3.104.34 \(\int \frac {-3+2 x^3+e^{2 x} (-3+6 x)}{(3 x+3 e^{2 x} x+1876 x^2+x^4) \log (\frac {3+3 e^{2 x}+1876 x+x^3}{x})} \, dx\)

Optimal. Leaf size=24 \[ \log \left (4 \log \left (1+3 \left (625+\frac {1+e^{2 x}}{x}\right )+x^2\right )\right ) \]

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Rubi [A]  time = 0.15, antiderivative size = 21, normalized size of antiderivative = 0.88, number of steps used = 1, number of rules used = 1, integrand size = 63, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {6684} \begin {gather*} \log \left (\log \left (\frac {x^3+1876 x+3 e^{2 x}+3}{x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 + 2*x^3 + E^(2*x)*(-3 + 6*x))/((3*x + 3*E^(2*x)*x + 1876*x^2 + x^4)*Log[(3 + 3*E^(2*x) + 1876*x + x^3)
/x]),x]

[Out]

Log[Log[(3 + 3*E^(2*x) + 1876*x + x^3)/x]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log \left (\log \left (\frac {3+3 e^{2 x}+1876 x+x^3}{x}\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.42, size = 21, normalized size = 0.88 \begin {gather*} \log \left (\log \left (\frac {3+3 e^{2 x}+1876 x+x^3}{x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + 2*x^3 + E^(2*x)*(-3 + 6*x))/((3*x + 3*E^(2*x)*x + 1876*x^2 + x^4)*Log[(3 + 3*E^(2*x) + 1876*x
+ x^3)/x]),x]

[Out]

Log[Log[(3 + 3*E^(2*x) + 1876*x + x^3)/x]]

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fricas [A]  time = 0.65, size = 20, normalized size = 0.83 \begin {gather*} \log \left (\log \left (\frac {x^{3} + 1876 \, x + 3 \, e^{\left (2 \, x\right )} + 3}{x}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x-3)*exp(2*x)+2*x^3-3)/(3*x*exp(2*x)+x^4+1876*x^2+3*x)/log((3*exp(2*x)+x^3+1876*x+3)/x),x, algor
ithm="fricas")

[Out]

log(log((x^3 + 1876*x + 3*e^(2*x) + 3)/x))

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giac [A]  time = 0.20, size = 20, normalized size = 0.83 \begin {gather*} \log \left (\log \left (\frac {x^{3} + 1876 \, x + 3 \, e^{\left (2 \, x\right )} + 3}{x}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x-3)*exp(2*x)+2*x^3-3)/(3*x*exp(2*x)+x^4+1876*x^2+3*x)/log((3*exp(2*x)+x^3+1876*x+3)/x),x, algor
ithm="giac")

[Out]

log(log((x^3 + 1876*x + 3*e^(2*x) + 3)/x))

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maple [A]  time = 0.12, size = 21, normalized size = 0.88

method	result	size
norman	\(\ln \left (\ln \left (\frac {3 \,{\mathrm e}^{2 x}+x^{3}+1876 x +3}{x}\right )\right )\)	\(21\)
risch	\(\ln \left (\ln \left (3 \,{\mathrm e}^{2 x}+x^{3}+1876 x +3\right )-\frac {i \left (\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (3 \,{\mathrm e}^{2 x}+x^{3}+1876 x +3\right )\right ) \mathrm {csgn}\left (\frac {i \left (3 \,{\mathrm e}^{2 x}+x^{3}+1876 x +3\right )}{x}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (3 \,{\mathrm e}^{2 x}+x^{3}+1876 x +3\right )}{x}\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (3 \,{\mathrm e}^{2 x}+x^{3}+1876 x +3\right )\right ) \mathrm {csgn}\left (\frac {i \left (3 \,{\mathrm e}^{2 x}+x^{3}+1876 x +3\right )}{x}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \left (3 \,{\mathrm e}^{2 x}+x^{3}+1876 x +3\right )}{x}\right )^{3}-2 i \ln \relax (x )\right )}{2}\right )\)	\(177\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6*x-3)*exp(2*x)+2*x^3-3)/(3*x*exp(2*x)+x^4+1876*x^2+3*x)/ln((3*exp(2*x)+x^3+1876*x+3)/x),x,method=_RETUR
NVERBOSE)

[Out]

ln(ln((3*exp(2*x)+x^3+1876*x+3)/x))

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maxima [A]  time = 0.40, size = 21, normalized size = 0.88 \begin {gather*} \log \left (\log \left (x^{3} + 1876 \, x + 3 \, e^{\left (2 \, x\right )} + 3\right ) - \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x-3)*exp(2*x)+2*x^3-3)/(3*x*exp(2*x)+x^4+1876*x^2+3*x)/log((3*exp(2*x)+x^3+1876*x+3)/x),x, algor
ithm="maxima")

[Out]

log(log(x^3 + 1876*x + 3*e^(2*x) + 3) - log(x))

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mupad [B]  time = 8.15, size = 20, normalized size = 0.83 \begin {gather*} \ln \left (\ln \left (\frac {1876\,x+3\,{\mathrm {e}}^{2\,x}+x^3+3}{x}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x)*(6*x - 3) + 2*x^3 - 3)/(log((1876*x + 3*exp(2*x) + x^3 + 3)/x)*(3*x + 3*x*exp(2*x) + 1876*x^2 +
x^4)),x)

[Out]

log(log((1876*x + 3*exp(2*x) + x^3 + 3)/x))

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sympy [A]  time = 0.49, size = 19, normalized size = 0.79 \begin {gather*} \log {\left (\log {\left (\frac {x^{3} + 1876 x + 3 e^{2 x} + 3}{x} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x-3)*exp(2*x)+2*x**3-3)/(3*x*exp(2*x)+x**4+1876*x**2+3*x)/ln((3*exp(2*x)+x**3+1876*x+3)/x),x)

[Out]

log(log((x**3 + 1876*x + 3*exp(2*x) + 3)/x))

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