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

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

________________________________________________________________________________________

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 = {} $$\begin{array}{c}\int \frac{392-49x+9x^3{\log }^2(3)+e^{2x}(72-81x+18x^2){\log }^2(3)+e^x((336-210x+42x^2)\log (3)+(72-45x+9x^2){\log }^2(3))}{-196x+49x^2+9x^4{\log }^2(3)+e^{2x}(-36x+9x^2){\log }^2(3)+e^x((-168x+42x^2)\log (3)+(-36x+9x^2){\log }^2(3))}dx\end{array}$$

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

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

________________________________________________________________________________________

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

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]]

________________________________________________________________________________________

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

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))

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

method	result	size
risch	$-2\ln (x)+\ln (x-4)+\ln ({\mathrm{e}}^{2x}+\frac{(3\ln (3)+14){\mathrm{e}}^x}{3\ln (3)}+\frac{9x^3\ln (3{)}^2+49x-196}{9\ln (3{)}^2(x-4)})$	$55$
norman	$-2\ln (x)+\ln (9x\ln (3{)}^2{\mathrm{e}}^{2x}+9x^3\ln (3{)}^2-36\ln (3{)}^2{\mathrm{e}}^{2x}+9x\ln (3{)}^2{\mathrm{e}}^x-36\ln (3{)}^2{\mathrm{e}}^x+42x\ln (3){\mathrm{e}}^x-168\ln (3){\mathrm{e}}^x+49x-196)$	$72$

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))

________________________________________________________________________________________

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

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))

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

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))

________________________________________________________________________________________