3.25.74 \(\int \frac {-4 x^5 \log (x)+(-3 x-x^5+(-6 x-2 x^5) \log (x)) \log (\frac {e^5}{3+x^4})+(24+24 x+8 x^4+8 x^5) \log ^2(\frac {e^5}{3+x^4})}{(12+4 x^4) \log ^2(\frac {e^5}{3+x^4})} \, dx\)

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

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Rubi [F]  time = 3.26, 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 {-4 x^5 \log (x)+\left (-3 x-x^5+\left (-6 x-2 x^5\right ) \log (x)\right ) \log \left (\frac {e^5}{3+x^4}\right )+\left (24+24 x+8 x^4+8 x^5\right ) \log ^2\left (\frac {e^5}{3+x^4}\right )}{\left (12+4 x^4\right ) \log ^2\left (\frac {e^5}{3+x^4}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-4*x^5*Log[x] + (-3*x - x^5 + (-6*x - 2*x^5)*Log[x])*Log[E^5/(3 + x^4)] + (24 + 24*x + 8*x^4 + 8*x^5)*Log
[E^5/(3 + x^4)]^2)/((12 + 4*x^4)*Log[E^5/(3 + x^4)]^2),x]

[Out]

(1 + x)^2 + (I/4)*Sqrt[3]*Defer[Int][Log[x]/(((-3)^(1/4) - x)*(5 + Log[(3 + x^4)^(-1)])^2), x] - (I/4)*Sqrt[3]
*Defer[Int][Log[x]/((-((-1)^(3/4)*3^(1/4)) - x)*(5 + Log[(3 + x^4)^(-1)])^2), x] - Defer[Int][(x*Log[x])/(5 +
Log[(3 + x^4)^(-1)])^2, x] - (I/4)*Sqrt[3]*Defer[Int][Log[x]/(((-3)^(1/4) + x)*(5 + Log[(3 + x^4)^(-1)])^2), x
] + (I/4)*Sqrt[3]*Defer[Int][Log[x]/((-((-1)^(3/4)*3^(1/4)) + x)*(5 + Log[(3 + x^4)^(-1)])^2), x] - Defer[Int]
[x/(5 + Log[(3 + x^4)^(-1)]), x]/4 - Defer[Int][(x*Log[x])/(5 + Log[(3 + x^4)^(-1)]), x]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2 (1+x)-\frac {x^5 \log (x)}{\left (3+x^4\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}-\frac {x (1+2 \log (x))}{4 \left (5+\log \left (\frac {1}{3+x^4}\right )\right )}\right ) \, dx\\ &=(1+x)^2-\frac {1}{4} \int \frac {x (1+2 \log (x))}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx-\int \frac {x^5 \log (x)}{\left (3+x^4\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx\\ &=(1+x)^2-\frac {1}{4} \int \left (\frac {x}{5+\log \left (\frac {1}{3+x^4}\right )}+\frac {2 x \log (x)}{5+\log \left (\frac {1}{3+x^4}\right )}\right ) \, dx-\int \left (\frac {x \log (x)}{\left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}-\frac {3 x \log (x)}{\left (3+x^4\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}\right ) \, dx\\ &=(1+x)^2-\frac {1}{4} \int \frac {x}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx-\frac {1}{2} \int \frac {x \log (x)}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx+3 \int \frac {x \log (x)}{\left (3+x^4\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx-\int \frac {x \log (x)}{\left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx\\ &=(1+x)^2-\frac {1}{4} \int \frac {x}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx-\frac {1}{2} \int \frac {x \log (x)}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx+3 \int \left (-\frac {i x \log (x)}{2 \sqrt {3} \left (-i \sqrt {3}+x^2\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}+\frac {i x \log (x)}{2 \sqrt {3} \left (i \sqrt {3}+x^2\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}\right ) \, dx-\int \frac {x \log (x)}{\left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx\\ &=(1+x)^2-\frac {1}{4} \int \frac {x}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx-\frac {1}{2} \int \frac {x \log (x)}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx-\frac {1}{2} \left (i \sqrt {3}\right ) \int \frac {x \log (x)}{\left (-i \sqrt {3}+x^2\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx+\frac {1}{2} \left (i \sqrt {3}\right ) \int \frac {x \log (x)}{\left (i \sqrt {3}+x^2\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx-\int \frac {x \log (x)}{\left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx\\ &=(1+x)^2-\frac {1}{4} \int \frac {x}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx-\frac {1}{2} \int \frac {x \log (x)}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx-\frac {1}{2} \left (i \sqrt {3}\right ) \int \left (-\frac {\log (x)}{2 \left (\sqrt [4]{-3}-x\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}+\frac {\log (x)}{2 \left (\sqrt [4]{-3}+x\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}\right ) \, dx+\frac {1}{2} \left (i \sqrt {3}\right ) \int \left (-\frac {\log (x)}{2 \left (-(-1)^{3/4} \sqrt [4]{3}-x\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}+\frac {\log (x)}{2 \left (-(-1)^{3/4} \sqrt [4]{3}+x\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2}\right ) \, dx-\int \frac {x \log (x)}{\left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx\\ &=(1+x)^2-\frac {1}{4} \int \frac {x}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx-\frac {1}{2} \int \frac {x \log (x)}{5+\log \left (\frac {1}{3+x^4}\right )} \, dx+\frac {1}{4} \left (i \sqrt {3}\right ) \int \frac {\log (x)}{\left (\sqrt [4]{-3}-x\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx-\frac {1}{4} \left (i \sqrt {3}\right ) \int \frac {\log (x)}{\left (-(-1)^{3/4} \sqrt [4]{3}-x\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx-\frac {1}{4} \left (i \sqrt {3}\right ) \int \frac {\log (x)}{\left (\sqrt [4]{-3}+x\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx+\frac {1}{4} \left (i \sqrt {3}\right ) \int \frac {\log (x)}{\left (-(-1)^{3/4} \sqrt [4]{3}+x\right ) \left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx-\int \frac {x \log (x)}{\left (5+\log \left (\frac {1}{3+x^4}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.84, size = 27, normalized size = 1.04 \begin {gather*} \frac {1}{4} x \left (8+4 x-\frac {x \log (x)}{5+\log \left (\frac {1}{3+x^4}\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*x^5*Log[x] + (-3*x - x^5 + (-6*x - 2*x^5)*Log[x])*Log[E^5/(3 + x^4)] + (24 + 24*x + 8*x^4 + 8*x^
5)*Log[E^5/(3 + x^4)]^2)/((12 + 4*x^4)*Log[E^5/(3 + x^4)]^2),x]

[Out]

(x*(8 + 4*x - (x*Log[x])/(5 + Log[(3 + x^4)^(-1)])))/4

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fricas [A]  time = 0.60, size = 42, normalized size = 1.62 \begin {gather*} -\frac {x^{2} \log \relax (x) - 4 \, {\left (x^{2} + 2 \, x\right )} \log \left (\frac {e^{5}}{x^{4} + 3}\right )}{4 \, \log \left (\frac {e^{5}}{x^{4} + 3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^5+8*x^4+24*x+24)*log(exp(5)/(x^4+3))^2+((-2*x^5-6*x)*log(x)-x^5-3*x)*log(exp(5)/(x^4+3))-4*x^5
*log(x))/(4*x^4+12)/log(exp(5)/(x^4+3))^2,x, algorithm="fricas")

[Out]

-1/4*(x^2*log(x) - 4*(x^2 + 2*x)*log(e^5/(x^4 + 3)))/log(e^5/(x^4 + 3))

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giac [A]  time = 0.46, size = 24, normalized size = 0.92 \begin {gather*} x^{2} + \frac {x^{2} \log \relax (x)}{4 \, {\left (\log \left (x^{4} + 3\right ) - 5\right )}} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^5+8*x^4+24*x+24)*log(exp(5)/(x^4+3))^2+((-2*x^5-6*x)*log(x)-x^5-3*x)*log(exp(5)/(x^4+3))-4*x^5
*log(x))/(4*x^4+12)/log(exp(5)/(x^4+3))^2,x, algorithm="giac")

[Out]

x^2 + 1/4*x^2*log(x)/(log(x^4 + 3) - 5) + 2*x

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maple [C]  time = 0.40, size = 30, normalized size = 1.15

method	result	size
risch	\(x^{2}+2 x -\frac {i x^{2} \ln \relax (x )}{2 \left (-2 i \ln \left (x^{4}+3\right )+10 i\right )}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x^5+8*x^4+24*x+24)*ln(exp(5)/(x^4+3))^2+((-2*x^5-6*x)*ln(x)-x^5-3*x)*ln(exp(5)/(x^4+3))-4*x^5*ln(x))/(
4*x^4+12)/ln(exp(5)/(x^4+3))^2,x,method=_RETURNVERBOSE)

[Out]

x^2+2*x-1/2*I*x^2*ln(x)/(-2*I*ln(x^4+3)+10*I)

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maxima [A]  time = 0.67, size = 42, normalized size = 1.62 \begin {gather*} \frac {x^{2} \log \relax (x) - 20 \, x^{2} + 4 \, {\left (x^{2} + 2 \, x\right )} \log \left (x^{4} + 3\right ) - 40 \, x}{4 \, {\left (\log \left (x^{4} + 3\right ) - 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^5+8*x^4+24*x+24)*log(exp(5)/(x^4+3))^2+((-2*x^5-6*x)*log(x)-x^5-3*x)*log(exp(5)/(x^4+3))-4*x^5
*log(x))/(4*x^4+12)/log(exp(5)/(x^4+3))^2,x, algorithm="maxima")

[Out]

1/4*(x^2*log(x) - 20*x^2 + 4*(x^2 + 2*x)*log(x^4 + 3) - 40*x)/(log(x^4 + 3) - 5)

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mupad [B]  time = 1.49, size = 77, normalized size = 2.96 \begin {gather*} 2\,x+\frac {3}{16\,x^2}+\frac {17\,x^2}{16}-\frac {\frac {x^2\,\ln \relax (x)}{4}+\frac {\ln \left (\frac {{\mathrm {e}}^5}{x^4+3}\right )\,\left (x^4+3\right )\,\left (2\,\ln \relax (x)+1\right )}{16\,x^2}}{\ln \left (\frac {{\mathrm {e}}^5}{x^4+3}\right )}+\frac {\ln \relax (x)\,\left (\frac {x^4}{8}+\frac {3}{8}\right )}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x^5*log(x) - log(exp(5)/(x^4 + 3))^2*(24*x + 8*x^4 + 8*x^5 + 24) + log(exp(5)/(x^4 + 3))*(3*x + log(x)
*(6*x + 2*x^5) + x^5))/(log(exp(5)/(x^4 + 3))^2*(4*x^4 + 12)),x)

[Out]

2*x + 3/(16*x^2) + (17*x^2)/16 - ((x^2*log(x))/4 + (log(exp(5)/(x^4 + 3))*(x^4 + 3)*(2*log(x) + 1))/(16*x^2))/
log(exp(5)/(x^4 + 3)) + (log(x)*(x^4/8 + 3/8))/x^2

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sympy [A]  time = 0.30, size = 24, normalized size = 0.92 \begin {gather*} - \frac {x^{2} \log {\relax (x )}}{4 \log {\left (\frac {e^{5}}{x^{4} + 3} \right )}} + x^{2} + 2 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x**5+8*x**4+24*x+24)*ln(exp(5)/(x**4+3))**2+((-2*x**5-6*x)*ln(x)-x**5-3*x)*ln(exp(5)/(x**4+3))-4
*x**5*ln(x))/(4*x**4+12)/ln(exp(5)/(x**4+3))**2,x)

[Out]

-x**2*log(x)/(4*log(exp(5)/(x**4 + 3))) + x**2 + 2*x

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