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

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) - Log[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 = 0.124666, 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, Rules used = {2151} \[ \frac{3 \log \left (d (2 c+d x)-d \sqrt [3]{2 c^3+d^3 x^3}\right )}{2 d}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 (2 c+d x)}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt{3}}\right )}{d}-\frac{\log (c+d x)}{d} \]

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

Rule 2151

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Simp[(Sqrt[3]*f*ArcTan
[(1 + (2*Rt[b, 3]*(2*c + d*x))/(d*(a + b*x^3)^(1/3)))/Sqrt[3]])/(Rt[b, 3]*d), x] + (Simp[(f*Log[c + d*x])/(Rt[
b, 3]*d), x] - Simp[(3*f*Log[Rt[b, 3]*(2*c + d*x) - d*(a + b*x^3)^(1/3)])/(2*Rt[b, 3]*d), x]) /; FreeQ[{a, b,
c, d, e, f}, x] && EqQ[d*e + c*f, 0] && EqQ[2*b*c^3 - a*d^3, 0]

Rubi steps

\begin{align*} \int \frac{c-d x}{(c+d x) \sqrt [3]{2 c^3+d^3 x^3}} \, dx &=-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 (2 c+d x)}{\sqrt [3]{2 c^3+d^3 x^3}}}{\sqrt{3}}\right )}{d}-\frac{\log (c+d x)}{d}+\frac{3 \log \left (d (2 c+d x)-d \sqrt [3]{2 c^3+d^3 x^3}\right )}{2 d}\\ \end{align*}

Mathematica [F]  time = 0.139853, 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 = 0.039, size = 0, normalized size = 0. \begin{align*} \int{\frac{-dx+c}{dx+c}{\frac{1}{\sqrt [3]{{d}^{3}{x}^{3}+2\,{c}^{3}}}}}\, dx \end{align*}

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. \begin{align*} -\int \frac{d x - c}{{\left (d^{3} x^{3} + 2 \, c^{3}\right )}^{\frac{1}{3}}{\left (d x + c\right )}}\,{d x} \end{align*}

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, 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. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

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, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{c}{c \sqrt [3]{2 c^{3} + d^{3} x^{3}} + d x \sqrt [3]{2 c^{3} + d^{3} x^{3}}}\, 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 \end{align*}

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

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, algorithm="giac")

[Out]

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