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

Optimal. Leaf size=23 \[ \frac {4 e^5 \left (e^2+x-\log (5)+\log (2 x)\right )}{3 x} \]

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Rubi [A]  time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.78, number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {12, 2304} \begin {gather*} \frac {4 e^5}{3 x}-\frac {4 \left (e^5 \left (1-e^2+\log (5)\right )-e^5 \log (2 x)\right )}{3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 27, normalized size = 1.17 \begin {gather*} \frac {4 e^7}{3 x}+\frac {4 e^5 \log \left (\frac {2 x}{5}\right )}{3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*E^7)/(3*x) + (4*E^5*Log[(2*x)/5])/(3*x)

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fricas [A]  time = 0.72, size = 23, normalized size = 1.00 \begin {gather*} -\frac {4 \, {\left (e^{5} \log \relax (5) - e^{5} \log \left (2 \, x\right ) - e^{7}\right )}}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*(e^5*log(5) - e^5*log(2*x) - e^7)/x

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giac [A]  time = 0.17, size = 23, normalized size = 1.00 \begin {gather*} -\frac {4 \, {\left (e^{5} \log \relax (5) - e^{5} \log \left (2 \, x\right ) - e^{7}\right )}}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*(e^5*log(5) - e^5*log(2*x) - e^7)/x

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maple [A]  time = 0.06, size = 28, normalized size = 1.22

method	result	size
norman	\(\frac {\frac {4 \,{\mathrm e}^{5} \ln \left (2 x \right )}{3}+\frac {4 \,{\mathrm e}^{2} {\mathrm e}^{5}}{3}-\frac {4 \,{\mathrm e}^{5} \ln \relax (5)}{3}}{x}\)	\(28\)
risch	\(\frac {4 \,{\mathrm e}^{5} \ln \left (2 x \right )}{3 x}+\frac {4 \,{\mathrm e}^{5} {\mathrm e}^{2}}{3 x}-\frac {4 \,{\mathrm e}^{5} \ln \relax (5)}{3 x}\)	\(31\)
derivativedivides	\(\frac {4 \,{\mathrm e}^{5} {\mathrm e}^{2}}{3 x}-\frac {4 \,{\mathrm e}^{5} \ln \relax (5)}{3 x}-\frac {8 \,{\mathrm e}^{5} \left (-\frac {\ln \left (2 x \right )}{2 x}-\frac {1}{2 x}\right )}{3}-\frac {4 \,{\mathrm e}^{5}}{3 x}\)	\(48\)
default	\(\frac {4 \,{\mathrm e}^{5} {\mathrm e}^{2}}{3 x}-\frac {4 \,{\mathrm e}^{5} \ln \relax (5)}{3 x}-\frac {8 \,{\mathrm e}^{5} \left (-\frac {\ln \left (2 x \right )}{2 x}-\frac {1}{2 x}\right )}{3}-\frac {4 \,{\mathrm e}^{5}}{3 x}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.38, size = 39, normalized size = 1.70 \begin {gather*} -\frac {4 \, e^{5} \log \relax (5)}{3 \, x} + \frac {4 \, {\left (e^{5} \log \left (2 \, x\right ) + e^{5}\right )}}{3 \, x} + \frac {4 \, e^{7}}{3 \, x} - \frac {4 \, e^{5}}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*e^5*log(5)/x + 4/3*(e^5*log(2*x) + e^5)/x + 4/3*e^7/x - 4/3*e^5/x

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mupad [B]  time = 4.45, size = 18, normalized size = 0.78 \begin {gather*} \frac {4\,{\mathrm {e}}^5\,\left (\ln \left (2\,x\right )+{\mathrm {e}}^2-\ln \relax (5)\right )}{3\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*exp(5)*(log(2*x) + exp(2) - log(5)))/(3*x)

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sympy [A]  time = 0.13, size = 31, normalized size = 1.35 \begin {gather*} \frac {4 e^{5} \log {\left (2 x \right )}}{3 x} - \frac {- \frac {4 e^{7}}{3} + \frac {4 e^{5} \log {\relax (5 )}}{3}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*exp(5)*log(2*x)/(3*x) - (-4*exp(7)/3 + 4*exp(5)*log(5)/3)/x

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