3.92.32 \(\int \frac {240-320 x^4+e^{2/3} (-5 x^2+20 x^6)}{-256 x+128 x^5-16 x^9+e^{2/3} (16 x^3-8 x^7+x^{11})+(-128 x+32 x^5+e^{2/3} (8 x^3-2 x^7)) \log (-\frac {x^3}{-16+e^{2/3} x^2})+(-16 x+e^{2/3} x^3) \log ^2(-\frac {x^3}{-16+e^{2/3} x^2})} \, dx\)

Optimal. Leaf size=29 \[ \frac {5}{4-x^4+\log \left (\frac {x}{-e^{2/3}+\frac {16}{x^2}}\right )} \]

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Rubi [A]  time = 0.37, antiderivative size = 30, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, integrand size = 142, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {6688, 12, 6686} \begin {gather*} \frac {5}{-x^4+\log \left (\frac {x^3}{16-e^{2/3} x^2}\right )+4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(240 - 320*x^4 + E^(2/3)*(-5*x^2 + 20*x^6))/(-256*x + 128*x^5 - 16*x^9 + E^(2/3)*(16*x^3 - 8*x^7 + x^11) +
(-128*x + 32*x^5 + E^(2/3)*(8*x^3 - 2*x^7))*Log[-(x^3/(-16 + E^(2/3)*x^2))] + (-16*x + E^(2/3)*x^3)*Log[-(x^3
/(-16 + E^(2/3)*x^2))]^2),x]

[Out]

5/(4 - x^4 + Log[x^3/(16 - E^(2/3)*x^2)])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 \left (-48+64 x^4-e^{2/3} x^2 \left (-1+4 x^4\right )\right )}{x \left (16-e^{2/3} x^2\right ) \left (4-x^4+\log \left (\frac {x^3}{16-e^{2/3} x^2}\right )\right )^2} \, dx\\ &=5 \int \frac {-48+64 x^4-e^{2/3} x^2 \left (-1+4 x^4\right )}{x \left (16-e^{2/3} x^2\right ) \left (4-x^4+\log \left (\frac {x^3}{16-e^{2/3} x^2}\right )\right )^2} \, dx\\ &=\frac {5}{4-x^4+\log \left (\frac {x^3}{16-e^{2/3} x^2}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 30, normalized size = 1.03 \begin {gather*} \frac {5}{4-x^4+\log \left (\frac {x^3}{16-e^{2/3} x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(240 - 320*x^4 + E^(2/3)*(-5*x^2 + 20*x^6))/(-256*x + 128*x^5 - 16*x^9 + E^(2/3)*(16*x^3 - 8*x^7 + x
^11) + (-128*x + 32*x^5 + E^(2/3)*(8*x^3 - 2*x^7))*Log[-(x^3/(-16 + E^(2/3)*x^2))] + (-16*x + E^(2/3)*x^3)*Log
[-(x^3/(-16 + E^(2/3)*x^2))]^2),x]

[Out]

5/(4 - x^4 + Log[x^3/(16 - E^(2/3)*x^2)])

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fricas [A]  time = 0.82, size = 27, normalized size = 0.93 \begin {gather*} -\frac {5}{x^{4} - \log \left (-\frac {x^{3}}{x^{2} e^{\frac {2}{3}} - 16}\right ) - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^6-5*x^2)*exp(2/3)-320*x^4+240)/((x^3*exp(2/3)-16*x)*log(-x^3/(x^2*exp(2/3)-16))^2+((-2*x^7+8*
x^3)*exp(2/3)+32*x^5-128*x)*log(-x^3/(x^2*exp(2/3)-16))+(x^11-8*x^7+16*x^3)*exp(2/3)-16*x^9+128*x^5-256*x),x,
algorithm="fricas")

[Out]

-5/(x^4 - log(-x^3/(x^2*e^(2/3) - 16)) - 4)

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giac [A]  time = 0.20, size = 27, normalized size = 0.93 \begin {gather*} -\frac {5}{x^{4} - \log \left (-\frac {x^{3}}{x^{2} e^{\frac {2}{3}} - 16}\right ) - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^6-5*x^2)*exp(2/3)-320*x^4+240)/((x^3*exp(2/3)-16*x)*log(-x^3/(x^2*exp(2/3)-16))^2+((-2*x^7+8*
x^3)*exp(2/3)+32*x^5-128*x)*log(-x^3/(x^2*exp(2/3)-16))+(x^11-8*x^7+16*x^3)*exp(2/3)-16*x^9+128*x^5-256*x),x,
algorithm="giac")

[Out]

-5/(x^4 - log(-x^3/(x^2*e^(2/3) - 16)) - 4)

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maple [A]  time = 0.05, size = 28, normalized size = 0.97

method	result	size
risch	\(-\frac {5}{x^{4}-\ln \left (-\frac {x^{3}}{x^{2} {\mathrm e}^{\frac {2}{3}}-16}\right )-4}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((20*x^6-5*x^2)*exp(2/3)-320*x^4+240)/((x^3*exp(2/3)-16*x)*ln(-x^3/(x^2*exp(2/3)-16))^2+((-2*x^7+8*x^3)*ex
p(2/3)+32*x^5-128*x)*ln(-x^3/(x^2*exp(2/3)-16))+(x^11-8*x^7+16*x^3)*exp(2/3)-16*x^9+128*x^5-256*x),x,method=_R
ETURNVERBOSE)

[Out]

-5/(x^4-ln(-x^3/(x^2*exp(2/3)-16))-4)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^6-5*x^2)*exp(2/3)-320*x^4+240)/((x^3*exp(2/3)-16*x)*log(-x^3/(x^2*exp(2/3)-16))^2+((-2*x^7+8*
x^3)*exp(2/3)+32*x^5-128*x)*log(-x^3/(x^2*exp(2/3)-16))+(x^11-8*x^7+16*x^3)*exp(2/3)-16*x^9+128*x^5-256*x),x,
algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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mupad [B]  time = 20.98, size = 27, normalized size = 0.93 \begin {gather*} \frac {5}{\ln \left (-\frac {x^3}{x^2\,{\mathrm {e}}^{2/3}-16}\right )-x^4+4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2/3)*(5*x^2 - 20*x^6) + 320*x^4 - 240)/(256*x - log(-x^3/(x^2*exp(2/3) - 16))*(exp(2/3)*(8*x^3 - 2*x^
7) - 128*x + 32*x^5) - exp(2/3)*(16*x^3 - 8*x^7 + x^11) + log(-x^3/(x^2*exp(2/3) - 16))^2*(16*x - x^3*exp(2/3)
) - 128*x^5 + 16*x^9),x)

[Out]

5/(log(-x^3/(x^2*exp(2/3) - 16)) - x^4 + 4)

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sympy [A]  time = 0.22, size = 22, normalized size = 0.76 \begin {gather*} \frac {5}{- x^{4} + \log {\left (- \frac {x^{3}}{x^{2} e^{\frac {2}{3}} - 16} \right )} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x**6-5*x**2)*exp(2/3)-320*x**4+240)/((x**3*exp(2/3)-16*x)*ln(-x**3/(x**2*exp(2/3)-16))**2+((-2*
x**7+8*x**3)*exp(2/3)+32*x**5-128*x)*ln(-x**3/(x**2*exp(2/3)-16))+(x**11-8*x**7+16*x**3)*exp(2/3)-16*x**9+128*
x**5-256*x),x)

[Out]

5/(-x**4 + log(-x**3/(x**2*exp(2/3) - 16)) + 4)

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