3.79.77 \(\int \frac {(4 x-4 x^3+e (2-4 x^2)) \log (6-x^2+\log (e x+x^2))}{6 x^2-x^4+e (6 x-x^3)+(e x+x^2) \log (e x+x^2)} \, dx\)

Optimal. Leaf size=16 \[ \log ^2\left (6-x^2+\log (x (e+x))\right ) \]

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Rubi [A]  time = 0.09, antiderivative size = 18, normalized size of antiderivative = 1.12, number of steps used = 1, number of rules used = 1, integrand size = 75, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {6686} \begin {gather*} \log ^2\left (-x^2+\log \left (x^2+e x\right )+6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((4*x - 4*x^3 + E*(2 - 4*x^2))*Log[6 - x^2 + Log[E*x + x^2]])/(6*x^2 - x^4 + E*(6*x - x^3) + (E*x + x^2)*L
og[E*x + x^2]),x]

[Out]

Log[6 - x^2 + Log[E*x + x^2]]^2

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log ^2\left (6-x^2+\log \left (e x+x^2\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 16, normalized size = 1.00 \begin {gather*} \log ^2\left (6-x^2+\log (x (e+x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((4*x - 4*x^3 + E*(2 - 4*x^2))*Log[6 - x^2 + Log[E*x + x^2]])/(6*x^2 - x^4 + E*(6*x - x^3) + (E*x +
x^2)*Log[E*x + x^2]),x]

[Out]

Log[6 - x^2 + Log[x*(E + x)]]^2

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fricas [A]  time = 0.57, size = 19, normalized size = 1.19 \begin {gather*} \log \left (-x^{2} + \log \left (x^{2} + x e\right ) + 6\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2+2)*exp(1)-4*x^3+4*x)*log(log(x*exp(1)+x^2)-x^2+6)/((x*exp(1)+x^2)*log(x*exp(1)+x^2)+(-x^3+6
*x)*exp(1)-x^4+6*x^2),x, algorithm="fricas")

[Out]

log(-x^2 + log(x^2 + x*e) + 6)^2

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giac [A]  time = 0.21, size = 19, normalized size = 1.19 \begin {gather*} \log \left (-x^{2} + \log \left (x^{2} + x e\right ) + 6\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2+2)*exp(1)-4*x^3+4*x)*log(log(x*exp(1)+x^2)-x^2+6)/((x*exp(1)+x^2)*log(x*exp(1)+x^2)+(-x^3+6
*x)*exp(1)-x^4+6*x^2),x, algorithm="giac")

[Out]

log(-x^2 + log(x^2 + x*e) + 6)^2

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (-4 x^{2}+2\right ) {\mathrm e}-4 x^{3}+4 x \right ) \ln \left (\ln \left (x \,{\mathrm e}+x^{2}\right )-x^{2}+6\right )}{\left (x \,{\mathrm e}+x^{2}\right ) \ln \left (x \,{\mathrm e}+x^{2}\right )+\left (-x^{3}+6 x \right ) {\mathrm e}-x^{4}+6 x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^2+2)*exp(1)-4*x^3+4*x)*ln(ln(x*exp(1)+x^2)-x^2+6)/((x*exp(1)+x^2)*ln(x*exp(1)+x^2)+(-x^3+6*x)*exp(1
)-x^4+6*x^2),x)

[Out]

int(((-4*x^2+2)*exp(1)-4*x^3+4*x)*ln(ln(x*exp(1)+x^2)-x^2+6)/((x*exp(1)+x^2)*ln(x*exp(1)+x^2)+(-x^3+6*x)*exp(1
)-x^4+6*x^2),x)

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maxima [A]  time = 0.40, size = 17, normalized size = 1.06 \begin {gather*} \log \left (-x^{2} + \log \left (x + e\right ) + \log \relax (x) + 6\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2+2)*exp(1)-4*x^3+4*x)*log(log(x*exp(1)+x^2)-x^2+6)/((x*exp(1)+x^2)*log(x*exp(1)+x^2)+(-x^3+6
*x)*exp(1)-x^4+6*x^2),x, algorithm="maxima")

[Out]

log(-x^2 + log(x + e) + log(x) + 6)^2

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mupad [B]  time = 8.05, size = 19, normalized size = 1.19 \begin {gather*} {\ln \left (\ln \left (x^2+\mathrm {e}\,x\right )-x^2+6\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(log(x*exp(1) + x^2) - x^2 + 6)*(exp(1)*(4*x^2 - 2) - 4*x + 4*x^3))/(exp(1)*(6*x - x^3) + log(x*exp(1
) + x^2)*(x*exp(1) + x^2) + 6*x^2 - x^4),x)

[Out]

log(log(x*exp(1) + x^2) - x^2 + 6)^2

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sympy [A]  time = 0.64, size = 17, normalized size = 1.06 \begin {gather*} \log {\left (- x^{2} + \log {\left (x^{2} + e x \right )} + 6 \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**2+2)*exp(1)-4*x**3+4*x)*ln(ln(x*exp(1)+x**2)-x**2+6)/((x*exp(1)+x**2)*ln(x*exp(1)+x**2)+(-x*
*3+6*x)*exp(1)-x**4+6*x**2),x)

[Out]

log(-x**2 + log(x**2 + E*x) + 6)**2

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