3.7.84 \(\int \frac {(-36 x^3-24 x^4-4 x^5) \log ^2(x)+e^{\frac {1}{(3 x^3+x^4) \log (x)}} (3+x+(9+4 x) \log (x)+(9 x^3+6 x^4+x^5) \log ^2(x))}{e^{\frac {1}{(3 x^3+x^4) \log (x)}} (-72 x^3-48 x^4-8 x^5) \log ^2(x)+e^{\frac {2}{(3 x^3+x^4) \log (x)}} (9 x^3+6 x^4+x^5) \log ^2(x)+(144 x^3+96 x^4+16 x^5) \log ^2(x)} \, dx\)

Optimal. Leaf size=26 \[ 2-\frac {x}{4-e^{\frac {1}{x^3 (3+x) \log (x)}}} \]

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Rubi [F]  time = 26.21, 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 (-36 x^3-24 x^4-4 x^5\right ) \log ^2(x)+e^{\frac {1}{\left (3 x^3+x^4\right ) \log (x)}} \left (3+x+(9+4 x) \log (x)+\left (9 x^3+6 x^4+x^5\right ) \log ^2(x)\right )}{e^{\frac {1}{\left (3 x^3+x^4\right ) \log (x)}} \left (-72 x^3-48 x^4-8 x^5\right ) \log ^2(x)+e^{\frac {2}{\left (3 x^3+x^4\right ) \log (x)}} \left (9 x^3+6 x^4+x^5\right ) \log ^2(x)+\left (144 x^3+96 x^4+16 x^5\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-36*x^3 - 24*x^4 - 4*x^5)*Log[x]^2 + E^(1/((3*x^3 + x^4)*Log[x]))*(3 + x + (9 + 4*x)*Log[x] + (9*x^3 + 6
*x^4 + x^5)*Log[x]^2))/(E^(1/((3*x^3 + x^4)*Log[x]))*(-72*x^3 - 48*x^4 - 8*x^5)*Log[x]^2 + E^(2/((3*x^3 + x^4)
*Log[x]))*(9*x^3 + 6*x^4 + x^5)*Log[x]^2 + (144*x^3 + 96*x^4 + 16*x^5)*Log[x]^2),x]

[Out]

Defer[Int][(-4 + E^(1/(x^3*(3 + x)*Log[x])))^(-1), x] + (4*Defer[Int][1/((-4 + E^(1/(x^3*(3 + x)*Log[x])))^2*x
^3*Log[x]^2), x])/3 + Defer[Int][1/((-4 + E^(1/(x^3*(3 + x)*Log[x])))*x^3*Log[x]^2), x]/3 - (4*Defer[Int][1/((
-4 + E^(1/(x^3*(3 + x)*Log[x])))^2*x^2*Log[x]^2), x])/9 - Defer[Int][1/((-4 + E^(1/(x^3*(3 + x)*Log[x])))*x^2*
Log[x]^2), x]/9 + (4*Defer[Int][1/((-4 + E^(1/(x^3*(3 + x)*Log[x])))^2*x*Log[x]^2), x])/27 + Defer[Int][1/((-4
+ E^(1/(x^3*(3 + x)*Log[x])))*x*Log[x]^2), x]/27 - (4*Defer[Int][1/((-4 + E^(1/(x^3*(3 + x)*Log[x])))^2*(3 +
x)*Log[x]^2), x])/27 - Defer[Int][1/((-4 + E^(1/(x^3*(3 + x)*Log[x])))*(3 + x)*Log[x]^2), x]/27 + 4*Defer[Int]
[1/((-4 + E^(1/(x^3*(3 + x)*Log[x])))^2*x^3*Log[x]), x] + Defer[Int][1/((-4 + E^(1/(x^3*(3 + x)*Log[x])))*x^3*
Log[x]), x] - (8*Defer[Int][1/((-4 + E^(1/(x^3*(3 + x)*Log[x])))^2*x^2*Log[x]), x])/9 - (2*Defer[Int][1/((-4 +
E^(1/(x^3*(3 + x)*Log[x])))*x^2*Log[x]), x])/9 + (4*Defer[Int][1/((-4 + E^(1/(x^3*(3 + x)*Log[x])))^2*x*Log[x
]), x])/27 + Defer[Int][1/((-4 + E^(1/(x^3*(3 + x)*Log[x])))*x*Log[x]), x]/27 + (4*Defer[Int][1/((-4 + E^(1/(x
^3*(3 + x)*Log[x])))^2*(3 + x)^2*Log[x]), x])/9 + Defer[Int][1/((-4 + E^(1/(x^3*(3 + x)*Log[x])))*(3 + x)^2*Lo
g[x]), x]/9 - (4*Defer[Int][1/((-4 + E^(1/(x^3*(3 + x)*Log[x])))^2*(3 + x)*Log[x]), x])/27 - Defer[Int][1/((-4
+ E^(1/(x^3*(3 + x)*Log[x])))*(3 + x)*Log[x]), x]/27

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {1}{x^3 (3+x) \log (x)}} (3+x)+e^{\frac {1}{x^3 (3+x) \log (x)}} (9+4 x) \log (x)+\left (-4+e^{\frac {1}{x^3 (3+x) \log (x)}}\right ) x^3 (3+x)^2 \log ^2(x)}{\left (4-e^{\frac {1}{x^3 (3+x) \log (x)}}\right )^2 x^3 (3+x)^2 \log ^2(x)} \, dx\\ &=\int \left (\frac {4 (3+x+9 \log (x)+4 x \log (x))}{\left (-4+e^{\frac {1}{x^3 (3+x) \log (x)}}\right )^2 x^3 (3+x)^2 \log ^2(x)}+\frac {3+x+9 \log (x)+4 x \log (x)+9 x^3 \log ^2(x)+6 x^4 \log ^2(x)+x^5 \log ^2(x)}{\left (-4+e^{\frac {1}{x^3 (3+x) \log (x)}}\right ) x^3 (3+x)^2 \log ^2(x)}\right ) \, dx\\ &=4 \int \frac {3+x+9 \log (x)+4 x \log (x)}{\left (-4+e^{\frac {1}{x^3 (3+x) \log (x)}}\right )^2 x^3 (3+x)^2 \log ^2(x)} \, dx+\int \frac {3+x+9 \log (x)+4 x \log (x)+9 x^3 \log ^2(x)+6 x^4 \log ^2(x)+x^5 \log ^2(x)}{\left (-4+e^{\frac {1}{x^3 (3+x) \log (x)}}\right ) x^3 (3+x)^2 \log ^2(x)} \, dx\\ &=4 \int \left (\frac {3+x+9 \log (x)+4 x \log (x)}{9 \left (-4+e^{\frac {1}{x^3 (3+x) \log (x)}}\right )^2 x^3 \log ^2(x)}-\frac {2 (3+x+9 \log (x)+4 x \log (x))}{27 \left (-4+e^{\frac {1}{x^3 (3+x) \log (x)}}\right )^2 x^2 \log ^2(x)}+\frac {3+x+9 \log (x)+4 x \log (x)}{27 \left (-4+e^{\frac {1}{x^3 (3+x) \log (x)}}\right )^2 x \log ^2(x)}-\frac {3+x+9 \log (x)+4 x \log (x)}{27 \left (-4+e^{\frac {1}{x^3 (3+x) \log (x)}}\right )^2 (3+x)^2 \log ^2(x)}-\frac {3+x+9 \log (x)+4 x \log (x)}{27 \left (-4+e^{\frac {1}{x^3 (3+x) \log (x)}}\right )^2 (3+x) \log ^2(x)}\right ) \, dx+\int \left (\frac {3+x+9 \log (x)+4 x \log (x)+9 x^3 \log ^2(x)+6 x^4 \log ^2(x)+x^5 \log ^2(x)}{9 \left (-4+e^{\frac {1}{x^3 (3+x) \log (x)}}\right ) x^3 \log ^2(x)}-\frac {2 \left (3+x+9 \log (x)+4 x \log (x)+9 x^3 \log ^2(x)+6 x^4 \log ^2(x)+x^5 \log ^2(x)\right )}{27 \left (-4+e^{\frac {1}{x^3 (3+x) \log (x)}}\right ) x^2 \log ^2(x)}+\frac {3+x+9 \log (x)+4 x \log (x)+9 x^3 \log ^2(x)+6 x^4 \log ^2(x)+x^5 \log ^2(x)}{27 \left (-4+e^{\frac {1}{x^3 (3+x) \log (x)}}\right ) x \log ^2(x)}-\frac {3+x+9 \log (x)+4 x \log (x)+9 x^3 \log ^2(x)+6 x^4 \log ^2(x)+x^5 \log ^2(x)}{27 \left (-4+e^{\frac {1}{x^3 (3+x) \log (x)}}\right ) (3+x)^2 \log ^2(x)}-\frac {3+x+9 \log (x)+4 x \log (x)+9 x^3 \log ^2(x)+6 x^4 \log ^2(x)+x^5 \log ^2(x)}{27 \left (-4+e^{\frac {1}{x^3 (3+x) \log (x)}}\right ) (3+x) \log ^2(x)}\right ) \, dx\\ &=\frac {1}{27} \int \frac {3+x+9 \log (x)+4 x \log (x)+9 x^3 \log ^2(x)+6 x^4 \log ^2(x)+x^5 \log ^2(x)}{\left (-4+e^{\frac {1}{x^3 (3+x) \log (x)}}\right ) x \log ^2(x)} \, dx-\frac {1}{27} \int \frac {3+x+9 \log (x)+4 x \log (x)+9 x^3 \log ^2(x)+6 x^4 \log ^2(x)+x^5 \log ^2(x)}{\left (-4+e^{\frac {1}{x^3 (3+x) \log (x)}}\right ) (3+x)^2 \log ^2(x)} \, dx-\frac {1}{27} \int \frac {3+x+9 \log (x)+4 x \log (x)+9 x^3 \log ^2(x)+6 x^4 \log ^2(x)+x^5 \log ^2(x)}{\left (-4+e^{\frac {1}{x^3 (3+x) \log (x)}}\right ) (3+x) \log ^2(x)} \, dx-\frac {2}{27} \int \frac {3+x+9 \log (x)+4 x \log (x)+9 x^3 \log ^2(x)+6 x^4 \log ^2(x)+x^5 \log ^2(x)}{\left (-4+e^{\frac {1}{x^3 (3+x) \log (x)}}\right ) x^2 \log ^2(x)} \, dx+\frac {1}{9} \int \frac {3+x+9 \log (x)+4 x \log (x)+9 x^3 \log ^2(x)+6 x^4 \log ^2(x)+x^5 \log ^2(x)}{\left (-4+e^{\frac {1}{x^3 (3+x) \log (x)}}\right ) x^3 \log ^2(x)} \, dx+\frac {4}{27} \int \frac {3+x+9 \log (x)+4 x \log (x)}{\left (-4+e^{\frac {1}{x^3 (3+x) \log (x)}}\right )^2 x \log ^2(x)} \, dx-\frac {4}{27} \int \frac {3+x+9 \log (x)+4 x \log (x)}{\left (-4+e^{\frac {1}{x^3 (3+x) \log (x)}}\right )^2 (3+x)^2 \log ^2(x)} \, dx-\frac {4}{27} \int \frac {3+x+9 \log (x)+4 x \log (x)}{\left (-4+e^{\frac {1}{x^3 (3+x) \log (x)}}\right )^2 (3+x) \log ^2(x)} \, dx-\frac {8}{27} \int \frac {3+x+9 \log (x)+4 x \log (x)}{\left (-4+e^{\frac {1}{x^3 (3+x) \log (x)}}\right )^2 x^2 \log ^2(x)} \, dx+\frac {4}{9} \int \frac {3+x+9 \log (x)+4 x \log (x)}{\left (-4+e^{\frac {1}{x^3 (3+x) \log (x)}}\right )^2 x^3 \log ^2(x)} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 21, normalized size = 0.81 \begin {gather*} \frac {x}{-4+e^{\frac {1}{x^3 (3+x) \log (x)}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-36*x^3 - 24*x^4 - 4*x^5)*Log[x]^2 + E^(1/((3*x^3 + x^4)*Log[x]))*(3 + x + (9 + 4*x)*Log[x] + (9*x
^3 + 6*x^4 + x^5)*Log[x]^2))/(E^(1/((3*x^3 + x^4)*Log[x]))*(-72*x^3 - 48*x^4 - 8*x^5)*Log[x]^2 + E^(2/((3*x^3
+ x^4)*Log[x]))*(9*x^3 + 6*x^4 + x^5)*Log[x]^2 + (144*x^3 + 96*x^4 + 16*x^5)*Log[x]^2),x]

[Out]

x/(-4 + E^(1/(x^3*(3 + x)*Log[x])))

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fricas [A]  time = 0.74, size = 23, normalized size = 0.88 \begin {gather*} \frac {x}{e^{\left (\frac {1}{{\left (x^{4} + 3 \, x^{3}\right )} \log \relax (x)}\right )} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^5+6*x^4+9*x^3)*log(x)^2+(4*x+9)*log(x)+3+x)*exp(1/(x^4+3*x^3)/log(x))+(-4*x^5-24*x^4-36*x^3)*lo
g(x)^2)/((x^5+6*x^4+9*x^3)*log(x)^2*exp(1/(x^4+3*x^3)/log(x))^2+(-8*x^5-48*x^4-72*x^3)*log(x)^2*exp(1/(x^4+3*x
^3)/log(x))+(16*x^5+96*x^4+144*x^3)*log(x)^2),x, algorithm="fricas")

[Out]

x/(e^(1/((x^4 + 3*x^3)*log(x))) - 4)

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giac [A]  time = 0.73, size = 23, normalized size = 0.88 \begin {gather*} \frac {x}{e^{\left (\frac {1}{x^{4} \log \relax (x) + 3 \, x^{3} \log \relax (x)}\right )} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^5+6*x^4+9*x^3)*log(x)^2+(4*x+9)*log(x)+3+x)*exp(1/(x^4+3*x^3)/log(x))+(-4*x^5-24*x^4-36*x^3)*lo
g(x)^2)/((x^5+6*x^4+9*x^3)*log(x)^2*exp(1/(x^4+3*x^3)/log(x))^2+(-8*x^5-48*x^4-72*x^3)*log(x)^2*exp(1/(x^4+3*x
^3)/log(x))+(16*x^5+96*x^4+144*x^3)*log(x)^2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 21, normalized size = 0.81

method	result	size
risch	\(\frac {x}{{\mathrm e}^{\frac {1}{x^{3} \ln \relax (x ) \left (3+x \right )}}-4}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^5+6*x^4+9*x^3)*ln(x)^2+(4*x+9)*ln(x)+3+x)*exp(1/(x^4+3*x^3)/ln(x))+(-4*x^5-24*x^4-36*x^3)*ln(x)^2)/((
x^5+6*x^4+9*x^3)*ln(x)^2*exp(1/(x^4+3*x^3)/ln(x))^2+(-8*x^5-48*x^4-72*x^3)*ln(x)^2*exp(1/(x^4+3*x^3)/ln(x))+(1
6*x^5+96*x^4+144*x^3)*ln(x)^2),x,method=_RETURNVERBOSE)

[Out]

x/(exp(1/x^3/ln(x)/(3+x))-4)

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maxima [B]  time = 0.89, size = 74, normalized size = 2.85 \begin {gather*} -\frac {x e^{\left (\frac {1}{27 \, {\left (x + 3\right )} \log \relax (x)} + \frac {1}{9 \, x^{2} \log \relax (x)}\right )}}{4 \, e^{\left (\frac {1}{27 \, {\left (x + 3\right )} \log \relax (x)} + \frac {1}{9 \, x^{2} \log \relax (x)}\right )} - e^{\left (\frac {1}{27 \, x \log \relax (x)} + \frac {1}{3 \, x^{3} \log \relax (x)}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^5+6*x^4+9*x^3)*log(x)^2+(4*x+9)*log(x)+3+x)*exp(1/(x^4+3*x^3)/log(x))+(-4*x^5-24*x^4-36*x^3)*lo
g(x)^2)/((x^5+6*x^4+9*x^3)*log(x)^2*exp(1/(x^4+3*x^3)/log(x))^2+(-8*x^5-48*x^4-72*x^3)*log(x)^2*exp(1/(x^4+3*x
^3)/log(x))+(16*x^5+96*x^4+144*x^3)*log(x)^2),x, algorithm="maxima")

[Out]

-x*e^(1/27/((x + 3)*log(x)) + 1/9/(x^2*log(x)))/(4*e^(1/27/((x + 3)*log(x)) + 1/9/(x^2*log(x))) - e^(1/27/(x*l
og(x)) + 1/3/(x^3*log(x))))

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mupad [B]  time = 1.04, size = 23, normalized size = 0.88 \begin {gather*} \frac {x}{{\mathrm {e}}^{\frac {1}{3\,x^3\,\ln \relax (x)+x^4\,\ln \relax (x)}}-4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)^2*(36*x^3 + 24*x^4 + 4*x^5) - exp(1/(log(x)*(3*x^3 + x^4)))*(x + log(x)^2*(9*x^3 + 6*x^4 + x^5) +
log(x)*(4*x + 9) + 3))/(log(x)^2*(144*x^3 + 96*x^4 + 16*x^5) + exp(2/(log(x)*(3*x^3 + x^4)))*log(x)^2*(9*x^3
+ 6*x^4 + x^5) - exp(1/(log(x)*(3*x^3 + x^4)))*log(x)^2*(72*x^3 + 48*x^4 + 8*x^5)),x)

[Out]

x/(exp(1/(3*x^3*log(x) + x^4*log(x))) - 4)

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sympy [A]  time = 0.46, size = 17, normalized size = 0.65 \begin {gather*} \frac {x}{e^{\frac {1}{\left (x^{4} + 3 x^{3}\right ) \log {\relax (x )}}} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**5+6*x**4+9*x**3)*ln(x)**2+(4*x+9)*ln(x)+3+x)*exp(1/(x**4+3*x**3)/ln(x))+(-4*x**5-24*x**4-36*x*
*3)*ln(x)**2)/((x**5+6*x**4+9*x**3)*ln(x)**2*exp(1/(x**4+3*x**3)/ln(x))**2+(-8*x**5-48*x**4-72*x**3)*ln(x)**2*
exp(1/(x**4+3*x**3)/ln(x))+(16*x**5+96*x**4+144*x**3)*ln(x)**2),x)

[Out]

x/(exp(1/((x**4 + 3*x**3)*log(x))) - 4)

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