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]

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

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Rubi [A]  time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, 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 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])

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 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]

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*}

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Mathematica [A]  time = 0.03, size = 29, normalized size = 1.00 \[ \frac {2 \tan ^{-1}\left (\frac {c+2 d x}{\sqrt {3} c}\right )}{\sqrt {3} c d} \]

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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fricas [A]  time = 0.59, size = 26, normalized size = 0.90 \[ \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, d x + c\right )}}{3 \, c}\right )}{3 \, c d} \]

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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giac [A]  time = 0.17, size = 26, normalized size = 0.90 \[ \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, d x + c\right )}}{3 \, c}\right )}{3 \, c d} \]

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)

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maple [A]  time = 0.04, size = 34, normalized size = 1.17 \[ \frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 d^{2} x +c d \right ) \sqrt {3}}{3 c d}\right )}{3 c d} \]

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 = 2.98, size = 33, normalized size = 1.14 \[ \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, d^{2} x + c d\right )}}{3 \, c d}\right )}{3 \, c d} \]

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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mupad [B]  time = 0.04, size = 28, normalized size = 0.97 \[ \frac {2\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}}{3}+\frac {2\,\sqrt {3}\,d\,x}{3\,c}\right )}{3\,c\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c - d*x)/(c^3 - d^3*x^3),x)

[Out]

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

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sympy [C]  time = 0.47, size = 53, normalized size = 1.83 \[ \frac {- \frac {\sqrt {3} i \log {\left (x + \frac {c - \sqrt {3} i c}{2 d} \right )}}{3} + \frac {\sqrt {3} i \log {\left (x + \frac {c + \sqrt {3} i c}{2 d} \right )}}{3}}{c d} \]

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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