3.97.8 \(\int \frac {4 x^2+e^8 (-3+3 x)}{3 e^8 x^2+4 x^3+(3 e^8 x+4 x^2) \log (\frac {3 e^8+4 x}{x})} \, dx\)

Optimal. Leaf size=16 \[ \log \left (12 \left (x+\log \left (4+\frac {3 e^8}{x}\right )\right )\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.21, antiderivative size = 14, normalized size of antiderivative = 0.88, number of steps used = 2, number of rules used = 2, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {6741, 6684} \begin {gather*} \log \left (x+\log \left (\frac {3 e^8}{x}+4\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*x^2 + E^8*(-3 + 3*x))/(3*E^8*x^2 + 4*x^3 + (3*E^8*x + 4*x^2)*Log[(3*E^8 + 4*x)/x]),x]

[Out]

Log[x + Log[4 + (3*E^8)/x]]

Rule 6684

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

Rule 6741

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-3 e^8+3 e^8 x+4 x^2}{x \left (3 e^8+4 x\right ) \left (x+\log \left (4+\frac {3 e^8}{x}\right )\right )} \, dx\\ &=\log \left (x+\log \left (4+\frac {3 e^8}{x}\right )\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 5.03, size = 14, normalized size = 0.88 \begin {gather*} \log \left (x+\log \left (4+\frac {3 e^8}{x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*x^2 + E^8*(-3 + 3*x))/(3*E^8*x^2 + 4*x^3 + (3*E^8*x + 4*x^2)*Log[(3*E^8 + 4*x)/x]),x]

[Out]

Log[x + Log[4 + (3*E^8)/x]]

________________________________________________________________________________________

fricas [A]  time = 0.59, size = 16, normalized size = 1.00 \begin {gather*} \log \left (x + \log \left (\frac {4 \, x + 3 \, e^{8}}{x}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x-3)*exp(4)^2+4*x^2)/((3*x*exp(4)^2+4*x^2)*log((3*exp(4)^2+4*x)/x)+3*x^2*exp(4)^2+4*x^3),x, algo
rithm="fricas")

[Out]

log(x + log((4*x + 3*e^8)/x))

________________________________________________________________________________________

giac [B]  time = 0.18, size = 72, normalized size = 4.50 \begin {gather*} {\left (e^{8} \log \left (\frac {{\left (4 \, x + 3 \, e^{8}\right )} \log \left (\frac {4 \, x + 3 \, e^{8}}{x}\right )}{x} + 3 \, e^{8} - 4 \, \log \left (\frac {4 \, x + 3 \, e^{8}}{x}\right )\right ) - e^{8} \log \left (\frac {4 \, x + 3 \, e^{8}}{x} - 4\right )\right )} e^{\left (-8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x-3)*exp(4)^2+4*x^2)/((3*x*exp(4)^2+4*x^2)*log((3*exp(4)^2+4*x)/x)+3*x^2*exp(4)^2+4*x^3),x, algo
rithm="giac")

[Out]

(e^8*log((4*x + 3*e^8)*log((4*x + 3*e^8)/x)/x + 3*e^8 - 4*log((4*x + 3*e^8)/x)) - e^8*log((4*x + 3*e^8)/x - 4)
)*e^(-8)

________________________________________________________________________________________

maple [A]  time = 0.11, size = 17, normalized size = 1.06

method	result	size
risch	\(\ln \left (\ln \left (\frac {3 \,{\mathrm e}^{8}+4 x}{x}\right )+x \right )\)	\(17\)
norman	\(\ln \left (\ln \left (\frac {3 \,{\mathrm e}^{8}+4 x}{x}\right )+x \right )\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x-3)*exp(4)^2+4*x^2)/((3*x*exp(4)^2+4*x^2)*ln((3*exp(4)^2+4*x)/x)+3*x^2*exp(4)^2+4*x^3),x,method=_RETU
RNVERBOSE)

[Out]

ln(ln((3*exp(8)+4*x)/x)+x)

________________________________________________________________________________________

maxima [A]  time = 0.40, size = 16, normalized size = 1.00 \begin {gather*} \log \left (x + \log \left (4 \, x + 3 \, e^{8}\right ) - \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x-3)*exp(4)^2+4*x^2)/((3*x*exp(4)^2+4*x^2)*log((3*exp(4)^2+4*x)/x)+3*x^2*exp(4)^2+4*x^3),x, algo
rithm="maxima")

[Out]

log(x + log(4*x + 3*e^8) - log(x))

________________________________________________________________________________________

mupad [B]  time = 5.76, size = 13, normalized size = 0.81 \begin {gather*} \ln \left (x+\ln \left (\frac {3\,{\mathrm {e}}^8}{x}+4\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2 + exp(8)*(3*x - 3))/(3*x^2*exp(8) + log((4*x + 3*exp(8))/x)*(3*x*exp(8) + 4*x^2) + 4*x^3),x)

[Out]

log(x + log((3*exp(8))/x + 4))

________________________________________________________________________________________

sympy [A]  time = 0.22, size = 14, normalized size = 0.88 \begin {gather*} \log {\left (x + \log {\left (\frac {4 x + 3 e^{8}}{x} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + log((4*x + 3*exp(8))/x))

________________________________________________________________________________________