3.18.35 \(\int \frac {6 e^x \log (5)+6 e^x x \log (5) \log (x)}{x+6 e^x x \log (5) \log (x)} \, dx\)

Optimal. Leaf size=14 \[ 4+\log \left (-1-6 e^x \log (5) \log (x)\right ) \]

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Rubi [F]  time = 0.73, 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 {6 e^x \log (5)+6 e^x x \log (5) \log (x)}{x+6 e^x x \log (5) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(6*E^x*Log[5] + 6*E^x*x*Log[5]*Log[x])/(x + 6*E^x*x*Log[5]*Log[x]),x]

[Out]

(6*x*Log[5])/Log[15625] - Defer[Int][(1 + 6*E^x*Log[5]*Log[x])^(-1), x] + 6*Log[5]*Defer[Int][E^x/(x*(1 + 6*E^
x*Log[5]*Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6 e^x \log (5) (1+x \log (x))}{x+6 e^x x \log (5) \log (x)} \, dx\\ &=(6 \log (5)) \int \frac {e^x (1+x \log (x))}{x+6 e^x x \log (5) \log (x)} \, dx\\ &=(6 \log (5)) \int \left (\frac {e^x}{x \left (1+6 e^x \log (5) \log (x)\right )}+\frac {e^x \log (x)}{1+6 e^x \log (5) \log (x)}\right ) \, dx\\ &=(6 \log (5)) \int \frac {e^x}{x \left (1+6 e^x \log (5) \log (x)\right )} \, dx+(6 \log (5)) \int \frac {e^x \log (x)}{1+6 e^x \log (5) \log (x)} \, dx\\ &=(6 \log (5)) \int \frac {e^x}{x \left (1+6 e^x \log (5) \log (x)\right )} \, dx+(6 \log (5)) \int \left (\frac {1}{\log (15625)}-\frac {1}{6 \log (5) \left (1+6 e^x \log (5) \log (x)\right )}\right ) \, dx\\ &=\frac {6 x \log (5)}{\log (15625)}+(6 \log (5)) \int \frac {e^x}{x \left (1+6 e^x \log (5) \log (x)\right )} \, dx-\int \frac {1}{1+6 e^x \log (5) \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 12, normalized size = 0.86 \begin {gather*} \log \left (1+6 e^x \log (5) \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6*E^x*Log[5] + 6*E^x*x*Log[5]*Log[x])/(x + 6*E^x*x*Log[5]*Log[x]),x]

[Out]

Log[1 + 6*E^x*Log[5]*Log[x]]

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fricas [A]  time = 0.81, size = 18, normalized size = 1.29 \begin {gather*} x + \log \left ({\left (6 \, e^{x} \log \relax (5) \log \relax (x) + 1\right )} e^{\left (-x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*log(5)*exp(x)*log(x)+6*exp(x)*log(5))/(6*x*log(5)*exp(x)*log(x)+x),x, algorithm="fricas")

[Out]

x + log((6*e^x*log(5)*log(x) + 1)*e^(-x))

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giac [A]  time = 0.35, size = 11, normalized size = 0.79 \begin {gather*} \log \left (6 \, e^{x} \log \relax (5) \log \relax (x) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*log(5)*exp(x)*log(x)+6*exp(x)*log(5))/(6*x*log(5)*exp(x)*log(x)+x),x, algorithm="giac")

[Out]

log(6*e^x*log(5)*log(x) + 1)

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maple [A]  time = 0.06, size = 12, normalized size = 0.86

method	result	size
norman	\(\ln \left (6 \ln \relax (5) {\mathrm e}^{x} \ln \relax (x )+1\right )\)	\(12\)
risch	\(x +\ln \left (\ln \relax (x )+\frac {{\mathrm e}^{-x}}{6 \ln \relax (5)}\right )\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x*ln(5)*exp(x)*ln(x)+6*exp(x)*ln(5))/(6*x*ln(5)*exp(x)*ln(x)+x),x,method=_RETURNVERBOSE)

[Out]

ln(6*ln(5)*exp(x)*ln(x)+1)

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maxima [A]  time = 0.55, size = 25, normalized size = 1.79 \begin {gather*} \log \left (\frac {6 \, e^{x} \log \relax (5) \log \relax (x) + 1}{6 \, \log \relax (5) \log \relax (x)}\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*log(5)*exp(x)*log(x)+6*exp(x)*log(5))/(6*x*log(5)*exp(x)*log(x)+x),x, algorithm="maxima")

[Out]

log(1/6*(6*e^x*log(5)*log(x) + 1)/(log(5)*log(x))) + log(log(x))

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mupad [B]  time = 1.23, size = 11, normalized size = 0.79 \begin {gather*} \ln \left (6\,{\mathrm {e}}^x\,\ln \relax (5)\,\ln \relax (x)+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*exp(x)*log(5) + 6*x*exp(x)*log(5)*log(x))/(x + 6*x*exp(x)*log(5)*log(x)),x)

[Out]

log(6*exp(x)*log(5)*log(x) + 1)

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sympy [A]  time = 0.36, size = 19, normalized size = 1.36 \begin {gather*} \log {\left (e^{x} + \frac {1}{6 \log {\relax (5 )} \log {\relax (x )}} \right )} + \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*ln(5)*exp(x)*ln(x)+6*exp(x)*ln(5))/(6*x*ln(5)*exp(x)*ln(x)+x),x)

[Out]

log(exp(x) + 1/(6*log(5)*log(x))) + log(log(x))

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