3.82.34 \(\int \frac {(e^{2 x} (-24+8 x) \log ^2(3)+e^x (96 x-32 x^2+48 x^4-16 x^5) \log ^2(3)+(-96 x^2+32 x^3-96 x^5+32 x^6-24 x^8+8 x^9) \log ^2(3)) \log (x)+(e^{2 x} (-32 x+8 x^2) \log ^2(3)+e^x (48 x+64 x^2-16 x^3+96 x^4+8 x^5-8 x^6) \log ^2(3)+(-96 x^2-240 x^5+48 x^6-96 x^8+24 x^9) \log ^2(3)) \log ^2(x)}{-27 x+27 x^2-9 x^3+x^4} \, dx\)

Optimal. Leaf size=29 \[ \frac {4 \left (-e^x+2 x+x^4\right )^2 \log ^2(3) \log ^2(x)}{(-3+x)^2} \]

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Rubi [F]  time = 12.84, 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 {\left (e^{2 x} (-24+8 x) \log ^2(3)+e^x \left (96 x-32 x^2+48 x^4-16 x^5\right ) \log ^2(3)+\left (-96 x^2+32 x^3-96 x^5+32 x^6-24 x^8+8 x^9\right ) \log ^2(3)\right ) \log (x)+\left (e^{2 x} \left (-32 x+8 x^2\right ) \log ^2(3)+e^x \left (48 x+64 x^2-16 x^3+96 x^4+8 x^5-8 x^6\right ) \log ^2(3)+\left (-96 x^2-240 x^5+48 x^6-96 x^8+24 x^9\right ) \log ^2(3)\right ) \log ^2(x)}{-27 x+27 x^2-9 x^3+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((E^(2*x)*(-24 + 8*x)*Log[3]^2 + E^x*(96*x - 32*x^2 + 48*x^4 - 16*x^5)*Log[3]^2 + (-96*x^2 + 32*x^3 - 96*x
^5 + 32*x^6 - 24*x^8 + 8*x^9)*Log[3]^2)*Log[x] + (E^(2*x)*(-32*x + 8*x^2)*Log[3]^2 + E^x*(48*x + 64*x^2 - 16*x
^3 + 96*x^4 + 8*x^5 - 8*x^6)*Log[3]^2 + (-96*x^2 - 240*x^5 + 48*x^6 - 96*x^8 + 24*x^9)*Log[3]^2)*Log[x]^2)/(-2
7*x + 27*x^2 - 9*x^3 + x^4),x]

[Out]

16*E^x*Log[3]^2 + (8*E^6*ExpIntegralEi[-2*(3 - x)]*Log[3]^2)/9 - (464*E^3*ExpIntegralEi[-3 + x]*Log[3]^2)/3 +
(704*ExpIntegralEi[x]*Log[3]^2)/3 - (8*ExpIntegralEi[2*x]*Log[3]^2)/9 - (16*x*HypergeometricPFQ[{1, 1, 1}, {2,
2, 2}, 2*x]*Log[3]^2)/9 - (4*Log[3]^2*Log[-2*x]^2)/9 - 80*E^x*Log[3]^2*Log[x] - (8*EulerGamma*Log[3]^2*Log[x]
)/9 - (464*E^x*Log[3]^2*Log[x])/(3 - x) + (8*E^(2*x)*Log[3]^2*Log[x])/(3*(3 - x)) - 16*E^x*x*Log[3]^2*Log[x] +
(40*E^6*ExpIntegralEi[-2*(3 - x)]*Log[3]^2*Log[x])/9 - 896*E^3*ExpIntegralEi[-3 + x]*Log[3]^2*Log[x] + (8*Exp
IntegralEi[2*x]*Log[3]^2*Log[x])/9 - (8*(ExpIntegralE[1, -2*x] + ExpIntegralEi[2*x])*Log[3]^2*Log[x])/9 - 3364
*Log[3]^2*Log[x]^2 + (30276*Log[3]^2*Log[x]^2)/(3 - x)^2 + 6264*x*Log[3]^2*Log[x]^2 - (25520*x*Log[3]^2*Log[x]
^2)/(3 - x) + 1716*x^2*Log[3]^2*Log[x]^2 + 448*x^3*Log[3]^2*Log[x]^2 + 108*x^4*Log[3]^2*Log[x]^2 + 24*x^5*Log[
3]^2*Log[x]^2 + 4*x^6*Log[3]^2*Log[x]^2 + 896*E^3*Log[3]^2*Defer[Int][ExpIntegralEi[-3 + x]/x, x] - (40*E^6*Lo
g[3]^2*Defer[Int][ExpIntegralEi[-6 + 2*x]/x, x])/9 - 264*Log[3]^2*Defer[Int][E^x*Log[x]^2, x] + 1392*Log[3]^2*
Defer[Int][(E^x*Log[x]^2)/(-3 + x)^3, x] - 8*Log[3]^2*Defer[Int][(E^(2*x)*Log[x]^2)/(-3 + x)^3, x] + 184*Log[3
]^2*Defer[Int][(E^x*Log[x]^2)/(-3 + x)^2, x] + 8*Log[3]^2*Defer[Int][(E^(2*x)*Log[x]^2)/(-3 + x)^2, x] - 880*L
og[3]^2*Defer[Int][(E^x*Log[x]^2)/(-3 + x), x] - 64*Log[3]^2*Defer[Int][E^x*x*Log[x]^2, x] - 8*Log[3]^2*Defer[
Int][E^x*x^2*Log[x]^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 \left (e^x-x \left (2+x^3\right )\right ) \log ^2(3) \log (x) \left (-\left ((-3+x) \left (e^x-x \left (2+x^3\right )\right )\right )-x \left (6+e^x (-4+x)+12 x^3-3 x^4\right ) \log (x)\right )}{(3-x)^3 x} \, dx\\ &=\left (8 \log ^2(3)\right ) \int \frac {\left (e^x-x \left (2+x^3\right )\right ) \log (x) \left (-\left ((-3+x) \left (e^x-x \left (2+x^3\right )\right )\right )-x \left (6+e^x (-4+x)+12 x^3-3 x^4\right ) \log (x)\right )}{(3-x)^3 x} \, dx\\ &=\left (8 \log ^2(3)\right ) \int \left (\frac {x \left (2+x^3\right )^2 \log (x)}{(-3+x)^2}-\frac {6 x \left (2+x^3\right ) \log ^2(x)}{(-3+x)^3}-\frac {12 x^4 \left (2+x^3\right ) \log ^2(x)}{(-3+x)^3}+\frac {3 x^5 \left (2+x^3\right ) \log ^2(x)}{(-3+x)^3}+\frac {e^{2 x} \log (x) \left (-3+x-4 x \log (x)+x^2 \log (x)\right )}{(-3+x)^3 x}-\frac {e^x \log (x) \left (-12+4 x-6 x^3+2 x^4-6 \log (x)-8 x \log (x)+2 x^2 \log (x)-12 x^3 \log (x)-x^4 \log (x)+x^5 \log (x)\right )}{(-3+x)^3}\right ) \, dx\\ &=\left (8 \log ^2(3)\right ) \int \frac {x \left (2+x^3\right )^2 \log (x)}{(-3+x)^2} \, dx+\left (8 \log ^2(3)\right ) \int \frac {e^{2 x} \log (x) \left (-3+x-4 x \log (x)+x^2 \log (x)\right )}{(-3+x)^3 x} \, dx-\left (8 \log ^2(3)\right ) \int \frac {e^x \log (x) \left (-12+4 x-6 x^3+2 x^4-6 \log (x)-8 x \log (x)+2 x^2 \log (x)-12 x^3 \log (x)-x^4 \log (x)+x^5 \log (x)\right )}{(-3+x)^3} \, dx+\left (24 \log ^2(3)\right ) \int \frac {x^5 \left (2+x^3\right ) \log ^2(x)}{(-3+x)^3} \, dx-\left (48 \log ^2(3)\right ) \int \frac {x \left (2+x^3\right ) \log ^2(x)}{(-3+x)^3} \, dx-\left (96 \log ^2(3)\right ) \int \frac {x^4 \left (2+x^3\right ) \log ^2(x)}{(-3+x)^3} \, dx\\ &=\left (8 \log ^2(3)\right ) \int \left (1566 \log (x)+\frac {2523 \log (x)}{(-3+x)^2}+\frac {5539 \log (x)}{-3+x}+429 x \log (x)+112 x^2 \log (x)+27 x^3 \log (x)+6 x^4 \log (x)+x^5 \log (x)\right ) \, dx+\left (8 \log ^2(3)\right ) \int \left (\frac {e^{2 x} \log (x)}{(-3+x)^2 x}+\frac {e^{2 x} (-4+x) \log ^2(x)}{(-3+x)^3}\right ) \, dx-\left (8 \log ^2(3)\right ) \int \left (\frac {2 e^x \left (2+x^3\right ) \log (x)}{(-3+x)^2}+\frac {e^x \left (-6-8 x+2 x^2-12 x^3-x^4+x^5\right ) \log ^2(x)}{(-3+x)^3}\right ) \, dx+\left (24 \log ^2(3)\right ) \int \left (5211 \log ^2(x)+\frac {7047 \log ^2(x)}{(-3+x)^3}+\frac {18306 \log ^2(x)}{(-3+x)^2}+\frac {20952 \log ^2(x)}{-3+x}+1233 x \log ^2(x)+272 x^2 \log ^2(x)+54 x^3 \log ^2(x)+9 x^4 \log ^2(x)+x^5 \log ^2(x)\right ) \, dx-\left (48 \log ^2(3)\right ) \int \left (9 \log ^2(x)+\frac {87 \log ^2(x)}{(-3+x)^3}+\frac {110 \log ^2(x)}{(-3+x)^2}+\frac {54 \log ^2(x)}{-3+x}+x \log ^2(x)\right ) \, dx-\left (96 \log ^2(3)\right ) \int \left (1233 \log ^2(x)+\frac {2349 \log ^2(x)}{(-3+x)^3}+\frac {5319 \log ^2(x)}{(-3+x)^2}+\frac {5211 \log ^2(x)}{-3+x}+272 x \log ^2(x)+54 x^2 \log ^2(x)+9 x^3 \log ^2(x)+x^4 \log ^2(x)\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 2.23, size = 29, normalized size = 1.00 \begin {gather*} \frac {4 \left (-e^x+2 x+x^4\right )^2 \log ^2(3) \log ^2(x)}{(-3+x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((E^(2*x)*(-24 + 8*x)*Log[3]^2 + E^x*(96*x - 32*x^2 + 48*x^4 - 16*x^5)*Log[3]^2 + (-96*x^2 + 32*x^3
- 96*x^5 + 32*x^6 - 24*x^8 + 8*x^9)*Log[3]^2)*Log[x] + (E^(2*x)*(-32*x + 8*x^2)*Log[3]^2 + E^x*(48*x + 64*x^2
- 16*x^3 + 96*x^4 + 8*x^5 - 8*x^6)*Log[3]^2 + (-96*x^2 - 240*x^5 + 48*x^6 - 96*x^8 + 24*x^9)*Log[3]^2)*Log[x]^
2)/(-27*x + 27*x^2 - 9*x^3 + x^4),x]

[Out]

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

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fricas [B]  time = 0.67, size = 62, normalized size = 2.14 \begin {gather*} -\frac {4 \, {\left (2 \, {\left (x^{4} + 2 \, x\right )} e^{x} \log \relax (3)^{2} - {\left (x^{8} + 4 \, x^{5} + 4 \, x^{2}\right )} \log \relax (3)^{2} - e^{\left (2 \, x\right )} \log \relax (3)^{2}\right )} \log \relax (x)^{2}}{x^{2} - 6 \, x + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^2-32*x)*log(3)^2*exp(x)^2+(-8*x^6+8*x^5+96*x^4-16*x^3+64*x^2+48*x)*log(3)^2*exp(x)+(24*x^9-96
*x^8+48*x^6-240*x^5-96*x^2)*log(3)^2)*log(x)^2+((8*x-24)*log(3)^2*exp(x)^2+(-16*x^5+48*x^4-32*x^2+96*x)*log(3)
^2*exp(x)+(8*x^9-24*x^8+32*x^6-96*x^5+32*x^3-96*x^2)*log(3)^2)*log(x))/(x^4-9*x^3+27*x^2-27*x),x, algorithm="f
ricas")

[Out]

-4*(2*(x^4 + 2*x)*e^x*log(3)^2 - (x^8 + 4*x^5 + 4*x^2)*log(3)^2 - e^(2*x)*log(3)^2)*log(x)^2/(x^2 - 6*x + 9)

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giac [B]  time = 0.16, size = 92, normalized size = 3.17 \begin {gather*} \frac {4 \, {\left (x^{8} \log \relax (3)^{2} \log \relax (x)^{2} + 4 \, x^{5} \log \relax (3)^{2} \log \relax (x)^{2} - 2 \, x^{4} e^{x} \log \relax (3)^{2} \log \relax (x)^{2} + 4 \, x^{2} \log \relax (3)^{2} \log \relax (x)^{2} - 4 \, x e^{x} \log \relax (3)^{2} \log \relax (x)^{2} + e^{\left (2 \, x\right )} \log \relax (3)^{2} \log \relax (x)^{2}\right )}}{x^{2} - 6 \, x + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^2-32*x)*log(3)^2*exp(x)^2+(-8*x^6+8*x^5+96*x^4-16*x^3+64*x^2+48*x)*log(3)^2*exp(x)+(24*x^9-96
*x^8+48*x^6-240*x^5-96*x^2)*log(3)^2)*log(x)^2+((8*x-24)*log(3)^2*exp(x)^2+(-16*x^5+48*x^4-32*x^2+96*x)*log(3)
^2*exp(x)+(8*x^9-24*x^8+32*x^6-96*x^5+32*x^3-96*x^2)*log(3)^2)*log(x))/(x^4-9*x^3+27*x^2-27*x),x, algorithm="g
iac")

[Out]

4*(x^8*log(3)^2*log(x)^2 + 4*x^5*log(3)^2*log(x)^2 - 2*x^4*e^x*log(3)^2*log(x)^2 + 4*x^2*log(3)^2*log(x)^2 - 4
*x*e^x*log(3)^2*log(x)^2 + e^(2*x)*log(3)^2*log(x)^2)/(x^2 - 6*x + 9)

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maple [A]  time = 0.11, size = 46, normalized size = 1.59

method	result	size
risch	\(\frac {4 \left (x^{8}+4 x^{5}-2 \,{\mathrm e}^{x} x^{4}+4 x^{2}-4 \,{\mathrm e}^{x} x +{\mathrm e}^{2 x}\right ) \ln \relax (3)^{2} \ln \relax (x )^{2}}{\left (x -3\right )^{2}}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((8*x^2-32*x)*ln(3)^2*exp(x)^2+(-8*x^6+8*x^5+96*x^4-16*x^3+64*x^2+48*x)*ln(3)^2*exp(x)+(24*x^9-96*x^8+48*
x^6-240*x^5-96*x^2)*ln(3)^2)*ln(x)^2+((8*x-24)*ln(3)^2*exp(x)^2+(-16*x^5+48*x^4-32*x^2+96*x)*ln(3)^2*exp(x)+(8
*x^9-24*x^8+32*x^6-96*x^5+32*x^3-96*x^2)*ln(3)^2)*ln(x))/(x^4-9*x^3+27*x^2-27*x),x,method=_RETURNVERBOSE)

[Out]

4*(x^8+4*x^5-2*exp(x)*x^4+4*x^2-4*exp(x)*x+exp(2*x))*ln(3)^2/(x-3)^2*ln(x)^2

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maxima [B]  time = 0.54, size = 160, normalized size = 5.52 \begin {gather*} 16 \, {\left (\frac {3 \, {\left (2 \, x - 3\right )} \log \relax (x)}{x^{2} - 6 \, x + 9} + \frac {3}{x - 3} - \log \left (x - 3\right ) + \log \relax (x)\right )} \log \relax (3)^{2} + 16 \, \log \relax (3)^{2} \log \left (x - 3\right ) - \frac {4 \, {\left (4 \, x^{2} \log \relax (3)^{2} \log \relax (x) - e^{\left (2 \, x\right )} \log \relax (3)^{2} \log \relax (x)^{2} + 2 \, {\left (x^{4} \log \relax (3)^{2} + 2 \, x \log \relax (3)^{2}\right )} e^{x} \log \relax (x)^{2} + 12 \, x \log \relax (3)^{2} - {\left (x^{8} \log \relax (3)^{2} + 4 \, x^{5} \log \relax (3)^{2} + 4 \, x^{2} \log \relax (3)^{2}\right )} \log \relax (x)^{2} - 36 \, \log \relax (3)^{2}\right )}}{x^{2} - 6 \, x + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^2-32*x)*log(3)^2*exp(x)^2+(-8*x^6+8*x^5+96*x^4-16*x^3+64*x^2+48*x)*log(3)^2*exp(x)+(24*x^9-96
*x^8+48*x^6-240*x^5-96*x^2)*log(3)^2)*log(x)^2+((8*x-24)*log(3)^2*exp(x)^2+(-16*x^5+48*x^4-32*x^2+96*x)*log(3)
^2*exp(x)+(8*x^9-24*x^8+32*x^6-96*x^5+32*x^3-96*x^2)*log(3)^2)*log(x))/(x^4-9*x^3+27*x^2-27*x),x, algorithm="m
axima")

[Out]

16*(3*(2*x - 3)*log(x)/(x^2 - 6*x + 9) + 3/(x - 3) - log(x - 3) + log(x))*log(3)^2 + 16*log(3)^2*log(x - 3) -
4*(4*x^2*log(3)^2*log(x) - e^(2*x)*log(3)^2*log(x)^2 + 2*(x^4*log(3)^2 + 2*x*log(3)^2)*e^x*log(x)^2 + 12*x*log
(3)^2 - (x^8*log(3)^2 + 4*x^5*log(3)^2 + 4*x^2*log(3)^2)*log(x)^2 - 36*log(3)^2)/(x^2 - 6*x + 9)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\ln \relax (x)}^2\,\left ({\ln \relax (3)}^2\,\left (-24\,x^9+96\,x^8-48\,x^6+240\,x^5+96\,x^2\right )-{\mathrm {e}}^x\,{\ln \relax (3)}^2\,\left (-8\,x^6+8\,x^5+96\,x^4-16\,x^3+64\,x^2+48\,x\right )+{\mathrm {e}}^{2\,x}\,{\ln \relax (3)}^2\,\left (32\,x-8\,x^2\right )\right )-\ln \relax (x)\,\left ({\mathrm {e}}^{2\,x}\,{\ln \relax (3)}^2\,\left (8\,x-24\right )-{\ln \relax (3)}^2\,\left (-8\,x^9+24\,x^8-32\,x^6+96\,x^5-32\,x^3+96\,x^2\right )+{\mathrm {e}}^x\,{\ln \relax (3)}^2\,\left (-16\,x^5+48\,x^4-32\,x^2+96\,x\right )\right )}{-x^4+9\,x^3-27\,x^2+27\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)^2*(log(3)^2*(96*x^2 + 240*x^5 - 48*x^6 + 96*x^8 - 24*x^9) - exp(x)*log(3)^2*(48*x + 64*x^2 - 16*x^
3 + 96*x^4 + 8*x^5 - 8*x^6) + exp(2*x)*log(3)^2*(32*x - 8*x^2)) - log(x)*(exp(2*x)*log(3)^2*(8*x - 24) - log(3
)^2*(96*x^2 - 32*x^3 + 96*x^5 - 32*x^6 + 24*x^8 - 8*x^9) + exp(x)*log(3)^2*(96*x - 32*x^2 + 48*x^4 - 16*x^5)))
/(27*x - 27*x^2 + 9*x^3 - x^4),x)

[Out]

int((log(x)^2*(log(3)^2*(96*x^2 + 240*x^5 - 48*x^6 + 96*x^8 - 24*x^9) - exp(x)*log(3)^2*(48*x + 64*x^2 - 16*x^
3 + 96*x^4 + 8*x^5 - 8*x^6) + exp(2*x)*log(3)^2*(32*x - 8*x^2)) - log(x)*(exp(2*x)*log(3)^2*(8*x - 24) - log(3
)^2*(96*x^2 - 32*x^3 + 96*x^5 - 32*x^6 + 24*x^8 - 8*x^9) + exp(x)*log(3)^2*(96*x - 32*x^2 + 48*x^4 - 16*x^5)))
/(27*x - 27*x^2 + 9*x^3 - x^4), x)

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sympy [B]  time = 0.81, size = 201, normalized size = 6.93 \begin {gather*} \frac {\left (4 x^{2} \log {\relax (3 )}^{2} \log {\relax (x )}^{2} - 24 x \log {\relax (3 )}^{2} \log {\relax (x )}^{2} + 36 \log {\relax (3 )}^{2} \log {\relax (x )}^{2}\right ) e^{2 x} + \left (- 8 x^{6} \log {\relax (3 )}^{2} \log {\relax (x )}^{2} + 48 x^{5} \log {\relax (3 )}^{2} \log {\relax (x )}^{2} - 72 x^{4} \log {\relax (3 )}^{2} \log {\relax (x )}^{2} - 16 x^{3} \log {\relax (3 )}^{2} \log {\relax (x )}^{2} + 96 x^{2} \log {\relax (3 )}^{2} \log {\relax (x )}^{2} - 144 x \log {\relax (3 )}^{2} \log {\relax (x )}^{2}\right ) e^{x}}{x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81} + \frac {\left (4 x^{8} \log {\relax (3 )}^{2} + 16 x^{5} \log {\relax (3 )}^{2} + 16 x^{2} \log {\relax (3 )}^{2}\right ) \log {\relax (x )}^{2}}{x^{2} - 6 x + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x**2-32*x)*ln(3)**2*exp(x)**2+(-8*x**6+8*x**5+96*x**4-16*x**3+64*x**2+48*x)*ln(3)**2*exp(x)+(24
*x**9-96*x**8+48*x**6-240*x**5-96*x**2)*ln(3)**2)*ln(x)**2+((8*x-24)*ln(3)**2*exp(x)**2+(-16*x**5+48*x**4-32*x
**2+96*x)*ln(3)**2*exp(x)+(8*x**9-24*x**8+32*x**6-96*x**5+32*x**3-96*x**2)*ln(3)**2)*ln(x))/(x**4-9*x**3+27*x*
*2-27*x),x)

[Out]

((4*x**2*log(3)**2*log(x)**2 - 24*x*log(3)**2*log(x)**2 + 36*log(3)**2*log(x)**2)*exp(2*x) + (-8*x**6*log(3)**
2*log(x)**2 + 48*x**5*log(3)**2*log(x)**2 - 72*x**4*log(3)**2*log(x)**2 - 16*x**3*log(3)**2*log(x)**2 + 96*x**
2*log(3)**2*log(x)**2 - 144*x*log(3)**2*log(x)**2)*exp(x))/(x**4 - 12*x**3 + 54*x**2 - 108*x + 81) + (4*x**8*l
og(3)**2 + 16*x**5*log(3)**2 + 16*x**2*log(3)**2)*log(x)**2/(x**2 - 6*x + 9)

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