3.100.57 \(\int \frac {14 e^5+63 x+42 x^2+(-63 x-84 x^2) \log (x)-6 e^5 x \log ^2(x)}{7 e^5 x \log ^2(x)} \, dx\)

Optimal. Leaf size=25 \[ -\frac {6 x}{7}-\frac {2+\frac {3 x (3+2 x)}{e^5}}{\log (x)} \]

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Rubi [F]  time = 0.22, 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 {14 e^5+63 x+42 x^2+\left (-63 x-84 x^2\right ) \log (x)-6 e^5 x \log ^2(x)}{7 e^5 x \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(14*E^5 + 63*x + 42*x^2 + (-63*x - 84*x^2)*Log[x] - 6*E^5*x*Log[x]^2)/(7*E^5*x*Log[x]^2),x]

[Out]

(-6*x)/7 - (12*ExpIntegralEi[2*Log[x]])/E^5 - (9*LogIntegral[x])/E^5 + Defer[Int][(2*E^5 + 9*x + 6*x^2)/(x*Log
[x]^2), x]/E^5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {14 e^5+63 x+42 x^2+\left (-63 x-84 x^2\right ) \log (x)-6 e^5 x \log ^2(x)}{x \log ^2(x)} \, dx}{7 e^5}\\ &=\frac {\int \left (-6 e^5+\frac {7 \left (2 e^5+9 x+6 x^2\right )}{x \log ^2(x)}-\frac {21 (3+4 x)}{\log (x)}\right ) \, dx}{7 e^5}\\ &=-\frac {6 x}{7}+\frac {\int \frac {2 e^5+9 x+6 x^2}{x \log ^2(x)} \, dx}{e^5}-\frac {3 \int \frac {3+4 x}{\log (x)} \, dx}{e^5}\\ &=-\frac {6 x}{7}+\frac {\int \frac {2 e^5+9 x+6 x^2}{x \log ^2(x)} \, dx}{e^5}-\frac {3 \int \left (\frac {3}{\log (x)}+\frac {4 x}{\log (x)}\right ) \, dx}{e^5}\\ &=-\frac {6 x}{7}+\frac {\int \frac {2 e^5+9 x+6 x^2}{x \log ^2(x)} \, dx}{e^5}-\frac {9 \int \frac {1}{\log (x)} \, dx}{e^5}-\frac {12 \int \frac {x}{\log (x)} \, dx}{e^5}\\ &=-\frac {6 x}{7}-\frac {9 \text {li}(x)}{e^5}+\frac {\int \frac {2 e^5+9 x+6 x^2}{x \log ^2(x)} \, dx}{e^5}-\frac {12 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )}{e^5}\\ &=-\frac {6 x}{7}-\frac {12 \text {Ei}(2 \log (x))}{e^5}-\frac {9 \text {li}(x)}{e^5}+\frac {\int \frac {2 e^5+9 x+6 x^2}{x \log ^2(x)} \, dx}{e^5}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 28, normalized size = 1.12 \begin {gather*} -\frac {6 x}{7}+\frac {-2 e^5-9 x-6 x^2}{e^5 \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(14*E^5 + 63*x + 42*x^2 + (-63*x - 84*x^2)*Log[x] - 6*E^5*x*Log[x]^2)/(7*E^5*x*Log[x]^2),x]

[Out]

(-6*x)/7 + (-2*E^5 - 9*x - 6*x^2)/(E^5*Log[x])

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fricas [A]  time = 0.62, size = 28, normalized size = 1.12 \begin {gather*} -\frac {{\left (6 \, x e^{5} \log \relax (x) + 42 \, x^{2} + 63 \, x + 14 \, e^{5}\right )} e^{\left (-5\right )}}{7 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/7*(-6*x*exp(5)*log(x)^2+(-84*x^2-63*x)*log(x)+14*exp(5)+42*x^2+63*x)/x/exp(5)/log(x)^2,x, algorith
m="fricas")

[Out]

-1/7*(6*x*e^5*log(x) + 42*x^2 + 63*x + 14*e^5)*e^(-5)/log(x)

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giac [A]  time = 0.16, size = 28, normalized size = 1.12 \begin {gather*} -\frac {{\left (6 \, x e^{5} \log \relax (x) + 42 \, x^{2} + 63 \, x + 14 \, e^{5}\right )} e^{\left (-5\right )}}{7 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/7*(-6*x*exp(5)*log(x)^2+(-84*x^2-63*x)*log(x)+14*exp(5)+42*x^2+63*x)/x/exp(5)/log(x)^2,x, algorith
m="giac")

[Out]

-1/7*(6*x*e^5*log(x) + 42*x^2 + 63*x + 14*e^5)*e^(-5)/log(x)

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maple [A]  time = 0.05, size = 26, normalized size = 1.04

method	result	size
risch	\(-\frac {6 x}{7}-\frac {{\mathrm e}^{-5} \left (6 x^{2}+2 \,{\mathrm e}^{5}+9 x \right )}{\ln \relax (x )}\)	\(26\)
norman	\(\frac {-2-9 x \,{\mathrm e}^{-5}-\frac {6 x \ln \relax (x )}{7}-6 \,{\mathrm e}^{-5} x^{2}}{\ln \relax (x )}\)	\(29\)
default	\(\frac {{\mathrm e}^{-5} \left (-6 x \,{\mathrm e}^{5}-\frac {42 x^{2}}{\ln \relax (x )}-\frac {14 \,{\mathrm e}^{5}}{\ln \relax (x )}-\frac {63 x}{\ln \relax (x )}\right )}{7}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/7*(-6*x*exp(5)*ln(x)^2+(-84*x^2-63*x)*ln(x)+14*exp(5)+42*x^2+63*x)/x/exp(5)/ln(x)^2,x,method=_RETURNVERB
OSE)

[Out]

-6/7*x-exp(-5)*(6*x^2+2*exp(5)+9*x)/ln(x)

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maxima [C]  time = 0.38, size = 46, normalized size = 1.84 \begin {gather*} -\frac {1}{7} \, {\left (6 \, x e^{5} + \frac {14 \, e^{5}}{\log \relax (x)} + 84 \, {\rm Ei}\left (2 \, \log \relax (x)\right ) + 63 \, {\rm Ei}\left (\log \relax (x)\right ) - 63 \, \Gamma \left (-1, -\log \relax (x)\right ) - 84 \, \Gamma \left (-1, -2 \, \log \relax (x)\right )\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/7*(-6*x*exp(5)*log(x)^2+(-84*x^2-63*x)*log(x)+14*exp(5)+42*x^2+63*x)/x/exp(5)/log(x)^2,x, algorith
m="maxima")

[Out]

-1/7*(6*x*e^5 + 14*e^5/log(x) + 84*Ei(2*log(x)) + 63*Ei(log(x)) - 63*gamma(-1, -log(x)) - 84*gamma(-1, -2*log(
x)))*e^(-5)

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mupad [B]  time = 7.44, size = 24, normalized size = 0.96 \begin {gather*} -\frac {6\,x}{7}-\frac {6\,{\mathrm {e}}^{-5}\,x^2+9\,{\mathrm {e}}^{-5}\,x+2}{\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-5)*(9*x + 2*exp(5) - (log(x)*(63*x + 84*x^2))/7 + 6*x^2 - (6*x*exp(5)*log(x)^2)/7))/(x*log(x)^2),x)

[Out]

- (6*x)/7 - (9*x*exp(-5) + 6*x^2*exp(-5) + 2)/log(x)

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sympy [A]  time = 0.10, size = 26, normalized size = 1.04 \begin {gather*} - \frac {6 x}{7} + \frac {- 6 x^{2} - 9 x - 2 e^{5}}{e^{5} \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/7*(-6*x*exp(5)*ln(x)**2+(-84*x**2-63*x)*ln(x)+14*exp(5)+42*x**2+63*x)/x/exp(5)/ln(x)**2,x)

[Out]

-6*x/7 + (-6*x**2 - 9*x - 2*exp(5))*exp(-5)/log(x)

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