3.80.9 \(\int \frac {-1-4 x^2-2 x^2 \log (4)}{x-2 e^7 x^2-2 x^3-2 x^3 \log (4)+2 x^2 \log (6 e^{-x})} \, dx\)

Optimal. Leaf size=30 \[ \log \left (x+\frac {e^7-\frac {1}{2 x}+x-\log \left (6 e^{-x}\right )}{\log (4)}\right ) \]

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Rubi [F]  time = 0.37, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-1-4 x^2-2 x^2 \log (4)}{x-2 e^7 x^2-2 x^3-2 x^3 \log (4)+2 x^2 \log \left (6 e^{-x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1 - 4*x^2 - 2*x^2*Log[4])/(x - 2*E^7*x^2 - 2*x^3 - 2*x^3*Log[4] + 2*x^2*Log[6/E^x]),x]

[Out]

Defer[Int][1/(x*(-1 + 2*E^7*x + 2*x^2*(1 + Log[4]) - 2*x*Log[6/E^x])), x] - 2*(2 + Log[4])*Defer[Int][x/(1 - 2
*E^7*x - 2*x^2*(1 + Log[4]) + 2*x*Log[6/E^x]), x]

Rubi steps

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

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Mathematica [A]  time = 0.28, size = 37, normalized size = 1.23 \begin {gather*} -\log (x)+\log \left (1-2 e^7 x-2 x^2-x^2 \log (16)+2 x \log \left (6 e^{-x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - 4*x^2 - 2*x^2*Log[4])/(x - 2*E^7*x^2 - 2*x^3 - 2*x^3*Log[4] + 2*x^2*Log[6/E^x]),x]

[Out]

-Log[x] + Log[1 - 2*E^7*x - 2*x^2 - x^2*Log[16] + 2*x*Log[6/E^x]]

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fricas [A]  time = 1.26, size = 30, normalized size = 1.00 \begin {gather*} \log \left (4 \, x^{2} \log \relax (2) + 4 \, x^{2} + 2 \, x e^{7} - 2 \, x \log \relax (6) - 1\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2*log(2)-4*x^2-1)/(2*x^2*log(6/exp(x))-4*x^3*log(2)-2*x^2*exp(7)-2*x^3+x),x, algorithm="fricas
")

[Out]

log(4*x^2*log(2) + 4*x^2 + 2*x*e^7 - 2*x*log(6) - 1) - log(x)

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giac [A]  time = 1.81, size = 32, normalized size = 1.07 \begin {gather*} \log \left ({\left | 4 \, x^{2} \log \relax (2) + 4 \, x^{2} + 2 \, x e^{7} - 2 \, x \log \relax (6) - 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2*log(2)-4*x^2-1)/(2*x^2*log(6/exp(x))-4*x^3*log(2)-2*x^2*exp(7)-2*x^3+x),x, algorithm="giac")

[Out]

log(abs(4*x^2*log(2) + 4*x^2 + 2*x*e^7 - 2*x*log(6) - 1)) - log(abs(x))

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maple [A]  time = 0.12, size = 36, normalized size = 1.20

method	result	size
norman	\(-\ln \relax (x )+\ln \left (4 x^{2} \ln \relax (2)+2 x \,{\mathrm e}^{7}+2 x^{2}-2 x \ln \left (6 \,{\mathrm e}^{-x}\right )-1\right )\)	\(36\)
risch	\(\ln \left (\ln \left ({\mathrm e}^{x}\right )-\frac {i \left (4 i x^{2} \ln \relax (2)-2 i x \ln \relax (3)+2 i x \,{\mathrm e}^{7}-2 i x \ln \relax (2)+2 i x^{2}-i\right )}{2 x}\right )\)	\(47\)
default	\(-\ln \relax (x )+\ln \left (4 x^{2} \ln \relax (2)+2 x \,{\mathrm e}^{7}+4 x^{2}-2 x \left (\ln \left (6 \,{\mathrm e}^{-x}\right )+\ln \left ({\mathrm e}^{x}\right )\right )+2 x \left (\ln \left ({\mathrm e}^{x}\right )-x \right )-1\right )\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x^2*ln(2)-4*x^2-1)/(2*x^2*ln(6/exp(x))-4*x^3*ln(2)-2*x^2*exp(7)-2*x^3+x),x,method=_RETURNVERBOSE)

[Out]

-ln(x)+ln(4*x^2*ln(2)+2*x*exp(7)+2*x^2-2*x*ln(6/exp(x))-1)

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maxima [B]  time = 0.45, size = 579, normalized size = 19.30 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2*log(2)-4*x^2-1)/(2*x^2*log(6/exp(x))-4*x^3*log(2)-2*x^2*exp(7)-2*x^3+x),x, algorithm="maxima
")

[Out]

-1/2*((e^7 - log(3) - log(2))*log((4*x*(log(2) + 1) - sqrt(-2*(e^7 - log(2))*log(3) + log(3)^2 - 2*(e^7 - 2)*l
og(2) + log(2)^2 + e^14 + 4) + e^7 - log(3) - log(2))/(4*x*(log(2) + 1) + sqrt(-2*(e^7 - log(2))*log(3) + log(
3)^2 - 2*(e^7 - 2)*log(2) + log(2)^2 + e^14 + 4) + e^7 - log(3) - log(2)))/(sqrt(-2*(e^7 - log(2))*log(3) + lo
g(3)^2 - 2*(e^7 - 2)*log(2) + log(2)^2 + e^14 + 4)*(log(2) + 1)) - log(4*x^2*(log(2) + 1) + 2*x*(e^7 - log(3)
- log(2)) - 1)/(log(2) + 1))*log(2) + 1/2*(e^7 - log(3) - log(2))*log((4*x*(log(2) + 1) - sqrt(-2*(e^7 - log(2
))*log(3) + log(3)^2 - 2*(e^7 - 2)*log(2) + log(2)^2 + e^14 + 4) + e^7 - log(3) - log(2))/(4*x*(log(2) + 1) +
sqrt(-2*(e^7 - log(2))*log(3) + log(3)^2 - 2*(e^7 - 2)*log(2) + log(2)^2 + e^14 + 4) + e^7 - log(3) - log(2)))
/sqrt(-2*(e^7 - log(2))*log(3) + log(3)^2 - 2*(e^7 - 2)*log(2) + log(2)^2 + e^14 + 4) - 1/2*(e^7 - log(3) - lo
g(2))*log((4*x*(log(2) + 1) - sqrt(-2*(e^7 - log(2))*log(3) + log(3)^2 - 2*(e^7 - 2)*log(2) + log(2)^2 + e^14
+ 4) + e^7 - log(3) - log(2))/(4*x*(log(2) + 1) + sqrt(-2*(e^7 - log(2))*log(3) + log(3)^2 - 2*(e^7 - 2)*log(2
) + log(2)^2 + e^14 + 4) + e^7 - log(3) - log(2)))/(sqrt(-2*(e^7 - log(2))*log(3) + log(3)^2 - 2*(e^7 - 2)*log
(2) + log(2)^2 + e^14 + 4)*(log(2) + 1)) + 1/2*log(4*x^2*(log(2) + 1) + 2*x*(e^7 - log(3) - log(2)) - 1)/(log(
2) + 1) + 1/2*log(4*x^2*(log(2) + 1) + 2*x*(e^7 - log(3) - log(2)) - 1) - log(x)

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mupad [B]  time = 1.23, size = 30, normalized size = 1.00 \begin {gather*} \ln \left (4\,x\,\ln \relax (6)-4\,x\,{\mathrm {e}}^7-8\,x^2\,\ln \relax (2)-8\,x^2+2\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2*log(2) + 4*x^2 + 1)/(2*x^2*exp(7) - 2*x^2*log(6*exp(-x)) - x + 4*x^3*log(2) + 2*x^3),x)

[Out]

log(4*x*log(6) - 4*x*exp(7) - 8*x^2*log(2) - 8*x^2 + 2) - log(x)

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sympy [A]  time = 1.18, size = 31, normalized size = 1.03 \begin {gather*} - \log {\relax (x )} + \log {\left (x^{2} + \frac {x \left (- \log {\relax (6 )} + e^{7}\right )}{2 \log {\relax (2 )} + 2} - \frac {1}{4 \log {\relax (2 )} + 4} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x**2*ln(2)-4*x**2-1)/(2*x**2*ln(6/exp(x))-4*x**3*ln(2)-2*x**2*exp(7)-2*x**3+x),x)

[Out]

-log(x) + log(x**2 + x*(-log(6) + exp(7))/(2*log(2) + 2) - 1/(4*log(2) + 4))

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