3.268 \(\int \frac{5+3 x}{1-x-x^2+x^3} \, dx\)

Optimal. Leaf size=12 \[ \frac{4}{1-x}+\tanh ^{-1}(x) \]

[Out]

4/(1 - x) + ArcTanh[x]

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Rubi [A]  time = 0.0207068, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2074, 206} \[ \frac{4}{1-x}+\tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(5 + 3*x)/(1 - x - x^2 + x^3),x]

[Out]

4/(1 - x) + ArcTanh[x]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rule 206

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

Rubi steps

\begin{align*} \int \frac{5+3 x}{1-x-x^2+x^3} \, dx &=\int \left (\frac{4}{(-1+x)^2}+\frac{1}{1-x^2}\right ) \, dx\\ &=\frac{4}{1-x}+\int \frac{1}{1-x^2} \, dx\\ &=\frac{4}{1-x}+\tanh ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0107631, size = 24, normalized size = 2. \[ -\frac{4}{x-1}-\frac{1}{2} \log (x-1)+\frac{1}{2} \log (x+1) \]

Antiderivative was successfully verified.

[In]

Integrate[(5 + 3*x)/(1 - x - x^2 + x^3),x]

[Out]

-4/(-1 + x) - Log[-1 + x]/2 + Log[1 + x]/2

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Maple [A]  time = 0.007, size = 21, normalized size = 1.8 \begin{align*} -4\, \left ( x-1 \right ) ^{-1}-{\frac{\ln \left ( x-1 \right ) }{2}}+{\frac{\ln \left ( 1+x \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5+3*x)/(x^3-x^2-x+1),x)

[Out]

-4/(x-1)-1/2*ln(x-1)+1/2*ln(1+x)

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Maxima [A]  time = 0.98613, size = 27, normalized size = 2.25 \begin{align*} -\frac{4}{x - 1} + \frac{1}{2} \, \log \left (x + 1\right ) - \frac{1}{2} \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+3*x)/(x^3-x^2-x+1),x, algorithm="maxima")

[Out]

-4/(x - 1) + 1/2*log(x + 1) - 1/2*log(x - 1)

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Fricas [B]  time = 1.22273, size = 80, normalized size = 6.67 \begin{align*} \frac{{\left (x - 1\right )} \log \left (x + 1\right ) -{\left (x - 1\right )} \log \left (x - 1\right ) - 8}{2 \,{\left (x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+3*x)/(x^3-x^2-x+1),x, algorithm="fricas")

[Out]

1/2*((x - 1)*log(x + 1) - (x - 1)*log(x - 1) - 8)/(x - 1)

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Sympy [B]  time = 0.09273, size = 17, normalized size = 1.42 \begin{align*} - \frac{\log{\left (x - 1 \right )}}{2} + \frac{\log{\left (x + 1 \right )}}{2} - \frac{4}{x - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+3*x)/(x**3-x**2-x+1),x)

[Out]

-log(x - 1)/2 + log(x + 1)/2 - 4/(x - 1)

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Giac [B]  time = 1.11756, size = 30, normalized size = 2.5 \begin{align*} -\frac{4}{x - 1} + \frac{1}{2} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+3*x)/(x^3-x^2-x+1),x, algorithm="giac")

[Out]

-4/(x - 1) + 1/2*log(abs(x + 1)) - 1/2*log(abs(x - 1))