3.19 \(\int \frac{c-d x}{c^3-d^3 x^3} \, dx\)

Optimal. Leaf size=29 \[ \frac{2 \tan ^{-1}\left (\frac{c+2 d x}{\sqrt{3} c}\right )}{\sqrt{3} c d} \]

[Out]

(2*ArcTan[(c + 2*d*x)/(Sqrt[3]*c)])/(Sqrt[3]*c*d)

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Rubi [A]  time = 0.0203174, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {1586, 617, 204} \[ \frac{2 \tan ^{-1}\left (\frac{c+2 d x}{\sqrt{3} c}\right )}{\sqrt{3} c d} \]

Antiderivative was successfully verified.

[In]

Int[(c - d*x)/(c^3 - d^3*x^3),x]

[Out]

(2*ArcTan[(c + 2*d*x)/(Sqrt[3]*c)])/(Sqrt[3]*c*d)

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0074607, size = 29, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\frac{c+2 d x}{\sqrt{3} c}\right )}{\sqrt{3} c d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcTan[(c + 2*d*x)/(Sqrt[3]*c)])/(Sqrt[3]*c*d)

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Maple [A]  time = 0.002, size = 34, normalized size = 1.2 \begin{align*}{\frac{2\,\sqrt{3}}{3\,cd}\arctan \left ({\frac{ \left ( 2\,{d}^{2}x+cd \right ) \sqrt{3}}{3\,cd}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.43697, size = 45, normalized size = 1.55 \begin{align*} \frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \, d^{2} x + c d\right )}}{3 \, c d}\right )}{3 \, c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-d*x+c)/(-d^3*x^3+c^3),x, algorithm="maxima")

[Out]

2/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*d^2*x + c*d)/(c*d))/(c*d)

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Fricas [A]  time = 1.02726, size = 72, normalized size = 2.48 \begin{align*} \frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \, d x + c\right )}}{3 \, c}\right )}{3 \, c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-d*x+c)/(-d^3*x^3+c^3),x, algorithm="fricas")

[Out]

2/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*d*x + c)/c)/(c*d)

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Sympy [C]  time = 0.213179, size = 53, normalized size = 1.83 \begin{align*} \frac{- \frac{\sqrt{3} i \log{\left (x + \frac{c - \sqrt{3} i c}{2 d} \right )}}{3} + \frac{\sqrt{3} i \log{\left (x + \frac{c + \sqrt{3} i c}{2 d} \right )}}{3}}{c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-sqrt(3)*I*log(x + (c - sqrt(3)*I*c)/(2*d))/3 + sqrt(3)*I*log(x + (c + sqrt(3)*I*c)/(2*d))/3)/(c*d)

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Giac [A]  time = 1.05908, size = 35, normalized size = 1.21 \begin{align*} \frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \, d x + c\right )}}{3 \, c}\right )}{3 \, c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-d*x+c)/(-d^3*x^3+c^3),x, algorithm="giac")

[Out]

2/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*d*x + c)/c)/(c*d)