3.53.65 \(\int \frac {36-e^{3 x} x+12 \log (2)+\log ^2(2)+e^{2 x} (1-12 x-2 x \log (2))+e^x (12-32 x+(2-12 x) \log (2)-x \log ^2(2))+(72+2 e^{2 x}+24 \log (2)+2 \log ^2(2)+e^x (24+4 \log (2))) \log (x)}{36 x+e^{2 x} x+12 x \log (2)+x \log ^2(2)+e^x (12 x+2 x \log (2))} \, dx\) [5265]

Optimal. Leaf size=25 \[ -e^x-\frac {4}{6+e^x+\log (2)}+\log ^2(x)+\log (2 x) \]

[Out]

ln(2*x)-exp(x)-4/(6+ln(2)+exp(x))+ln(x)^2

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Rubi [A]
time = 1.65, antiderivative size = 29, normalized size of antiderivative = 1.16, number of steps used = 14, number of rules used = 10, integrand size = 128, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.078, Rules used = {6, 6873, 6874, 2225, 2320, 36, 29, 31, 46, 2338} \begin {gather*} -e^x+\frac {1}{4} (2 \log (x)+1)^2-\frac {4}{e^x+6+\log (2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(36 - E^(3*x)*x + 12*Log[2] + Log[2]^2 + E^(2*x)*(1 - 12*x - 2*x*Log[2]) + E^x*(12 - 32*x + (2 - 12*x)*Log
[2] - x*Log[2]^2) + (72 + 2*E^(2*x) + 24*Log[2] + 2*Log[2]^2 + E^x*(24 + 4*Log[2]))*Log[x])/(36*x + E^(2*x)*x
+ 12*x*Log[2] + x*Log[2]^2 + E^x*(12*x + 2*x*Log[2])),x]

[Out]

-E^x - 4/(6 + E^x + Log[2]) + (1 + 2*Log[x])^2/4

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 2225

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 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 6873

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

Rule 6874

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {36-e^{3 x} x+12 \log (2)+\log ^2(2)+e^{2 x} (1-12 x-2 x \log (2))+e^x \left (12-32 x+(2-12 x) \log (2)-x \log ^2(2)\right )+\left (72+2 e^{2 x}+24 \log (2)+2 \log ^2(2)+e^x (24+4 \log (2))\right ) \log (x)}{e^{2 x} x+x \log ^2(2)+x (36+12 \log (2))+e^x (12 x+2 x \log (2))} \, dx\\ &=\int \frac {36-e^{3 x} x+12 \log (2)+\log ^2(2)+e^{2 x} (1-12 x-2 x \log (2))+e^x \left (12-32 x+(2-12 x) \log (2)-x \log ^2(2)\right )+\left (72+2 e^{2 x}+24 \log (2)+2 \log ^2(2)+e^x (24+4 \log (2))\right ) \log (x)}{e^{2 x} x+e^x (12 x+2 x \log (2))+x \left (36+12 \log (2)+\log ^2(2)\right )} \, dx\\ &=\int \frac {-e^{3 x} x+e^{2 x} (1-12 x-2 x \log (2))+e^x \left (12-32 x+(2-12 x) \log (2)-x \log ^2(2)\right )+36 \left (1+\frac {1}{36} \log (2) (12+\log (2))\right )+\left (72+2 e^{2 x}+24 \log (2)+2 \log ^2(2)+e^x (24+4 \log (2))\right ) \log (x)}{x \left (e^x+6 \left (1+\frac {\log (2)}{6}\right )\right )^2} \, dx\\ &=\int \left (-e^x+\frac {4}{e^x+6 \left (1+\frac {\log (2)}{6}\right )}+\frac {4 (-6-\log (2))}{\left (e^x+6 \left (1+\frac {\log (2)}{6}\right )\right )^2}+\frac {1+2 \log (x)}{x}\right ) \, dx\\ &=4 \int \frac {1}{e^x+6 \left (1+\frac {\log (2)}{6}\right )} \, dx-(4 (6+\log (2))) \int \frac {1}{\left (e^x+6 \left (1+\frac {\log (2)}{6}\right )\right )^2} \, dx-\int e^x \, dx+\int \frac {1+2 \log (x)}{x} \, dx\\ &=-e^x+\frac {1}{4} (1+2 \log (x))^2+4 \text {Subst}\left (\int \frac {1}{x (6+x+\log (2))} \, dx,x,e^x\right )-(4 (6+\log (2))) \text {Subst}\left (\int \frac {1}{x (6+x+\log (2))^2} \, dx,x,e^x\right )\\ &=-e^x+\frac {1}{4} (1+2 \log (x))^2+\frac {4 \text {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )}{6+\log (2)}-\frac {4 \text {Subst}\left (\int \frac {1}{6+x+\log (2)} \, dx,x,e^x\right )}{6+\log (2)}-(4 (6+\log (2))) \text {Subst}\left (\int \left (\frac {1}{x (6+\log (2))^2}-\frac {1}{(6+\log (2)) (6+x+\log (2))^2}-\frac {1}{(6+\log (2))^2 (6+x+\log (2))}\right ) \, dx,x,e^x\right )\\ &=-e^x-\frac {4}{6+e^x+\log (2)}+\frac {1}{4} (1+2 \log (x))^2\\ \end {aligned} \end {gather*}

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Mathematica [F]
time = 0.86, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {36-e^{3 x} x+12 \log (2)+\log ^2(2)+e^{2 x} (1-12 x-2 x \log (2))+e^x \left (12-32 x+(2-12 x) \log (2)-x \log ^2(2)\right )+\left (72+2 e^{2 x}+24 \log (2)+2 \log ^2(2)+e^x (24+4 \log (2))\right ) \log (x)}{36 x+e^{2 x} x+12 x \log (2)+x \log ^2(2)+e^x (12 x+2 x \log (2))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(36 - E^(3*x)*x + 12*Log[2] + Log[2]^2 + E^(2*x)*(1 - 12*x - 2*x*Log[2]) + E^x*(12 - 32*x + (2 - 12*
x)*Log[2] - x*Log[2]^2) + (72 + 2*E^(2*x) + 24*Log[2] + 2*Log[2]^2 + E^x*(24 + 4*Log[2]))*Log[x])/(36*x + E^(2
*x)*x + 12*x*Log[2] + x*Log[2]^2 + E^x*(12*x + 2*x*Log[2])),x]

[Out]

Integrate[(36 - E^(3*x)*x + 12*Log[2] + Log[2]^2 + E^(2*x)*(1 - 12*x - 2*x*Log[2]) + E^x*(12 - 32*x + (2 - 12*
x)*Log[2] - x*Log[2]^2) + (72 + 2*E^(2*x) + 24*Log[2] + 2*Log[2]^2 + E^x*(24 + 4*Log[2]))*Log[x])/(36*x + E^(2
*x)*x + 12*x*Log[2] + x*Log[2]^2 + E^x*(12*x + 2*x*Log[2])), x]

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Maple [A]
time = 0.50, size = 47, normalized size = 1.88

method	result	size
risch	\(\ln \left (x \right )^{2}+\frac {\ln \left (2\right ) \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )-{\mathrm e}^{x} \ln \left (2\right )-{\mathrm e}^{2 x}+6 \ln \left (x \right )-6 \,{\mathrm e}^{x}-4}{6+\ln \left (2\right )+{\mathrm e}^{x}}\)	\(47\)
norman	\(\frac {\left (\ln \left (2\right )+6\right ) \ln \left (x \right )^{2}+\left (\ln \left (2\right )+6\right ) \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )^{2}+{\mathrm e}^{x} \ln \left (x \right )-{\mathrm e}^{2 x}+\ln \left (2\right )^{2}+12 \ln \left (2\right )+32}{6+\ln \left (2\right )+{\mathrm e}^{x}}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*exp(x)^2+(4*ln(2)+24)*exp(x)+2*ln(2)^2+24*ln(2)+72)*ln(x)-x*exp(x)^3+(-2*x*ln(2)-12*x+1)*exp(x)^2+(-x*
ln(2)^2+(-12*x+2)*ln(2)-32*x+12)*exp(x)+ln(2)^2+12*ln(2)+36)/(x*exp(x)^2+(2*x*ln(2)+12*x)*exp(x)+x*ln(2)^2+12*
x*ln(2)+36*x),x,method=_RETURNVERBOSE)

[Out]

ln(x)^2+(ln(2)*ln(x)+exp(x)*ln(x)-exp(x)*ln(2)-exp(2*x)+6*ln(x)-6*exp(x)-4)/(6+ln(2)+exp(x))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (23) = 46\).
time = 0.54, size = 48, normalized size = 1.92 \begin {gather*} \frac {{\left (\log \left (2\right ) + 6\right )} \log \left (x\right )^{2} + {\left (\log \left (x\right )^{2} - \log \left (2\right ) + \log \left (x\right ) - 6\right )} e^{x} + {\left (\log \left (2\right ) + 6\right )} \log \left (x\right ) - e^{\left (2 \, x\right )} - 4}{e^{x} + \log \left (2\right ) + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(x)^2+(4*log(2)+24)*exp(x)+2*log(2)^2+24*log(2)+72)*log(x)-x*exp(x)^3+(-2*x*log(2)-12*x+1)*ex
p(x)^2+(-x*log(2)^2+(-12*x+2)*log(2)-32*x+12)*exp(x)+log(2)^2+12*log(2)+36)/(x*exp(x)^2+(2*x*log(2)+12*x)*exp(
x)+x*log(2)^2+12*x*log(2)+36*x),x, algorithm="maxima")

[Out]

((log(2) + 6)*log(x)^2 + (log(x)^2 - log(2) + log(x) - 6)*e^x + (log(2) + 6)*log(x) - e^(2*x) - 4)/(e^x + log(
2) + 6)

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Fricas [A]
time = 0.36, size = 45, normalized size = 1.80 \begin {gather*} \frac {{\left (e^{x} + \log \left (2\right ) + 6\right )} \log \left (x\right )^{2} - {\left (\log \left (2\right ) + 6\right )} e^{x} + {\left (e^{x} + \log \left (2\right ) + 6\right )} \log \left (x\right ) - e^{\left (2 \, x\right )} - 4}{e^{x} + \log \left (2\right ) + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(x)^2+(4*log(2)+24)*exp(x)+2*log(2)^2+24*log(2)+72)*log(x)-x*exp(x)^3+(-2*x*log(2)-12*x+1)*ex
p(x)^2+(-x*log(2)^2+(-12*x+2)*log(2)-32*x+12)*exp(x)+log(2)^2+12*log(2)+36)/(x*exp(x)^2+(2*x*log(2)+12*x)*exp(
x)+x*log(2)^2+12*x*log(2)+36*x),x, algorithm="fricas")

[Out]

((e^x + log(2) + 6)*log(x)^2 - (log(2) + 6)*e^x + (e^x + log(2) + 6)*log(x) - e^(2*x) - 4)/(e^x + log(2) + 6)

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Sympy [A]
time = 0.12, size = 20, normalized size = 0.80 \begin {gather*} - e^{x} + \log {\left (x \right )}^{2} + \log {\left (x \right )} - \frac {4}{e^{x} + \log {\left (2 \right )} + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(x)**2+(4*ln(2)+24)*exp(x)+2*ln(2)**2+24*ln(2)+72)*ln(x)-x*exp(x)**3+(-2*x*ln(2)-12*x+1)*exp(
x)**2+(-x*ln(2)**2+(-12*x+2)*ln(2)-32*x+12)*exp(x)+ln(2)**2+12*ln(2)+36)/(x*exp(x)**2+(2*x*ln(2)+12*x)*exp(x)+
x*ln(2)**2+12*x*ln(2)+36*x),x)

[Out]

-exp(x) + log(x)**2 + log(x) - 4/(exp(x) + log(2) + 6)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (23) = 46\).
time = 0.39, size = 61, normalized size = 2.44 \begin {gather*} \frac {e^{x} \log \left (x\right )^{2} + \log \left (2\right ) \log \left (x\right )^{2} - e^{x} \log \left (2\right ) + e^{x} \log \left (x\right ) + \log \left (2\right ) \log \left (x\right ) + 6 \, \log \left (x\right )^{2} - e^{\left (2 \, x\right )} - 6 \, e^{x} + 6 \, \log \left (x\right ) - 4}{e^{x} + \log \left (2\right ) + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(x)^2+(4*log(2)+24)*exp(x)+2*log(2)^2+24*log(2)+72)*log(x)-x*exp(x)^3+(-2*x*log(2)-12*x+1)*ex
p(x)^2+(-x*log(2)^2+(-12*x+2)*log(2)-32*x+12)*exp(x)+log(2)^2+12*log(2)+36)/(x*exp(x)^2+(2*x*log(2)+12*x)*exp(
x)+x*log(2)^2+12*x*log(2)+36*x),x, algorithm="giac")

[Out]

(e^x*log(x)^2 + log(2)*log(x)^2 - e^x*log(2) + e^x*log(x) + log(2)*log(x) + 6*log(x)^2 - e^(2*x) - 6*e^x + 6*l
og(x) - 4)/(e^x + log(2) + 6)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {12\,\ln \left (2\right )-x\,{\mathrm {e}}^{3\,x}-{\mathrm {e}}^{2\,x}\,\left (12\,x+2\,x\,\ln \left (2\right )-1\right )-{\mathrm {e}}^x\,\left (32\,x+\ln \left (2\right )\,\left (12\,x-2\right )+x\,{\ln \left (2\right )}^2-12\right )+\ln \left (x\right )\,\left (2\,{\mathrm {e}}^{2\,x}+24\,\ln \left (2\right )+{\mathrm {e}}^x\,\left (4\,\ln \left (2\right )+24\right )+2\,{\ln \left (2\right )}^2+72\right )+{\ln \left (2\right )}^2+36}{36\,x+x\,{\mathrm {e}}^{2\,x}+12\,x\,\ln \left (2\right )+x\,{\ln \left (2\right )}^2+{\mathrm {e}}^x\,\left (12\,x+2\,x\,\ln \left (2\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*log(2) - x*exp(3*x) - exp(2*x)*(12*x + 2*x*log(2) - 1) - exp(x)*(32*x + log(2)*(12*x - 2) + x*log(2)^2
- 12) + log(x)*(2*exp(2*x) + 24*log(2) + exp(x)*(4*log(2) + 24) + 2*log(2)^2 + 72) + log(2)^2 + 36)/(36*x + x
*exp(2*x) + 12*x*log(2) + x*log(2)^2 + exp(x)*(12*x + 2*x*log(2))),x)

[Out]

int((12*log(2) - x*exp(3*x) - exp(2*x)*(12*x + 2*x*log(2) - 1) - exp(x)*(32*x + log(2)*(12*x - 2) + x*log(2)^2
- 12) + log(x)*(2*exp(2*x) + 24*log(2) + exp(x)*(4*log(2) + 24) + 2*log(2)^2 + 72) + log(2)^2 + 36)/(36*x + x
*exp(2*x) + 12*x*log(2) + x*log(2)^2 + exp(x)*(12*x + 2*x*log(2))), x)

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