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

Optimal. Leaf size=19 \[ \frac{x^4}{4}+\frac{x^2}{2}+\log (x+1) \]

[Out]

x^2/2 + x^4/4 + Log[1 + x]

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Rubi [A]  time = 0.0132435, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {1810, 627, 31} \[ \frac{x^4}{4}+\frac{x^2}{2}+\log (x+1) \]

Antiderivative was successfully verified.

[In]

Int[(-1 + x^5)/(-1 + x^2),x]

[Out]

x^2/2 + x^4/4 + Log[1 + x]

Rule 1810

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 627

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0041398, size = 19, normalized size = 1. \[ \frac{x^4}{4}+\frac{x^2}{2}+\log (x+1) \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x^5)/(-1 + x^2),x]

[Out]

x^2/2 + x^4/4 + Log[1 + x]

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Maple [A]  time = 0.001, size = 16, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{2}}+{\frac{{x}^{4}}{4}}+\ln \left ( 1+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5-1)/(x^2-1),x)

[Out]

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

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Maxima [A]  time = 0.962015, size = 20, normalized size = 1.05 \begin{align*} \frac{1}{4} \, x^{4} + \frac{1}{2} \, x^{2} + \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4 + 1/2*x^2 + log(x + 1)

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Fricas [A]  time = 1.4384, size = 43, normalized size = 2.26 \begin{align*} \frac{1}{4} \, x^{4} + \frac{1}{2} \, x^{2} + \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4 + 1/2*x^2 + log(x + 1)

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Sympy [A]  time = 0.075318, size = 14, normalized size = 0.74 \begin{align*} \frac{x^{4}}{4} + \frac{x^{2}}{2} + \log{\left (x + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**4/4 + x**2/2 + log(x + 1)

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Giac [A]  time = 1.17694, size = 22, normalized size = 1.16 \begin{align*} \frac{1}{4} \, x^{4} + \frac{1}{2} \, x^{2} + \log \left ({\left | x + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4 + 1/2*x^2 + log(abs(x + 1))