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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]