3.52.25 \(\int \frac {384+576 e^x+288 e^{2 x}+48 e^{3 x}+(8 x^2+4 e^x x^2) \log ^3(x)+(4 x^2+e^x (2 x^2-2 x^3)) \log ^4(x)}{8 x+12 e^x x+6 e^{2 x} x+e^{3 x} x} \, dx\)

Optimal. Leaf size=26 \[ \frac {x^2 \log ^4(x)}{\left (2+e^x\right )^2}+16 \log \left (\frac {x^3}{2}\right ) \]

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Rubi [F]  time = 1.17, 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 {384+576 e^x+288 e^{2 x}+48 e^{3 x}+\left (8 x^2+4 e^x x^2\right ) \log ^3(x)+\left (4 x^2+e^x \left (2 x^2-2 x^3\right )\right ) \log ^4(x)}{8 x+12 e^x x+6 e^{2 x} x+e^{3 x} x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(384 + 576*E^x + 288*E^(2*x) + 48*E^(3*x) + (8*x^2 + 4*E^x*x^2)*Log[x]^3 + (4*x^2 + E^x*(2*x^2 - 2*x^3))*L
og[x]^4)/(8*x + 12*E^x*x + 6*E^(2*x)*x + E^(3*x)*x),x]

[Out]

48*Log[x] + 4*Defer[Int][(x*Log[x]^3)/(2 + E^x)^2, x] + 2*Defer[Int][(x*Log[x]^4)/(2 + E^x)^2, x] + 4*Defer[In
t][(x^2*Log[x]^4)/(2 + E^x)^3, x] - 2*Defer[Int][(x^2*Log[x]^4)/(2 + E^x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {48}{x}+\frac {4 x \log ^3(x)}{\left (2+e^x\right )^2}-\frac {2 \left (-2+e^x (-1+x)\right ) x \log ^4(x)}{\left (2+e^x\right )^3}\right ) \, dx\\ &=48 \log (x)-2 \int \frac {\left (-2+e^x (-1+x)\right ) x \log ^4(x)}{\left (2+e^x\right )^3} \, dx+4 \int \frac {x \log ^3(x)}{\left (2+e^x\right )^2} \, dx\\ &=48 \log (x)-2 \int \left (\frac {(-1+x) x \log ^4(x)}{\left (2+e^x\right )^2}-\frac {2 x^2 \log ^4(x)}{\left (2+e^x\right )^3}\right ) \, dx+4 \int \frac {x \log ^3(x)}{\left (2+e^x\right )^2} \, dx\\ &=48 \log (x)-2 \int \frac {(-1+x) x \log ^4(x)}{\left (2+e^x\right )^2} \, dx+4 \int \frac {x \log ^3(x)}{\left (2+e^x\right )^2} \, dx+4 \int \frac {x^2 \log ^4(x)}{\left (2+e^x\right )^3} \, dx\\ &=48 \log (x)-2 \int \left (-\frac {x \log ^4(x)}{\left (2+e^x\right )^2}+\frac {x^2 \log ^4(x)}{\left (2+e^x\right )^2}\right ) \, dx+4 \int \frac {x \log ^3(x)}{\left (2+e^x\right )^2} \, dx+4 \int \frac {x^2 \log ^4(x)}{\left (2+e^x\right )^3} \, dx\\ &=48 \log (x)+2 \int \frac {x \log ^4(x)}{\left (2+e^x\right )^2} \, dx-2 \int \frac {x^2 \log ^4(x)}{\left (2+e^x\right )^2} \, dx+4 \int \frac {x \log ^3(x)}{\left (2+e^x\right )^2} \, dx+4 \int \frac {x^2 \log ^4(x)}{\left (2+e^x\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 20, normalized size = 0.77 \begin {gather*} 48 \log (x)+\frac {x^2 \log ^4(x)}{\left (2+e^x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(384 + 576*E^x + 288*E^(2*x) + 48*E^(3*x) + (8*x^2 + 4*E^x*x^2)*Log[x]^3 + (4*x^2 + E^x*(2*x^2 - 2*x
^3))*Log[x]^4)/(8*x + 12*E^x*x + 6*E^(2*x)*x + E^(3*x)*x),x]

[Out]

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

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fricas [A]  time = 0.55, size = 36, normalized size = 1.38 \begin {gather*} \frac {x^{2} \log \relax (x)^{4} + 48 \, {\left (e^{\left (2 \, x\right )} + 4 \, e^{x} + 4\right )} \log \relax (x)}{e^{\left (2 \, x\right )} + 4 \, e^{x} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^3+2*x^2)*exp(x)+4*x^2)*log(x)^4+(4*exp(x)*x^2+8*x^2)*log(x)^3+48*exp(x)^3+288*exp(x)^2+576*e
xp(x)+384)/(x*exp(x)^3+6*x*exp(x)^2+12*exp(x)*x+8*x),x, algorithm="fricas")

[Out]

(x^2*log(x)^4 + 48*(e^(2*x) + 4*e^x + 4)*log(x))/(e^(2*x) + 4*e^x + 4)

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giac [A]  time = 0.15, size = 40, normalized size = 1.54 \begin {gather*} \frac {x^{2} \log \relax (x)^{4} + 48 \, e^{\left (2 \, x\right )} \log \relax (x) + 192 \, e^{x} \log \relax (x) + 192 \, \log \relax (x)}{e^{\left (2 \, x\right )} + 4 \, e^{x} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^3+2*x^2)*exp(x)+4*x^2)*log(x)^4+(4*exp(x)*x^2+8*x^2)*log(x)^3+48*exp(x)^3+288*exp(x)^2+576*e
xp(x)+384)/(x*exp(x)^3+6*x*exp(x)^2+12*exp(x)*x+8*x),x, algorithm="giac")

[Out]

(x^2*log(x)^4 + 48*e^(2*x)*log(x) + 192*e^x*log(x) + 192*log(x))/(e^(2*x) + 4*e^x + 4)

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maple [A]  time = 0.04, size = 20, normalized size = 0.77

method	result	size
risch	\(\frac {\ln \relax (x )^{4} x^{2}}{\left ({\mathrm e}^{x}+2\right )^{2}}+48 \ln \relax (x )\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-2*x^3+2*x^2)*exp(x)+4*x^2)*ln(x)^4+(4*exp(x)*x^2+8*x^2)*ln(x)^3+48*exp(x)^3+288*exp(x)^2+576*exp(x)+38
4)/(x*exp(x)^3+6*x*exp(x)^2+12*exp(x)*x+8*x),x,method=_RETURNVERBOSE)

[Out]

ln(x)^4*x^2/(exp(x)+2)^2+48*ln(x)

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maxima [A]  time = 0.40, size = 25, normalized size = 0.96 \begin {gather*} \frac {x^{2} \log \relax (x)^{4}}{e^{\left (2 \, x\right )} + 4 \, e^{x} + 4} + 48 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^3+2*x^2)*exp(x)+4*x^2)*log(x)^4+(4*exp(x)*x^2+8*x^2)*log(x)^3+48*exp(x)^3+288*exp(x)^2+576*e
xp(x)+384)/(x*exp(x)^3+6*x*exp(x)^2+12*exp(x)*x+8*x),x, algorithm="maxima")

[Out]

x^2*log(x)^4/(e^(2*x) + 4*e^x + 4) + 48*log(x)

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mupad [B]  time = 3.28, size = 29, normalized size = 1.12 \begin {gather*} \frac {\ln \relax (x)\,\left (48\,{\mathrm {e}}^{2\,x}+192\,{\mathrm {e}}^x+x^2\,{\ln \relax (x)}^3+192\right )}{{\left ({\mathrm {e}}^x+2\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((288*exp(2*x) + 48*exp(3*x) + 576*exp(x) + log(x)^4*(exp(x)*(2*x^2 - 2*x^3) + 4*x^2) + log(x)^3*(4*x^2*exp
(x) + 8*x^2) + 384)/(8*x + 6*x*exp(2*x) + x*exp(3*x) + 12*x*exp(x)),x)

[Out]

(log(x)*(48*exp(2*x) + 192*exp(x) + x^2*log(x)^3 + 192))/(exp(x) + 2)^2

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sympy [A]  time = 0.25, size = 24, normalized size = 0.92 \begin {gather*} \frac {x^{2} \log {\relax (x )}^{4}}{e^{2 x} + 4 e^{x} + 4} + 48 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x**3+2*x**2)*exp(x)+4*x**2)*ln(x)**4+(4*exp(x)*x**2+8*x**2)*ln(x)**3+48*exp(x)**3+288*exp(x)**
2+576*exp(x)+384)/(x*exp(x)**3+6*x*exp(x)**2+12*exp(x)*x+8*x),x)

[Out]

x**2*log(x)**4/(exp(2*x) + 4*exp(x) + 4) + 48*log(x)

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