3.37.92 \(\int \frac {32-48 x-8 x^2+e^5 (8-12 x-2 x^2)+(16+e^5 (4-x)-4 x) \log (x)+(-8 x^2-2 e^5 x^2) \log (x^2)}{576+432 x+108 x^2+9 x^3+(384+288 x+72 x^2+6 x^3) \log (x)+(64+48 x+12 x^2+x^3) \log ^2(x)+(384 x+288 x^2+72 x^3+6 x^4+(128 x+96 x^2+24 x^3+2 x^4) \log (x)) \log (x^2)+(64 x^2+48 x^3+12 x^4+x^5) \log ^2(x^2)} \, dx\)

Optimal. Leaf size=26 \[ \frac {\left (4+e^5\right ) x}{(-4-x)^2 \left (3+\log (x)+x \log \left (x^2\right )\right )} \]

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Rubi [F]  time = 1.02, 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 {32-48 x-8 x^2+e^5 \left (8-12 x-2 x^2\right )+\left (16+e^5 (4-x)-4 x\right ) \log (x)+\left (-8 x^2-2 e^5 x^2\right ) \log \left (x^2\right )}{576+432 x+108 x^2+9 x^3+\left (384+288 x+72 x^2+6 x^3\right ) \log (x)+\left (64+48 x+12 x^2+x^3\right ) \log ^2(x)+\left (384 x+288 x^2+72 x^3+6 x^4+\left (128 x+96 x^2+24 x^3+2 x^4\right ) \log (x)\right ) \log \left (x^2\right )+\left (64 x^2+48 x^3+12 x^4+x^5\right ) \log ^2\left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(32 - 48*x - 8*x^2 + E^5*(8 - 12*x - 2*x^2) + (16 + E^5*(4 - x) - 4*x)*Log[x] + (-8*x^2 - 2*E^5*x^2)*Log[x
^2])/(576 + 432*x + 108*x^2 + 9*x^3 + (384 + 288*x + 72*x^2 + 6*x^3)*Log[x] + (64 + 48*x + 12*x^2 + x^3)*Log[x
]^2 + (384*x + 288*x^2 + 72*x^3 + 6*x^4 + (128*x + 96*x^2 + 24*x^3 + 2*x^4)*Log[x])*Log[x^2] + (64*x^2 + 48*x^
3 + 12*x^4 + x^5)*Log[x^2]^2),x]

[Out]

10*(4 + E^5)*Defer[Int][1/((4 + x)^2*(3 + Log[x] + x*Log[x^2])^2), x] - 2*(4 + E^5)*Defer[Int][1/((4 + x)*(3 +
Log[x] + x*Log[x^2])^2), x] + (4 + E^5)*Defer[Int][Log[x]/((4 + x)^2*(3 + Log[x] + x*Log[x^2])^2), x] + 8*(4
+ E^5)*Defer[Int][1/((4 + x)^3*(3 + Log[x] + x*Log[x^2])), x] - 2*(4 + E^5)*Defer[Int][1/((4 + x)^2*(3 + Log[x
] + x*Log[x^2])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (4+e^5\right ) \left (-((-4+x) \log (x))-2 \left (-4+6 x+x^2+x^2 \log \left (x^2\right )\right )\right )}{(4+x)^3 \left (3+\log (x)+x \log \left (x^2\right )\right )^2} \, dx\\ &=\left (4+e^5\right ) \int \frac {-((-4+x) \log (x))-2 \left (-4+6 x+x^2+x^2 \log \left (x^2\right )\right )}{(4+x)^3 \left (3+\log (x)+x \log \left (x^2\right )\right )^2} \, dx\\ &=\left (4+e^5\right ) \int \left (\frac {2-2 x+\log (x)}{(4+x)^2 \left (3+\log (x)+x \log \left (x^2\right )\right )^2}-\frac {2 x}{(4+x)^3 \left (3+\log (x)+x \log \left (x^2\right )\right )}\right ) \, dx\\ &=\left (4+e^5\right ) \int \frac {2-2 x+\log (x)}{(4+x)^2 \left (3+\log (x)+x \log \left (x^2\right )\right )^2} \, dx-\left (2 \left (4+e^5\right )\right ) \int \frac {x}{(4+x)^3 \left (3+\log (x)+x \log \left (x^2\right )\right )} \, dx\\ &=\left (4+e^5\right ) \int \left (\frac {2}{(4+x)^2 \left (3+\log (x)+x \log \left (x^2\right )\right )^2}-\frac {2 x}{(4+x)^2 \left (3+\log (x)+x \log \left (x^2\right )\right )^2}+\frac {\log (x)}{(4+x)^2 \left (3+\log (x)+x \log \left (x^2\right )\right )^2}\right ) \, dx-\left (2 \left (4+e^5\right )\right ) \int \left (-\frac {4}{(4+x)^3 \left (3+\log (x)+x \log \left (x^2\right )\right )}+\frac {1}{(4+x)^2 \left (3+\log (x)+x \log \left (x^2\right )\right )}\right ) \, dx\\ &=\left (4+e^5\right ) \int \frac {\log (x)}{(4+x)^2 \left (3+\log (x)+x \log \left (x^2\right )\right )^2} \, dx+\left (2 \left (4+e^5\right )\right ) \int \frac {1}{(4+x)^2 \left (3+\log (x)+x \log \left (x^2\right )\right )^2} \, dx-\left (2 \left (4+e^5\right )\right ) \int \frac {x}{(4+x)^2 \left (3+\log (x)+x \log \left (x^2\right )\right )^2} \, dx-\left (2 \left (4+e^5\right )\right ) \int \frac {1}{(4+x)^2 \left (3+\log (x)+x \log \left (x^2\right )\right )} \, dx+\left (8 \left (4+e^5\right )\right ) \int \frac {1}{(4+x)^3 \left (3+\log (x)+x \log \left (x^2\right )\right )} \, dx\\ &=\left (4+e^5\right ) \int \frac {\log (x)}{(4+x)^2 \left (3+\log (x)+x \log \left (x^2\right )\right )^2} \, dx+\left (2 \left (4+e^5\right )\right ) \int \frac {1}{(4+x)^2 \left (3+\log (x)+x \log \left (x^2\right )\right )^2} \, dx-\left (2 \left (4+e^5\right )\right ) \int \frac {1}{(4+x)^2 \left (3+\log (x)+x \log \left (x^2\right )\right )} \, dx-\left (2 \left (4+e^5\right )\right ) \int \left (-\frac {4}{(4+x)^2 \left (3+\log (x)+x \log \left (x^2\right )\right )^2}+\frac {1}{(4+x) \left (3+\log (x)+x \log \left (x^2\right )\right )^2}\right ) \, dx+\left (8 \left (4+e^5\right )\right ) \int \frac {1}{(4+x)^3 \left (3+\log (x)+x \log \left (x^2\right )\right )} \, dx\\ &=\left (4+e^5\right ) \int \frac {\log (x)}{(4+x)^2 \left (3+\log (x)+x \log \left (x^2\right )\right )^2} \, dx+\left (2 \left (4+e^5\right )\right ) \int \frac {1}{(4+x)^2 \left (3+\log (x)+x \log \left (x^2\right )\right )^2} \, dx-\left (2 \left (4+e^5\right )\right ) \int \frac {1}{(4+x) \left (3+\log (x)+x \log \left (x^2\right )\right )^2} \, dx-\left (2 \left (4+e^5\right )\right ) \int \frac {1}{(4+x)^2 \left (3+\log (x)+x \log \left (x^2\right )\right )} \, dx+\left (8 \left (4+e^5\right )\right ) \int \frac {1}{(4+x)^2 \left (3+\log (x)+x \log \left (x^2\right )\right )^2} \, dx+\left (8 \left (4+e^5\right )\right ) \int \frac {1}{(4+x)^3 \left (3+\log (x)+x \log \left (x^2\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.56, size = 24, normalized size = 0.92 \begin {gather*} \frac {\left (4+e^5\right ) x}{(4+x)^2 \left (3+\log (x)+x \log \left (x^2\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(32 - 48*x - 8*x^2 + E^5*(8 - 12*x - 2*x^2) + (16 + E^5*(4 - x) - 4*x)*Log[x] + (-8*x^2 - 2*E^5*x^2)
*Log[x^2])/(576 + 432*x + 108*x^2 + 9*x^3 + (384 + 288*x + 72*x^2 + 6*x^3)*Log[x] + (64 + 48*x + 12*x^2 + x^3)
*Log[x]^2 + (384*x + 288*x^2 + 72*x^3 + 6*x^4 + (128*x + 96*x^2 + 24*x^3 + 2*x^4)*Log[x])*Log[x^2] + (64*x^2 +
48*x^3 + 12*x^4 + x^5)*Log[x^2]^2),x]

[Out]

((4 + E^5)*x)/((4 + x)^2*(3 + Log[x] + x*Log[x^2]))

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fricas [A]  time = 0.85, size = 39, normalized size = 1.50 \begin {gather*} \frac {x e^{5} + 4 \, x}{3 \, x^{2} + {\left (2 \, x^{3} + 17 \, x^{2} + 40 \, x + 16\right )} \log \relax (x) + 24 \, x + 48} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2*exp(5)-8*x^2)*log(x^2)+((-x+4)*exp(5)-4*x+16)*log(x)+(-2*x^2-12*x+8)*exp(5)-8*x^2-48*x+32)/
((x^5+12*x^4+48*x^3+64*x^2)*log(x^2)^2+((2*x^4+24*x^3+96*x^2+128*x)*log(x)+6*x^4+72*x^3+288*x^2+384*x)*log(x^2
)+(x^3+12*x^2+48*x+64)*log(x)^2+(6*x^3+72*x^2+288*x+384)*log(x)+9*x^3+108*x^2+432*x+576),x, algorithm="fricas"
)

[Out]

(x*e^5 + 4*x)/(3*x^2 + (2*x^3 + 17*x^2 + 40*x + 16)*log(x) + 24*x + 48)

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giac [A]  time = 0.25, size = 44, normalized size = 1.69 \begin {gather*} \frac {x e^{5} + 4 \, x}{2 \, x^{3} \log \relax (x) + 17 \, x^{2} \log \relax (x) + 3 \, x^{2} + 40 \, x \log \relax (x) + 24 \, x + 16 \, \log \relax (x) + 48} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2*exp(5)-8*x^2)*log(x^2)+((-x+4)*exp(5)-4*x+16)*log(x)+(-2*x^2-12*x+8)*exp(5)-8*x^2-48*x+32)/
((x^5+12*x^4+48*x^3+64*x^2)*log(x^2)^2+((2*x^4+24*x^3+96*x^2+128*x)*log(x)+6*x^4+72*x^3+288*x^2+384*x)*log(x^2
)+(x^3+12*x^2+48*x+64)*log(x)^2+(6*x^3+72*x^2+288*x+384)*log(x)+9*x^3+108*x^2+432*x+576),x, algorithm="giac")

[Out]

(x*e^5 + 4*x)/(2*x^3*log(x) + 17*x^2*log(x) + 3*x^2 + 40*x*log(x) + 24*x + 16*log(x) + 48)

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maple [C]  time = 0.13, size = 82, normalized size = 3.15

method	result	size
risch	\(\frac {2 i x \left (4+{\mathrm e}^{5}\right )}{\left (x^{2}+8 x +16\right ) \left (\pi x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi x \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i x \ln \relax (x )+2 i \ln \relax (x )+6 i\right )}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^2*exp(5)-8*x^2)*ln(x^2)+((-x+4)*exp(5)-4*x+16)*ln(x)+(-2*x^2-12*x+8)*exp(5)-8*x^2-48*x+32)/((x^5+12
*x^4+48*x^3+64*x^2)*ln(x^2)^2+((2*x^4+24*x^3+96*x^2+128*x)*ln(x)+6*x^4+72*x^3+288*x^2+384*x)*ln(x^2)+(x^3+12*x
^2+48*x+64)*ln(x)^2+(6*x^3+72*x^2+288*x+384)*ln(x)+9*x^3+108*x^2+432*x+576),x,method=_RETURNVERBOSE)

[Out]

2*I*x*(4+exp(5))/(x^2+8*x+16)/(Pi*x*csgn(I*x)^2*csgn(I*x^2)-2*Pi*x*csgn(I*x)*csgn(I*x^2)^2+Pi*x*csgn(I*x^2)^3+
4*I*x*ln(x)+2*I*ln(x)+6*I)

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maxima [A]  time = 0.42, size = 36, normalized size = 1.38 \begin {gather*} \frac {x {\left (e^{5} + 4\right )}}{3 \, x^{2} + {\left (2 \, x^{3} + 17 \, x^{2} + 40 \, x + 16\right )} \log \relax (x) + 24 \, x + 48} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2*exp(5)-8*x^2)*log(x^2)+((-x+4)*exp(5)-4*x+16)*log(x)+(-2*x^2-12*x+8)*exp(5)-8*x^2-48*x+32)/
((x^5+12*x^4+48*x^3+64*x^2)*log(x^2)^2+((2*x^4+24*x^3+96*x^2+128*x)*log(x)+6*x^4+72*x^3+288*x^2+384*x)*log(x^2
)+(x^3+12*x^2+48*x+64)*log(x)^2+(6*x^3+72*x^2+288*x+384)*log(x)+9*x^3+108*x^2+432*x+576),x, algorithm="maxima"
)

[Out]

x*(e^5 + 4)/(3*x^2 + (2*x^3 + 17*x^2 + 40*x + 16)*log(x) + 24*x + 48)

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mupad [B]  time = 2.88, size = 147, normalized size = 5.65 \begin {gather*} \frac {\ln \left (x^2\right )\,\left (\left ({\mathrm {e}}^5+4\right )\,x^4+\left (4\,{\mathrm {e}}^5+16\right )\,x^3\right )-\ln \relax (x)\,\left (\left (2\,{\mathrm {e}}^5+8\right )\,x^4+\left (8\,{\mathrm {e}}^5+32\right )\,x^3\right )+x^2\,\left (4\,{\mathrm {e}}^5+16\right )+x^5\,\left (4\,{\mathrm {e}}^5+16\right )-x^3\,\left (7\,{\mathrm {e}}^5+28\right )+x^4\,\left (14\,{\mathrm {e}}^5+56\right )}{{\left (x+4\right )}^3\,\left (\ln \relax (x)\,\left (2\,x+1\right )+x\,\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )+3\right )\,\left (x+x^2\,\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )-2\,x^2+4\,x^3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(48*x + log(x)*(4*x + exp(5)*(x - 4) - 16) + exp(5)*(12*x + 2*x^2 - 8) + log(x^2)*(2*x^2*exp(5) + 8*x^2)
+ 8*x^2 - 32)/(432*x + log(x^2)*(384*x + log(x)*(128*x + 96*x^2 + 24*x^3 + 2*x^4) + 288*x^2 + 72*x^3 + 6*x^4)
+ 108*x^2 + 9*x^3 + log(x^2)^2*(64*x^2 + 48*x^3 + 12*x^4 + x^5) + log(x)*(288*x + 72*x^2 + 6*x^3 + 384) + log(
x)^2*(48*x + 12*x^2 + x^3 + 64) + 576),x)

[Out]

(log(x^2)*(x^3*(4*exp(5) + 16) + x^4*(exp(5) + 4)) - log(x)*(x^4*(2*exp(5) + 8) + x^3*(8*exp(5) + 32)) + x^2*(
4*exp(5) + 16) + x^5*(4*exp(5) + 16) - x^3*(7*exp(5) + 28) + x^4*(14*exp(5) + 56))/((x + 4)^3*(log(x)*(2*x + 1
) + x*(log(x^2) - 2*log(x)) + 3)*(x + x^2*(log(x^2) - 2*log(x)) - 2*x^2 + 4*x^3))

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sympy [A]  time = 0.41, size = 36, normalized size = 1.38 \begin {gather*} \frac {4 x + x e^{5}}{3 x^{2} + 24 x + \left (2 x^{3} + 17 x^{2} + 40 x + 16\right ) \log {\relax (x )} + 48} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**2*exp(5)-8*x**2)*ln(x**2)+((-x+4)*exp(5)-4*x+16)*ln(x)+(-2*x**2-12*x+8)*exp(5)-8*x**2-48*x+3
2)/((x**5+12*x**4+48*x**3+64*x**2)*ln(x**2)**2+((2*x**4+24*x**3+96*x**2+128*x)*ln(x)+6*x**4+72*x**3+288*x**2+3
84*x)*ln(x**2)+(x**3+12*x**2+48*x+64)*ln(x)**2+(6*x**3+72*x**2+288*x+384)*ln(x)+9*x**3+108*x**2+432*x+576),x)

[Out]

(4*x + x*exp(5))/(3*x**2 + 24*x + (2*x**3 + 17*x**2 + 40*x + 16)*log(x) + 48)

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