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

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

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Rubi [A]  time = 0.12, antiderivative size = 95, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [F]  time = 0.14, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c-d x}{(c+d x) \sqrt [3]{2 c^3+d^3 x^3}} \, dx \end {gather*}

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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IntegrateAlgebraic [A]  time = 1.42, size = 159, normalized size = 1.67 \begin {gather*} \frac {\log \left (\sqrt [3]{2 c^3+d^3 x^3}-2 c-d x\right )}{d}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{2 c^3+d^3 x^3}}{\sqrt [3]{2 c^3+d^3 x^3}+4 c+2 d x}\right )}{d}-\frac {\log \left (\left (2 c^3+d^3 x^3\right )^{2/3}+(2 c+d x) \sqrt [3]{2 c^3+d^3 x^3}+4 c^2+4 c d x+d^2 x^2\right )}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (re
sidue poly has multiple non-linear factors)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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)

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-d x +c}{\left (d x +c \right ) \left (d^{3} x^{3}+2 c^{3}\right )^{\frac {1}{3}}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\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 {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {c-d\,x}{{\left (2\,c^3+d^3\,x^3\right )}^{1/3}\,\left (c+d\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \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 \end {gather*}

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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