3.101.66 \(\int \frac {-17+e^x (6 x+4 x^2+e^5 (2+4 x)) \log (2)+e^x (2 x+2 e^5 x+2 x^2) \log (2) \log (x)}{3 x} \, dx\)

Optimal. Leaf size=27 \[ \left (5+\frac {2}{3} \left (1-e^x \left (e^5+x\right ) \log (2)\right )\right ) (-2-\log (x)) \]

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Rubi [B]  time = 0.39, antiderivative size = 68, normalized size of antiderivative = 2.52, number of steps used = 16, number of rules used = 8, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 14, 6742, 2199, 2194, 2178, 2176, 2554} \begin {gather*} \frac {4}{3} e^x x \log (2)-\frac {2}{3} e^x \log (2) \log (x)+\frac {2}{3} e^x \left (x+e^5+1\right ) \log (2) \log (x)-\frac {17 \log (x)}{3}-2 e^x \log (2)+\frac {2}{3} \left (3+2 e^5\right ) e^x \log (2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-17 + E^x*(6*x + 4*x^2 + E^5*(2 + 4*x))*Log[2] + E^x*(2*x + 2*E^5*x + 2*x^2)*Log[2]*Log[x])/(3*x),x]

[Out]

-2*E^x*Log[2] + (2*E^x*(3 + 2*E^5)*Log[2])/3 + (4*E^x*x*Log[2])/3 - (17*Log[x])/3 - (2*E^x*Log[2]*Log[x])/3 +
(2*E^x*(1 + E^5 + x)*Log[2]*Log[x])/3

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2199

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !$UseGamma === True

Rule 2554

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x]
)/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {-17+e^x \left (6 x+4 x^2+e^5 (2+4 x)\right ) \log (2)+e^x \left (2 x+2 e^5 x+2 x^2\right ) \log (2) \log (x)}{x} \, dx\\ &=\frac {1}{3} \int \left (-\frac {17}{x}+\frac {2 e^x \log (2) \left (e^5+3 \left (1+\frac {2 e^5}{3}\right ) x+2 x^2+\left (1+e^5\right ) x \log (x)+x^2 \log (x)\right )}{x}\right ) \, dx\\ &=-\frac {17 \log (x)}{3}+\frac {1}{3} (2 \log (2)) \int \frac {e^x \left (e^5+3 \left (1+\frac {2 e^5}{3}\right ) x+2 x^2+\left (1+e^5\right ) x \log (x)+x^2 \log (x)\right )}{x} \, dx\\ &=-\frac {17 \log (x)}{3}+\frac {1}{3} (2 \log (2)) \int \left (\frac {e^x \left (e^5+\left (3+2 e^5\right ) x+2 x^2\right )}{x}+e^x \left (1+e^5+x\right ) \log (x)\right ) \, dx\\ &=-\frac {17 \log (x)}{3}+\frac {1}{3} (2 \log (2)) \int \frac {e^x \left (e^5+\left (3+2 e^5\right ) x+2 x^2\right )}{x} \, dx+\frac {1}{3} (2 \log (2)) \int e^x \left (1+e^5+x\right ) \log (x) \, dx\\ &=-\frac {17 \log (x)}{3}-\frac {2}{3} e^x \log (2) \log (x)+\frac {2}{3} e^x \left (1+e^5+x\right ) \log (2) \log (x)-\frac {1}{3} (2 \log (2)) \int \frac {e^x \left (e^5+x\right )}{x} \, dx+\frac {1}{3} (2 \log (2)) \int \left (e^x \left (3+2 e^5\right )+\frac {e^{5+x}}{x}+2 e^x x\right ) \, dx\\ &=-\frac {17 \log (x)}{3}-\frac {2}{3} e^x \log (2) \log (x)+\frac {2}{3} e^x \left (1+e^5+x\right ) \log (2) \log (x)-\frac {1}{3} (2 \log (2)) \int \left (e^x+\frac {e^{5+x}}{x}\right ) \, dx+\frac {1}{3} (2 \log (2)) \int \frac {e^{5+x}}{x} \, dx+\frac {1}{3} (4 \log (2)) \int e^x x \, dx+\frac {1}{3} \left (2 \left (3+2 e^5\right ) \log (2)\right ) \int e^x \, dx\\ &=\frac {2}{3} e^x \left (3+2 e^5\right ) \log (2)+\frac {4}{3} e^x x \log (2)+\frac {2}{3} e^5 \text {Ei}(x) \log (2)-\frac {17 \log (x)}{3}-\frac {2}{3} e^x \log (2) \log (x)+\frac {2}{3} e^x \left (1+e^5+x\right ) \log (2) \log (x)-\frac {1}{3} (2 \log (2)) \int e^x \, dx-\frac {1}{3} (2 \log (2)) \int \frac {e^{5+x}}{x} \, dx-\frac {1}{3} (4 \log (2)) \int e^x \, dx\\ &=-2 e^x \log (2)+\frac {2}{3} e^x \left (3+2 e^5\right ) \log (2)+\frac {4}{3} e^x x \log (2)-\frac {17 \log (x)}{3}-\frac {2}{3} e^x \log (2) \log (x)+\frac {2}{3} e^x \left (1+e^5+x\right ) \log (2) \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.26, size = 33, normalized size = 1.22 \begin {gather*} \frac {1}{3} \left (e^x \left (e^5+x\right ) \log (16)-17 \log (x)+e^x \left (e^5+x\right ) \log (4) \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-17 + E^x*(6*x + 4*x^2 + E^5*(2 + 4*x))*Log[2] + E^x*(2*x + 2*E^5*x + 2*x^2)*Log[2]*Log[x])/(3*x),x
]

[Out]

(E^x*(E^5 + x)*Log[16] - 17*Log[x] + E^x*(E^5 + x)*Log[4]*Log[x])/3

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fricas [A]  time = 0.63, size = 27, normalized size = 1.00 \begin {gather*} \frac {4}{3} \, {\left (x + e^{5}\right )} e^{x} \log \relax (2) + \frac {1}{3} \, {\left (2 \, {\left (x + e^{5}\right )} e^{x} \log \relax (2) - 17\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((2*x*exp(5)+2*x^2+2*x)*log(2)*exp(x)*log(x)+((4*x+2)*exp(5)+4*x^2+6*x)*log(2)*exp(x)-17)/x,x, a
lgorithm="fricas")

[Out]

4/3*(x + e^5)*e^x*log(2) + 1/3*(2*(x + e^5)*e^x*log(2) - 17)*log(x)

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giac [B]  time = 0.17, size = 39, normalized size = 1.44 \begin {gather*} \frac {2}{3} \, x e^{x} \log \relax (2) \log \relax (x) + \frac {4}{3} \, x e^{x} \log \relax (2) + \frac {2}{3} \, e^{\left (x + 5\right )} \log \relax (2) \log \relax (x) + \frac {4}{3} \, e^{\left (x + 5\right )} \log \relax (2) - \frac {17}{3} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((2*x*exp(5)+2*x^2+2*x)*log(2)*exp(x)*log(x)+((4*x+2)*exp(5)+4*x^2+6*x)*log(2)*exp(x)-17)/x,x, a
lgorithm="giac")

[Out]

2/3*x*e^x*log(2)*log(x) + 4/3*x*e^x*log(2) + 2/3*e^(x + 5)*log(2)*log(x) + 4/3*e^(x + 5)*log(2) - 17/3*log(x)

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maple [A]  time = 0.06, size = 33, normalized size = 1.22

method	result	size
risch	\(\frac {2 \ln \relax (2) \left ({\mathrm e}^{5}+x \right ) {\mathrm e}^{x} \ln \relax (x )}{3}+\frac {4 \ln \relax (2) {\mathrm e}^{5+x}}{3}+\frac {4 x \ln \relax (2) {\mathrm e}^{x}}{3}-\frac {17 \ln \relax (x )}{3}\)	\(33\)
default	\(\frac {4 x \ln \relax (2) {\mathrm e}^{x}}{3}+\frac {4 \,{\mathrm e}^{5} {\mathrm e}^{x} \ln \relax (2)}{3}+\frac {2 x \ln \relax (2) {\mathrm e}^{x} \ln \relax (x )}{3}+\frac {2 \,{\mathrm e}^{5} \ln \relax (2) {\mathrm e}^{x} \ln \relax (x )}{3}-\frac {17 \ln \relax (x )}{3}\)	\(40\)
norman	\(\frac {4 x \ln \relax (2) {\mathrm e}^{x}}{3}+\frac {4 \,{\mathrm e}^{5} {\mathrm e}^{x} \ln \relax (2)}{3}+\frac {2 x \ln \relax (2) {\mathrm e}^{x} \ln \relax (x )}{3}+\frac {2 \,{\mathrm e}^{5} \ln \relax (2) {\mathrm e}^{x} \ln \relax (x )}{3}-\frac {17 \ln \relax (x )}{3}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*((2*x*exp(5)+2*x^2+2*x)*ln(2)*exp(x)*ln(x)+((4*x+2)*exp(5)+4*x^2+6*x)*ln(2)*exp(x)-17)/x,x,method=_RET
URNVERBOSE)

[Out]

2/3*ln(2)*(exp(5)+x)*exp(x)*ln(x)+4/3*ln(2)*exp(5+x)+4/3*x*ln(2)*exp(x)-17/3*ln(x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {2}{3} \, {\rm Ei}\relax (x) e^{5} \log \relax (2) + \frac {4}{3} \, {\left (x - 1\right )} e^{x} \log \relax (2) + \frac {2}{3} \, {\left (x \log \relax (2) + e^{5} \log \relax (2) - \log \relax (2)\right )} e^{x} \log \relax (x) + \frac {2}{3} \, e^{x} \log \relax (2) \log \relax (x) - \frac {2}{3} \, {\rm Ei}\relax (x) \log \relax (2) + \frac {4}{3} \, e^{\left (x + 5\right )} \log \relax (2) + 2 \, e^{x} \log \relax (2) - \frac {1}{3} \, \int \frac {2 \, {\left (x \log \relax (2) + e^{5} \log \relax (2) - \log \relax (2)\right )} e^{x}}{x}\,{d x} - \frac {17}{3} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((2*x*exp(5)+2*x^2+2*x)*log(2)*exp(x)*log(x)+((4*x+2)*exp(5)+4*x^2+6*x)*log(2)*exp(x)-17)/x,x, a
lgorithm="maxima")

[Out]

2/3*Ei(x)*e^5*log(2) + 4/3*(x - 1)*e^x*log(2) + 2/3*(x*log(2) + e^5*log(2) - log(2))*e^x*log(x) + 2/3*e^x*log(
2)*log(x) - 2/3*Ei(x)*log(2) + 4/3*e^(x + 5)*log(2) + 2*e^x*log(2) - 1/3*integrate(2*(x*log(2) + e^5*log(2) -
log(2))*e^x/x, x) - 17/3*log(x)

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mupad [B]  time = 8.40, size = 39, normalized size = 1.44 \begin {gather*} \frac {4\,{\mathrm {e}}^{x+5}\,\ln \relax (2)}{3}-\frac {17\,\ln \relax (x)}{3}+\frac {4\,x\,{\mathrm {e}}^x\,\ln \relax (2)}{3}+\frac {2\,{\mathrm {e}}^{x+5}\,\ln \relax (2)\,\ln \relax (x)}{3}+\frac {2\,x\,{\mathrm {e}}^x\,\ln \relax (2)\,\ln \relax (x)}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(x)*log(2)*(6*x + 4*x^2 + exp(5)*(4*x + 2)))/3 + (exp(x)*log(2)*log(x)*(2*x + 2*x*exp(5) + 2*x^2))/3
- 17/3)/x,x)

[Out]

(4*exp(x + 5)*log(2))/3 - (17*log(x))/3 + (4*x*exp(x)*log(2))/3 + (2*exp(x + 5)*log(2)*log(x))/3 + (2*x*exp(x)
*log(2)*log(x))/3

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sympy [A]  time = 0.47, size = 48, normalized size = 1.78 \begin {gather*} \frac {\left (2 x \log {\relax (2 )} \log {\relax (x )} + 4 x \log {\relax (2 )} + 2 e^{5} \log {\relax (2 )} \log {\relax (x )} + 4 e^{5} \log {\relax (2 )}\right ) e^{x}}{3} - \frac {17 \log {\relax (x )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((2*x*exp(5)+2*x**2+2*x)*ln(2)*exp(x)*ln(x)+((4*x+2)*exp(5)+4*x**2+6*x)*ln(2)*exp(x)-17)/x,x)

[Out]

(2*x*log(2)*log(x) + 4*x*log(2) + 2*exp(5)*log(2)*log(x) + 4*exp(5)*log(2))*exp(x)/3 - 17*log(x)/3

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