∫ 392−49x+9x3 log2(3)+e2x(72−81x+18x2) log2(3)+ex((336−210x+42x2) log(3)+(72−45x+9x2) log2(3))__________________________________________________________________________ −196x+49x2+9x4 log2(3)+e2x(−36x+9x2) log2(3)+ex((−168x+42x2) log(3)+(−36x+9x2) log2(3)) dx

3.69.11 \(\int \frac {392-49 x+9 x^3 \log ^2(3)+e^{2 x} (72-81 x+18 x^2) \log ^2(3)+e^x ((336-210 x+42 x^2) \log (3)+(72-45 x+9 x^2) \log ^2(3))}{-196 x+49 x^2+9 x^4 \log ^2(3)+e^{2 x} (-36 x+9 x^2) \log ^2(3)+e^x ((-168 x+42 x^2) \log (3)+(-36 x+9 x^2) \log ^2(3))} \, dx\)

Optimal. Leaf size=31 \[ \log \left (x-\frac {(4-x) \left (e^x+\left (e^x+\frac {7}{3 \log (3)}\right )^2\right )}{x^2}\right ) \]

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Rubi [F] time = 28.69, 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 {392-49 x+9 x^3 \log ^2(3)+e^{2 x} \left (72-81 x+18 x^2\right ) \log ^2(3)+e^x \left (\left (336-210 x+42 x^2\right ) \log (3)+\left (72-45 x+9 x^2\right ) \log ^2(3)\right )}{-196 x+49 x^2+9 x^4 \log ^2(3)+e^{2 x} \left (-36 x+9 x^2\right ) \log ^2(3)+e^x \left (\left (-168 x+42 x^2\right ) \log (3)+\left (-36 x+9 x^2\right ) \log ^2(3)\right )} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Int[(392 - 49*x + 9*x^3*Log[3]^2 + E^(2*x)*(72 - 81*x + 18*x^2)*Log[3]^2 + E^x*((336 - 210*x + 42*x^2)*Log[3]

+ (72 - 45*x + 9*x^2)*Log[3]^2))/(-196*x + 49*x^2 + 9*x^4*Log[3]^2 + E^(2*x)*(-36*x + 9*x^2)*Log[3]^2 + E^x*((

-168*x + 42*x^2)*Log[3] + (-36*x + 9*x^2)*Log[3]^2)),x]

[Out]

2*x + Log[4 - x] - 2*Log[x] + 1152*Log[3]^2*Defer[Int][(196 - 49*x + 36*E^(2*x)*Log[3]^2 - 9*E^(2*x)*x*Log[3]^

2 - 9*x^3*Log[3]^2 + 168*E^x*Log[3]*(1 + Log[27]^2/(42*Log[3])) - 42*E^x*x*Log[3]*(1 + Log[27]^2/(42*Log[3])))

^(-1), x] + 3136*Defer[Int][1/((4 - x)*(196 - 49*x + 36*E^(2*x)*Log[3]^2 - 9*E^(2*x)*x*Log[3]^2 - 9*x^3*Log[3]

^2 + 168*E^x*Log[3]*(1 + Log[27]^2/(42*Log[3])) - 42*E^x*x*Log[3]*(1 + Log[27]^2/(42*Log[3])))), x] + 5760*Log

[3]^2*Defer[Int][1/((4 - x)*(196 - 49*x + 36*E^(2*x)*Log[3]^2 - 9*E^(2*x)*x*Log[3]^2 - 9*x^3*Log[3]^2 + 168*E^

x*Log[3]*(1 + Log[27]^2/(42*Log[3])) - 42*E^x*x*Log[3]*(1 + Log[27]^2/(42*Log[3])))), x] - 32*(49 + 54*Log[3]^

2)*Defer[Int][1/((4 - x)*(196 - 49*x + 36*E^(2*x)*Log[3]^2 - 9*E^(2*x)*x*Log[3]^2 - 9*x^3*Log[3]^2 + 168*E^x*L

og[3]*(1 + Log[27]^2/(42*Log[3])) - 42*E^x*x*Log[3]*(1 + Log[27]^2/(42*Log[3])))), x] + 1568*Defer[Int][1/((-4

+ x)*(196 - 49*x + 36*E^(2*x)*Log[3]^2 - 9*E^(2*x)*x*Log[3]^2 - 9*x^3*Log[3]^2 + 168*E^x*Log[3]*(1 + Log[27]^

2/(42*Log[3])) - 42*E^x*x*Log[3]*(1 + Log[27]^2/(42*Log[3])))), x] + 4608*Log[3]^2*Defer[Int][1/((-4 + x)*(196

- 49*x + 36*E^(2*x)*Log[3]^2 - 9*E^(2*x)*x*Log[3]^2 - 9*x^3*Log[3]^2 + 168*E^x*Log[3]*(1 + Log[27]^2/(42*Log[

3])) - 42*E^x*x*Log[3]*(1 + Log[27]^2/(42*Log[3])))), x] + 288*Log[3]^2*Defer[Int][x/(196 - 49*x + 36*E^(2*x)*

Log[3]^2 - 9*E^(2*x)*x*Log[3]^2 - 9*x^3*Log[3]^2 + 168*E^x*Log[3]*(1 + Log[27]^2/(42*Log[3])) - 42*E^x*x*Log[3

]*(1 + Log[27]^2/(42*Log[3]))), x] + 72*Log[3]^2*Defer[Int][x^2/(196 - 49*x + 36*E^(2*x)*Log[3]^2 - 9*E^(2*x)*

x*Log[3]^2 - 9*x^3*Log[3]^2 + 168*E^x*Log[3]*(1 + Log[27]^2/(42*Log[3])) - 42*E^x*x*Log[3]*(1 + Log[27]^2/(42*

Log[3]))), x] + 18*Log[3]^2*Defer[Int][x^3/(196 - 49*x + 36*E^(2*x)*Log[3]^2 - 9*E^(2*x)*x*Log[3]^2 - 9*x^3*Lo

g[3]^2 + 168*E^x*Log[3]*(1 + Log[27]^2/(42*Log[3])) - 42*E^x*x*Log[3]*(1 + Log[27]^2/(42*Log[3]))), x] + 784*D

efer[Int][(-196 + 49*x - 36*E^(2*x)*Log[3]^2 + 9*E^(2*x)*x*Log[3]^2 + 9*x^3*Log[3]^2 - 168*E^x*Log[3]*(1 + Log

[27]^2/(42*Log[3])) + 42*E^x*x*Log[3]*(1 + Log[27]^2/(42*Log[3])))^(-1), x] + 1440*Log[3]^2*Defer[Int][(-196 +

49*x - 36*E^(2*x)*Log[3]^2 + 9*E^(2*x)*x*Log[3]^2 + 9*x^3*Log[3]^2 - 168*E^x*Log[3]*(1 + Log[27]^2/(42*Log[3]

)) + 42*E^x*x*Log[3]*(1 + Log[27]^2/(42*Log[3])))^(-1), x] - 8*(49 + 54*Log[3]^2)*Defer[Int][(-196 + 49*x - 36

*E^(2*x)*Log[3]^2 + 9*E^(2*x)*x*Log[3]^2 + 9*x^3*Log[3]^2 - 168*E^x*Log[3]*(1 + Log[27]^2/(42*Log[3])) + 42*E^

x*x*Log[3]*(1 + Log[27]^2/(42*Log[3])))^(-1), x] + 4*(42*Log[3] + Log[27]^2)*Defer[Int][E^x/(-196 + 49*x - 36*

E^(2*x)*Log[3]^2 + 9*E^(2*x)*x*Log[3]^2 + 9*x^3*Log[3]^2 - 168*E^x*Log[3]*(1 + Log[27]^2/(42*Log[3])) + 42*E^x

*x*Log[3]*(1 + Log[27]^2/(42*Log[3]))), x] + 360*Log[3]^2*Defer[Int][x/(-196 + 49*x - 36*E^(2*x)*Log[3]^2 + 9*

E^(2*x)*x*Log[3]^2 + 9*x^3*Log[3]^2 - 168*E^x*Log[3]*(1 + Log[27]^2/(42*Log[3])) + 42*E^x*x*Log[3]*(1 + Log[27

]^2/(42*Log[3]))), x] - 2*(49 + 54*Log[3]^2)*Defer[Int][x/(-196 + 49*x - 36*E^(2*x)*Log[3]^2 + 9*E^(2*x)*x*Log

[3]^2 + 9*x^3*Log[3]^2 - 168*E^x*Log[3]*(1 + Log[27]^2/(42*Log[3])) + 42*E^x*x*Log[3]*(1 + Log[27]^2/(42*Log[3

]))), x] - (42*Log[3] + Log[27]^2)*Defer[Int][(E^x*x)/(-196 + 49*x - 36*E^(2*x)*Log[3]^2 + 9*E^(2*x)*x*Log[3]^

2 + 9*x^3*Log[3]^2 - 168*E^x*Log[3]*(1 + Log[27]^2/(42*Log[3])) + 42*E^x*x*Log[3]*(1 + Log[27]^2/(42*Log[3])))

, x] + 90*Log[3]^2*Defer[Int][x^2/(-196 + 49*x - 36*E^(2*x)*Log[3]^2 + 9*E^(2*x)*x*Log[3]^2 + 9*x^3*Log[3]^2 -

168*E^x*Log[3]*(1 + Log[27]^2/(42*Log[3])) + 42*E^x*x*Log[3]*(1 + Log[27]^2/(42*Log[3]))), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-392+49 x-9 x^3 \log ^2(3)-9 e^{2 x} \left (8-9 x+2 x^2\right ) \log ^2(3)-3 e^x \left (8-5 x+x^2\right ) \log (3) (14+\log (27))}{x \left (196-49 x-9 e^{2 x} (-4+x) \log ^2(3)-9 x^3 \log ^2(3)-3 e^x (-4+x) \log (3) (14+\log (27))\right )} \, dx\\ &=\int \left (\frac {8-9 x+2 x^2}{(-4+x) x}+\frac {-1568+784 x+90 x^3 \log ^2(3)-18 x^4 \log ^2(3)-98 x^2 \left (1+\frac {54 \log ^2(3)}{49}\right )-672 e^x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )+336 e^x x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )-42 e^x x^2 \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )}{(4-x) \left (196-49 x+36 e^{2 x} \log ^2(3)-9 e^{2 x} x \log ^2(3)-9 x^3 \log ^2(3)+168 e^x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )-42 e^x x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )\right )}\right ) \, dx\\ &=\int \frac {8-9 x+2 x^2}{(-4+x) x} \, dx+\int \frac {-1568+784 x+90 x^3 \log ^2(3)-18 x^4 \log ^2(3)-98 x^2 \left (1+\frac {54 \log ^2(3)}{49}\right )-672 e^x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )+336 e^x x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )-42 e^x x^2 \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )}{(4-x) \left (196-49 x+36 e^{2 x} \log ^2(3)-9 e^{2 x} x \log ^2(3)-9 x^3 \log ^2(3)+168 e^x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )-42 e^x x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B] time = 0.28, size = 77, normalized size = 2.48 \begin {gather*} -2 \log (x)+\log \left (196-49 x+168 e^x \log (3)-42 e^x x \log (3)+36 e^{2 x} \log ^2(3)-9 e^{2 x} x \log ^2(3)-9 x^3 \log ^2(3)+12 e^x \log (3) \log (27)-3 e^x x \log (3) \log (27)\right ) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(392 - 49*x + 9*x^3*Log[3]^2 + E^(2*x)*(72 - 81*x + 18*x^2)*Log[3]^2 + E^x*((336 - 210*x + 42*x^2)*L

og[3] + (72 - 45*x + 9*x^2)*Log[3]^2))/(-196*x + 49*x^2 + 9*x^4*Log[3]^2 + E^(2*x)*(-36*x + 9*x^2)*Log[3]^2 +

E^x*((-168*x + 42*x^2)*Log[3] + (-36*x + 9*x^2)*Log[3]^2)),x]

[Out]

-2*Log[x] + Log[196 - 49*x + 168*E^x*Log[3] - 42*E^x*x*Log[3] + 36*E^(2*x)*Log[3]^2 - 9*E^(2*x)*x*Log[3]^2 - 9

*x^3*Log[3]^2 + 12*E^x*Log[3]*Log[27] - 3*E^x*x*Log[3]*Log[27]]

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fricas [B] time = 0.63, size = 64, normalized size = 2.06 \begin {gather*} \log \left (x - 4\right ) - 2 \, \log \relax (x) + \log \left (\frac {9 \, x^{3} \log \relax (3)^{2} + 9 \, {\left (x - 4\right )} e^{\left (2 \, x\right )} \log \relax (3)^{2} + 3 \, {\left (3 \, {\left (x - 4\right )} \log \relax (3)^{2} + 14 \, {\left (x - 4\right )} \log \relax (3)\right )} e^{x} + 49 \, x - 196}{x - 4}\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x^2-81*x+72)*log(3)^2*exp(x)^2+((9*x^2-45*x+72)*log(3)^2+(42*x^2-210*x+336)*log(3))*exp(x)+9*x^

3*log(3)^2-49*x+392)/((9*x^2-36*x)*log(3)^2*exp(x)^2+((9*x^2-36*x)*log(3)^2+(42*x^2-168*x)*log(3))*exp(x)+9*x^

4*log(3)^2+49*x^2-196*x),x, algorithm="fricas")

[Out]

log(x - 4) - 2*log(x) + log((9*x^3*log(3)^2 + 9*(x - 4)*e^(2*x)*log(3)^2 + 3*(3*(x - 4)*log(3)^2 + 14*(x - 4)*

log(3))*e^x + 49*x - 196)/(x - 4))

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giac [B] time = 0.40, size = 71, normalized size = 2.29 \begin {gather*} \log \left (9 \, x^{3} \log \relax (3)^{2} + 9 \, x e^{\left (2 \, x\right )} \log \relax (3)^{2} + 9 \, x e^{x} \log \relax (3)^{2} + 42 \, x e^{x} \log \relax (3) - 36 \, e^{\left (2 \, x\right )} \log \relax (3)^{2} - 36 \, e^{x} \log \relax (3)^{2} - 168 \, e^{x} \log \relax (3) + 49 \, x - 196\right ) - 2 \, \log \relax (x) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x^2-81*x+72)*log(3)^2*exp(x)^2+((9*x^2-45*x+72)*log(3)^2+(42*x^2-210*x+336)*log(3))*exp(x)+9*x^

3*log(3)^2-49*x+392)/((9*x^2-36*x)*log(3)^2*exp(x)^2+((9*x^2-36*x)*log(3)^2+(42*x^2-168*x)*log(3))*exp(x)+9*x^

4*log(3)^2+49*x^2-196*x),x, algorithm="giac")

[Out]

log(9*x^3*log(3)^2 + 9*x*e^(2*x)*log(3)^2 + 9*x*e^x*log(3)^2 + 42*x*e^x*log(3) - 36*e^(2*x)*log(3)^2 - 36*e^x*

log(3)^2 - 168*e^x*log(3) + 49*x - 196) - 2*log(x)

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maple [A] time = 0.14, size = 55, normalized size = 1.77


method	result	size

risch	\(-2 \ln \relax (x )+\ln \left (x -4\right )+\ln \left ({\mathrm e}^{2 x}+\frac {\left (3 \ln \relax (3)+14\right ) {\mathrm e}^{x}}{3 \ln \relax (3)}+\frac {9 x^{3} \ln \relax (3)^{2}+49 x -196}{9 \ln \relax (3)^{2} \left (x -4\right )}\right )\)	\(55\)
norman	\(-2 \ln \relax (x )+\ln \left (9 x \ln \relax (3)^{2} {\mathrm e}^{2 x}+9 x^{3} \ln \relax (3)^{2}-36 \ln \relax (3)^{2} {\mathrm e}^{2 x}+9 x \ln \relax (3)^{2} {\mathrm e}^{x}-36 \ln \relax (3)^{2} {\mathrm e}^{x}+42 x \ln \relax (3) {\mathrm e}^{x}-168 \ln \relax (3) {\mathrm e}^{x}+49 x -196\right )\)	\(72\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((18*x^2-81*x+72)*ln(3)^2*exp(x)^2+((9*x^2-45*x+72)*ln(3)^2+(42*x^2-210*x+336)*ln(3))*exp(x)+9*x^3*ln(3)^2

-49*x+392)/((9*x^2-36*x)*ln(3)^2*exp(x)^2+((9*x^2-36*x)*ln(3)^2+(42*x^2-168*x)*ln(3))*exp(x)+9*x^4*ln(3)^2+49*

x^2-196*x),x,method=_RETURNVERBOSE)

[Out]

-2*ln(x)+ln(x-4)+ln(exp(2*x)+1/3/ln(3)*(3*ln(3)+14)*exp(x)+1/9*(9*x^3*ln(3)^2+49*x-196)/ln(3)^2/(x-4))

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maxima [B] time = 0.55, size = 88, normalized size = 2.84 \begin {gather*} \log \left (x - 4\right ) - 2 \, \log \relax (x) + \log \left (\frac {9 \, x^{3} \log \relax (3)^{2} + 9 \, {\left (x \log \relax (3)^{2} - 4 \, \log \relax (3)^{2}\right )} e^{\left (2 \, x\right )} + 3 \, {\left ({\left (3 \, \log \relax (3)^{2} + 14 \, \log \relax (3)\right )} x - 12 \, \log \relax (3)^{2} - 56 \, \log \relax (3)\right )} e^{x} + 49 \, x - 196}{9 \, {\left (x \log \relax (3)^{2} - 4 \, \log \relax (3)^{2}\right )}}\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x^2-81*x+72)*log(3)^2*exp(x)^2+((9*x^2-45*x+72)*log(3)^2+(42*x^2-210*x+336)*log(3))*exp(x)+9*x^

3*log(3)^2-49*x+392)/((9*x^2-36*x)*log(3)^2*exp(x)^2+((9*x^2-36*x)*log(3)^2+(42*x^2-168*x)*log(3))*exp(x)+9*x^

4*log(3)^2+49*x^2-196*x),x, algorithm="maxima")

[Out]

log(x - 4) - 2*log(x) + log(1/9*(9*x^3*log(3)^2 + 9*(x*log(3)^2 - 4*log(3)^2)*e^(2*x) + 3*((3*log(3)^2 + 14*lo

g(3))*x - 12*log(3)^2 - 56*log(3))*e^x + 49*x - 196)/(x*log(3)^2 - 4*log(3)^2))

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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {9\,x^3\,{\ln \relax (3)}^2-49\,x+{\mathrm {e}}^x\,\left (\ln \relax (3)\,\left (42\,x^2-210\,x+336\right )+{\ln \relax (3)}^2\,\left (9\,x^2-45\,x+72\right )\right )+{\mathrm {e}}^{2\,x}\,{\ln \relax (3)}^2\,\left (18\,x^2-81\,x+72\right )+392}{196\,x-9\,x^4\,{\ln \relax (3)}^2+{\mathrm {e}}^x\,\left (\ln \relax (3)\,\left (168\,x-42\,x^2\right )+{\ln \relax (3)}^2\,\left (36\,x-9\,x^2\right )\right )-49\,x^2+{\mathrm {e}}^{2\,x}\,{\ln \relax (3)}^2\,\left (36\,x-9\,x^2\right )} \,d x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(-(9*x^3*log(3)^2 - 49*x + exp(x)*(log(3)*(42*x^2 - 210*x + 336) + log(3)^2*(9*x^2 - 45*x + 72)) + exp(2*x)

*log(3)^2*(18*x^2 - 81*x + 72) + 392)/(196*x - 9*x^4*log(3)^2 + exp(x)*(log(3)*(168*x - 42*x^2) + log(3)^2*(36

*x - 9*x^2)) - 49*x^2 + exp(2*x)*log(3)^2*(36*x - 9*x^2)),x)

[Out]

-int((9*x^3*log(3)^2 - 49*x + exp(x)*(log(3)*(42*x^2 - 210*x + 336) + log(3)^2*(9*x^2 - 45*x + 72)) + exp(2*x)

*log(3)^2*(18*x^2 - 81*x + 72) + 392)/(196*x - 9*x^4*log(3)^2 + exp(x)*(log(3)*(168*x - 42*x^2) + log(3)^2*(36

*x - 9*x^2)) - 49*x^2 + exp(2*x)*log(3)^2*(36*x - 9*x^2)), x)

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sympy [B] time = 1.24, size = 61, normalized size = 1.97 \begin {gather*} - 2 \log {\relax (x )} + \log {\left (x - 4 \right )} + \log {\left (e^{2 x} + \frac {\left (3 \log {\relax (3 )} + 14\right ) e^{x}}{3 \log {\relax (3 )}} + \frac {9 x^{3} \log {\relax (3 )}^{2} + 49 x - 196}{9 x \log {\relax (3 )}^{2} - 36 \log {\relax (3 )}^{2}} \right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x**2-81*x+72)*ln(3)**2*exp(x)**2+((9*x**2-45*x+72)*ln(3)**2+(42*x**2-210*x+336)*ln(3))*exp(x)+9

*x**3*ln(3)**2-49*x+392)/((9*x**2-36*x)*ln(3)**2*exp(x)**2+((9*x**2-36*x)*ln(3)**2+(42*x**2-168*x)*ln(3))*exp(

x)+9*x**4*ln(3)**2+49*x**2-196*x),x)

[Out]

-2*log(x) + log(x - 4) + log(exp(2*x) + (3*log(3) + 14)*exp(x)/(3*log(3)) + (9*x**3*log(3)**2 + 49*x - 196)/(9

*x*log(3)**2 - 36*log(3)**2))

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