + d^3*x^3)^(1/3)]/(4*c*d) + (3*Log[d*(2*c +
d*x) - d*(2*c^3 + d^3*x^3)^(1/3)])/(4*c*d)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (c + d x\right ) \sqrt [3]{2 c^{3} + d^{3} x^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0778992, size = 0, normalized size = 0. \[ \int \frac{1}{(c+d x) \sqrt [3]{2 c^3+d^3 x^3}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.058, size = 0, normalized size = 0. \[ \int{\frac{1}{dx+c}{\frac{1}{\sqrt [3]{{d}^{3}{x}^{3}+2\,{c}^{3}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(1/(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. \[ \int \frac{1}{{\left (d^{3} x^{3} + 2 \, c^{3}\right )}^{\frac{1}{3}}{\left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((c + 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. \[ \int \frac{1}{{\left (d^{3} x^{3} + 2 \, c^{3}\right )}^{\frac{1}{3}}{\left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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