3.74.58 \(\int \frac {1-x+(3 x^3+3 e^4 x^3) \log (5)}{-x^2+(-3 x+x^4+e^4 x^4) \log (5)+x \log (x)} \, dx\)

Optimal. Leaf size=24 \[ \log \left (x+\left (3-x^2 \left (x+e^4 x\right )\right ) \log (5)-\log (x)\right ) \]

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Rubi [F]  time = 0.48, 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-x+\left (3 x^3+3 e^4 x^3\right ) \log (5)}{-x^2+\left (-3 x+x^4+e^4 x^4\right ) \log (5)+x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 - x + (3*x^3 + 3*E^4*x^3)*Log[5])/(-x^2 + (-3*x + x^4 + E^4*x^4)*Log[5] + x*Log[x]),x]

[Out]

-3*(1 + E^4)*Log[5]*Defer[Int][x^2/(x + 3*Log[5] - (1 + E^4)*x^3*Log[5] - Log[x]), x] + Defer[Int][(x - (1 + E
^4)*x^3*Log[5] + Log[125] - Log[x])^(-1), x] + Defer[Int][1/(x*(-x - 3*Log[5] + (1 + E^4)*x^3*Log[5] + Log[x])
), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1+x-3 \left (1+e^4\right ) x^3 \log (5)}{x^2-\left (-3 x+x^4+e^4 x^4\right ) \log (5)-x \log (x)} \, dx\\ &=\int \left (\frac {3 \left (-1-e^4\right ) x^2 \log (5)}{x+3 \log (5)-\left (1+e^4\right ) x^3 \log (5)-\log (x)}+\frac {1}{x-\left (1+e^4\right ) x^3 \log (5)+\log (125)-\log (x)}+\frac {1}{x \left (-x-3 \log (5)+\left (1+e^4\right ) x^3 \log (5)+\log (x)\right )}\right ) \, dx\\ &=-\left (\left (3 \left (1+e^4\right ) \log (5)\right ) \int \frac {x^2}{x+3 \log (5)-\left (1+e^4\right ) x^3 \log (5)-\log (x)} \, dx\right )+\int \frac {1}{x-\left (1+e^4\right ) x^3 \log (5)+\log (125)-\log (x)} \, dx+\int \frac {1}{x \left (-x-3 \log (5)+\left (1+e^4\right ) x^3 \log (5)+\log (x)\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.48, size = 26, normalized size = 1.08 \begin {gather*} \log \left (-x-3 \log (5)+x^3 \log (5)+e^4 x^3 \log (5)+\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - x + (3*x^3 + 3*E^4*x^3)*Log[5])/(-x^2 + (-3*x + x^4 + E^4*x^4)*Log[5] + x*Log[x]),x]

[Out]

Log[-x - 3*Log[5] + x^3*Log[5] + E^4*x^3*Log[5] + Log[x]]

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fricas [A]  time = 0.60, size = 21, normalized size = 0.88 \begin {gather*} \log \left ({\left (x^{3} e^{4} + x^{3} - 3\right )} \log \relax (5) - x + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^3*exp(4)+3*x^3)*log(5)-x+1)/(x*log(x)+(x^4*exp(4)+x^4-3*x)*log(5)-x^2),x, algorithm="fricas")

[Out]

log((x^3*e^4 + x^3 - 3)*log(5) - x + log(x))

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giac [A]  time = 0.17, size = 27, normalized size = 1.12 \begin {gather*} \log \left (-x^{3} e^{4} \log \relax (5) - x^{3} \log \relax (5) + x + 3 \, \log \relax (5) - \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^3*exp(4)+3*x^3)*log(5)-x+1)/(x*log(x)+(x^4*exp(4)+x^4-3*x)*log(5)-x^2),x, algorithm="giac")

[Out]

log(-x^3*e^4*log(5) - x^3*log(5) + x + 3*log(5) - log(x))

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maple [A]  time = 0.05, size = 26, normalized size = 1.08

method	result	size
norman	\(\ln \left (\ln \relax (5) {\mathrm e}^{4} x^{3}+x^{3} \ln \relax (5)+\ln \relax (x )-3 \ln \relax (5)-x \right )\)	\(26\)
risch	\(\ln \left (\ln \relax (5) {\mathrm e}^{4} x^{3}+x^{3} \ln \relax (5)+\ln \relax (x )-3 \ln \relax (5)-x \right )\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^3*exp(4)+3*x^3)*ln(5)-x+1)/(x*ln(x)+(x^4*exp(4)+x^4-3*x)*ln(5)-x^2),x,method=_RETURNVERBOSE)

[Out]

ln(ln(5)*exp(4)*x^3+x^3*ln(5)+ln(x)-3*ln(5)-x)

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maxima [A]  time = 0.48, size = 23, normalized size = 0.96 \begin {gather*} \log \left ({\left (e^{4} \log \relax (5) + \log \relax (5)\right )} x^{3} - x - 3 \, \log \relax (5) + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^3*exp(4)+3*x^3)*log(5)-x+1)/(x*log(x)+(x^4*exp(4)+x^4-3*x)*log(5)-x^2),x, algorithm="maxima")

[Out]

log((e^4*log(5) + log(5))*x^3 - x - 3*log(5) + log(x))

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mupad [B]  time = 5.76, size = 21, normalized size = 0.88 \begin {gather*} \ln \left (\ln \relax (x)-3\,\ln \relax (5)-x+x^3\,\ln \relax (5)\,\left ({\mathrm {e}}^4+1\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(5)*(3*x^3*exp(4) + 3*x^3) - x + 1)/(log(5)*(x^4*exp(4) - 3*x + x^4) + x*log(x) - x^2),x)

[Out]

log(log(x) - 3*log(5) - x + x^3*log(5)*(exp(4) + 1))

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sympy [A]  time = 0.20, size = 27, normalized size = 1.12 \begin {gather*} \log {\left (x^{3} \log {\relax (5 )} + x^{3} e^{4} \log {\relax (5 )} - x + \log {\relax (x )} - 3 \log {\relax (5 )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**3*exp(4)+3*x**3)*ln(5)-x+1)/(x*ln(x)+(x**4*exp(4)+x**4-3*x)*ln(5)-x**2),x)

[Out]

log(x**3*log(5) + x**3*exp(4)*log(5) - x + log(x) - 3*log(5))

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