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 = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 2.00 \[ -\frac {4}{x-1}-\frac {1}{2} \log (x-1)+\frac {1}{2} \log (x+1) \]

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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fricas [B]  time = 0.73, size = 26, normalized size = 2.17 \[ \frac {{\left (x - 1\right )} \log \left (x + 1\right ) - {\left (x - 1\right )} \log \left (x - 1\right ) - 8}{2 \, {\left (x - 1\right )}} \]

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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giac [B]  time = 0.30, size = 22, normalized size = 1.83 \[ -\frac {4}{x - 1} + \frac {1}{2} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \]

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))

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maple [A]  time = 0.01, size = 21, normalized size = 1.75 \[ -\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (x +1\right )}{2}-\frac {4}{x -1} \]

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(x+1)

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maxima [A]  time = 1.08, size = 20, normalized size = 1.67 \[ -\frac {4}{x - 1} + \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \]

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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mupad [B]  time = 0.07, size = 10, normalized size = 0.83 \[ \mathrm {atanh}\relax (x)-\frac {4}{x-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x + 5)/(x + x^2 - x^3 - 1),x)

[Out]

atanh(x) - 4/(x - 1)

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sympy [B]  time = 0.10, size = 17, normalized size = 1.42 \[ - \frac {\log {\left (x - 1 \right )}}{2} + \frac {\log {\left (x + 1 \right )}}{2} - \frac {4}{x - 1} \]

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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