3.17.3 \(\int \frac {-4 x+(3+x) \log (3+x) \log (\log ^4(3+x))}{(9+3 x) \log (3+x) \log ^2(\log ^4(3+x))} \, dx\)

Optimal. Leaf size=14 \[ \frac {x}{3 \log \left (\log ^4(3+x)\right )} \]

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Rubi [F]  time = 0.45, 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+(3+x) \log (3+x) \log \left (\log ^4(3+x)\right )}{(9+3 x) \log (3+x) \log ^2\left (\log ^4(3+x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-4*x + (3 + x)*Log[3 + x]*Log[Log[3 + x]^4])/((9 + 3*x)*Log[3 + x]*Log[Log[3 + x]^4]^2),x]

[Out]

-Log[Log[3 + x]^4]^(-1) - (4*Defer[Subst][Defer[Int][1/(Log[x]*Log[Log[x]^4]^2), x], x, 3 + x])/3 + Defer[Subs
t][Defer[Int][Log[Log[x]^4]^(-1), x], x, 3 + x]/3

Rubi steps

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

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Mathematica [A]  time = 0.13, size = 14, normalized size = 1.00 \begin {gather*} \frac {x}{3 \log \left (\log ^4(3+x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*x + (3 + x)*Log[3 + x]*Log[Log[3 + x]^4])/((9 + 3*x)*Log[3 + x]*Log[Log[3 + x]^4]^2),x]

[Out]

x/(3*Log[Log[3 + x]^4])

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fricas [A]  time = 0.62, size = 12, normalized size = 0.86 \begin {gather*} \frac {x}{3 \, \log \left (\log \left (x + 3\right )^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+x)*log(3+x)*log(log(3+x)^4)-4*x)/(3*x+9)/log(3+x)/log(log(3+x)^4)^2,x, algorithm="fricas")

[Out]

1/3*x/log(log(x + 3)^4)

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giac [A]  time = 0.31, size = 12, normalized size = 0.86 \begin {gather*} \frac {x}{3 \, \log \left (\log \left (x + 3\right )^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+x)*log(3+x)*log(log(3+x)^4)-4*x)/(3*x+9)/log(3+x)/log(log(3+x)^4)^2,x, algorithm="giac")

[Out]

1/3*x/log(log(x + 3)^4)

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maple [A]  time = 0.45, size = 13, normalized size = 0.93

method	result	size
norman	\(\frac {x}{3 \ln \left (\ln \left (3+x \right )^{4}\right )}\)	\(13\)
risch	\(\frac {2 i x}{3 \left (\pi \mathrm {csgn}\left (i \ln \left (3+x \right )\right )^{2} \mathrm {csgn}\left (i \ln \left (3+x \right )^{2}\right )-2 \pi \,\mathrm {csgn}\left (i \ln \left (3+x \right )\right ) \mathrm {csgn}\left (i \ln \left (3+x \right )^{2}\right )^{2}+\pi \,\mathrm {csgn}\left (i \ln \left (3+x \right )\right ) \mathrm {csgn}\left (i \ln \left (3+x \right )^{2}\right ) \mathrm {csgn}\left (i \ln \left (3+x \right )^{3}\right )-\pi \,\mathrm {csgn}\left (i \ln \left (3+x \right )\right ) \mathrm {csgn}\left (i \ln \left (3+x \right )^{3}\right )^{2}+\pi \,\mathrm {csgn}\left (i \ln \left (3+x \right )\right ) \mathrm {csgn}\left (i \ln \left (3+x \right )^{3}\right ) \mathrm {csgn}\left (i \ln \left (3+x \right )^{4}\right )-\pi \,\mathrm {csgn}\left (i \ln \left (3+x \right )\right ) \mathrm {csgn}\left (i \ln \left (3+x \right )^{4}\right )^{2}+\pi \mathrm {csgn}\left (i \ln \left (3+x \right )^{2}\right )^{3}-\pi \,\mathrm {csgn}\left (i \ln \left (3+x \right )^{2}\right ) \mathrm {csgn}\left (i \ln \left (3+x \right )^{3}\right )^{2}+\pi \mathrm {csgn}\left (i \ln \left (3+x \right )^{3}\right )^{3}-\pi \,\mathrm {csgn}\left (i \ln \left (3+x \right )^{3}\right ) \mathrm {csgn}\left (i \ln \left (3+x \right )^{4}\right )^{2}+\pi \mathrm {csgn}\left (i \ln \left (3+x \right )^{4}\right )^{3}+8 i \ln \left (\ln \left (3+x \right )\right )\right )}\)	\(259\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3+x)*ln(3+x)*ln(ln(3+x)^4)-4*x)/(3*x+9)/ln(3+x)/ln(ln(3+x)^4)^2,x,method=_RETURNVERBOSE)

[Out]

1/3*x/ln(ln(3+x)^4)

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maxima [A]  time = 0.43, size = 10, normalized size = 0.71 \begin {gather*} \frac {x}{12 \, \log \left (\log \left (x + 3\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+x)*log(3+x)*log(log(3+x)^4)-4*x)/(3*x+9)/log(3+x)/log(log(3+x)^4)^2,x, algorithm="maxima")

[Out]

1/12*x/log(log(x + 3))

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mupad [B]  time = 1.42, size = 12, normalized size = 0.86 \begin {gather*} \frac {x}{3\,\ln \left ({\ln \left (x+3\right )}^4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x - log(log(x + 3)^4)*log(x + 3)*(x + 3))/(log(log(x + 3)^4)^2*log(x + 3)*(3*x + 9)),x)

[Out]

x/(3*log(log(x + 3)^4))

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sympy [A]  time = 0.25, size = 10, normalized size = 0.71 \begin {gather*} \frac {x}{3 \log {\left (\log {\left (x + 3 \right )}^{4} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+x)*ln(3+x)*ln(ln(3+x)**4)-4*x)/(3*x+9)/ln(3+x)/ln(ln(3+x)**4)**2,x)

[Out]

x/(3*log(log(x + 3)**4))

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