3.94.13 \(\int \frac {-32-16 x+3 x^3+4 x^6-x^7}{16 x-3 x^3+x^7} \, dx\)

Optimal. Leaf size=21 \[ -8+\log \left (e^{-x} \left (3-\frac {16}{x^2}-x^4\right )\right ) \]

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Rubi [A]  time = 0.16, antiderivative size = 19, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {1594, 6742, 1587} \begin {gather*} \log \left (x^6-3 x^2+16\right )-x-2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-32 - 16*x + 3*x^3 + 4*x^6 - x^7)/(16*x - 3*x^3 + x^7),x]

[Out]

-x - 2*Log[x] + Log[16 - 3*x^2 + x^6]

Rule 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-32-16 x+3 x^3+4 x^6-x^7}{x \left (16-3 x^2+x^6\right )} \, dx\\ &=\int \left (-1-\frac {2}{x}+\frac {6 x \left (-1+x^4\right )}{16-3 x^2+x^6}\right ) \, dx\\ &=-x-2 \log (x)+6 \int \frac {x \left (-1+x^4\right )}{16-3 x^2+x^6} \, dx\\ &=-x-2 \log (x)+\log \left (16-3 x^2+x^6\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 0.90 \begin {gather*} -x-2 \log (x)+\log \left (16-3 x^2+x^6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-32 - 16*x + 3*x^3 + 4*x^6 - x^7)/(16*x - 3*x^3 + x^7),x]

[Out]

-x - 2*Log[x] + Log[16 - 3*x^2 + x^6]

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fricas [A]  time = 1.27, size = 19, normalized size = 0.90 \begin {gather*} -x + \log \left (x^{6} - 3 \, x^{2} + 16\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^7+4*x^6+3*x^3-16*x-32)/(x^7-3*x^3+16*x),x, algorithm="fricas")

[Out]

-x + log(x^6 - 3*x^2 + 16) - 2*log(x)

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giac [A]  time = 0.12, size = 20, normalized size = 0.95 \begin {gather*} -x + \log \left (x^{6} - 3 \, x^{2} + 16\right ) - 2 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^7+4*x^6+3*x^3-16*x-32)/(x^7-3*x^3+16*x),x, algorithm="giac")

[Out]

-x + log(x^6 - 3*x^2 + 16) - 2*log(abs(x))

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maple [A]  time = 0.04, size = 20, normalized size = 0.95

method	result	size
default	\(-x -2 \ln \relax (x )+\ln \left (x^{6}-3 x^{2}+16\right )\)	\(20\)
norman	\(-x -2 \ln \relax (x )+\ln \left (x^{6}-3 x^{2}+16\right )\)	\(20\)
risch	\(-x -2 \ln \relax (x )+\ln \left (x^{6}-3 x^{2}+16\right )\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^7+4*x^6+3*x^3-16*x-32)/(x^7-3*x^3+16*x),x,method=_RETURNVERBOSE)

[Out]

-x-2*ln(x)+ln(x^6-3*x^2+16)

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maxima [A]  time = 0.35, size = 19, normalized size = 0.90 \begin {gather*} -x + \log \left (x^{6} - 3 \, x^{2} + 16\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^7+4*x^6+3*x^3-16*x-32)/(x^7-3*x^3+16*x),x, algorithm="maxima")

[Out]

-x + log(x^6 - 3*x^2 + 16) - 2*log(x)

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mupad [B]  time = 7.57, size = 19, normalized size = 0.90 \begin {gather*} \ln \left (x^6-3\,x^2+16\right )-x-2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(16*x - 3*x^3 - 4*x^6 + x^7 + 32)/(16*x - 3*x^3 + x^7),x)

[Out]

log(x^6 - 3*x^2 + 16) - x - 2*log(x)

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sympy [A]  time = 0.12, size = 17, normalized size = 0.81 \begin {gather*} - x - 2 \log {\relax (x )} + \log {\left (x^{6} - 3 x^{2} + 16 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**7+4*x**6+3*x**3-16*x-32)/(x**7-3*x**3+16*x),x)

[Out]

-x - 2*log(x) + log(x**6 - 3*x**2 + 16)

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