\(\int \frac {c+d x}{c^3+d^3 x^3} \, dx\) [18]

Optimal result

Integrand size = 19, antiderivative size = 29 \[ \int \frac {c+d x}{c^3+d^3 x^3} \, dx=-\frac {2 \arctan \left (\frac {c-2 d x}{\sqrt {3} c}\right )}{\sqrt {3} c d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1600, 631, 210} \[ \int \frac {c+d x}{c^3+d^3 x^3} \, dx=-\frac {2 \arctan \left (\frac {c-2 d x}{\sqrt {3} c}\right )}{\sqrt {3} c d} \]

[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 210

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

Rule 631

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 1600

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*} \text {integral}& = \int \frac {1}{c^2-c d x+d^2 x^2} \, dx \\ & = \frac {2 \text {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] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {c+d x}{c^3+d^3 x^3} \, dx=\frac {2 \arctan \left (\frac {-c+2 d x}{\sqrt {3} c}\right )}{\sqrt {3} c d} \]

[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)

Maple [A] (verified)

Time = 1.52 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00

method	result	size
risch	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {2 d \sqrt {3}\, x}{3 c}-\frac {\sqrt {3}}{3}\right )}{3 d c}\)	\(29\)
default	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 x \,d^{2}-c d \right ) \sqrt {3}}{3 c d}\right )}{3 c d}\)	\(35\)

[In]

int((d*x+c)/(d^3*x^3+c^3),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x}{c^3+d^3 x^3} \, dx=\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, d x - c\right )}}{3 \, c}\right )}{3 \, c d} \]

[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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.86 \[ \int \frac {c+d x}{c^3+d^3 x^3} \, dx=\frac {- \frac {\sqrt {3} i \log {\left (x + \frac {- c - \sqrt {3} i c}{2 d} \right )}}{3} + \frac {\sqrt {3} i \log {\left (x + \frac {- c + \sqrt {3} i c}{2 d} \right )}}{3}}{c d} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17 \[ \int \frac {c+d x}{c^3+d^3 x^3} \, dx=\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, d^{2} x - c d\right )}}{3 \, c d}\right )}{3 \, c d} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x}{c^3+d^3 x^3} \, dx=\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, d x - c\right )}}{3 \, c}\right )}{3 \, c d} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x}{c^3+d^3 x^3} \, dx=-\frac {2\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}}{3}-\frac {2\,\sqrt {3}\,d\,x}{3\,c}\right )}{3\,c\,d} \]

[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)