3.70.78 \(\int \frac {-3 x^5+(1536+384 x+1536 x^2+384 x^3+28 x^5+7 x^6) \log (4+x) \log (\log (4+x))}{(24 x^5+6 x^6) \log (4+x) \log (\log (4+x))} \, dx\)

Optimal. Leaf size=33 \[ -4-\left (4+\frac {4}{x^2}\right )^2+x+\frac {1}{2} \left (4+\frac {x}{3}-\log (\log (\log (4+x)))\right ) \]

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Rubi [A]  time = 0.80, antiderivative size = 26, normalized size of antiderivative = 0.79, number of steps used = 6, number of rules used = 4, integrand size = 68, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {1593, 6742, 14, 6684} \begin {gather*} -\frac {16}{x^4}-\frac {32}{x^2}+\frac {7 x}{6}-\frac {1}{2} \log (\log (\log (x+4))) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3*x^5 + (1536 + 384*x + 1536*x^2 + 384*x^3 + 28*x^5 + 7*x^6)*Log[4 + x]*Log[Log[4 + x]])/((24*x^5 + 6*x^
6)*Log[4 + x]*Log[Log[4 + x]]),x]

[Out]

-16/x^4 - 32/x^2 + (7*x)/6 - Log[Log[Log[4 + x]]]/2

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-3 x^5+\left (1536+384 x+1536 x^2+384 x^3+28 x^5+7 x^6\right ) \log (4+x) \log (\log (4+x))}{x^5 (24+6 x) \log (4+x) \log (\log (4+x))} \, dx\\ &=\int \left (\frac {384+384 x^2+7 x^5}{6 x^5}-\frac {1}{2 (4+x) \log (4+x) \log (\log (4+x))}\right ) \, dx\\ &=\frac {1}{6} \int \frac {384+384 x^2+7 x^5}{x^5} \, dx-\frac {1}{2} \int \frac {1}{(4+x) \log (4+x) \log (\log (4+x))} \, dx\\ &=-\frac {1}{2} \log (\log (\log (4+x)))+\frac {1}{6} \int \left (7+\frac {384}{x^5}+\frac {384}{x^3}\right ) \, dx\\ &=-\frac {16}{x^4}-\frac {32}{x^2}+\frac {7 x}{6}-\frac {1}{2} \log (\log (\log (4+x)))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 26, normalized size = 0.79 \begin {gather*} \frac {1}{6} \left (-\frac {96}{x^4}-\frac {192}{x^2}+7 x-3 \log (\log (\log (4+x)))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3*x^5 + (1536 + 384*x + 1536*x^2 + 384*x^3 + 28*x^5 + 7*x^6)*Log[4 + x]*Log[Log[4 + x]])/((24*x^5
+ 6*x^6)*Log[4 + x]*Log[Log[4 + x]]),x]

[Out]

(-96/x^4 - 192/x^2 + 7*x - 3*Log[Log[Log[4 + x]]])/6

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fricas [A]  time = 0.61, size = 28, normalized size = 0.85 \begin {gather*} \frac {7 \, x^{5} - 3 \, x^{4} \log \left (\log \left (\log \left (x + 4\right )\right )\right ) - 192 \, x^{2} - 96}{6 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7*x^6+28*x^5+384*x^3+1536*x^2+384*x+1536)*log(4+x)*log(log(4+x))-3*x^5)/(6*x^6+24*x^5)/log(4+x)/lo
g(log(4+x)),x, algorithm="fricas")

[Out]

1/6*(7*x^5 - 3*x^4*log(log(log(x + 4))) - 192*x^2 - 96)/x^4

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giac [A]  time = 0.20, size = 24, normalized size = 0.73 \begin {gather*} \frac {7}{6} \, x - \frac {16 \, {\left (2 \, x^{2} + 1\right )}}{x^{4}} - \frac {1}{2} \, \log \left (\log \left (\log \left (x + 4\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7*x^6+28*x^5+384*x^3+1536*x^2+384*x+1536)*log(4+x)*log(log(4+x))-3*x^5)/(6*x^6+24*x^5)/log(4+x)/lo
g(log(4+x)),x, algorithm="giac")

[Out]

7/6*x - 16*(2*x^2 + 1)/x^4 - 1/2*log(log(log(x + 4)))

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maple [A]  time = 0.08, size = 23, normalized size = 0.70

method	result	size
default	\(\frac {7 x}{6}-\frac {16}{x^{4}}-\frac {32}{x^{2}}-\frac {\ln \left (\ln \left (\ln \left (4+x \right )\right )\right )}{2}\)	\(23\)
risch	\(\frac {7 x^{5}-192 x^{2}-96}{6 x^{4}}-\frac {\ln \left (\ln \left (\ln \left (4+x \right )\right )\right )}{2}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((7*x^6+28*x^5+384*x^3+1536*x^2+384*x+1536)*ln(4+x)*ln(ln(4+x))-3*x^5)/(6*x^6+24*x^5)/ln(4+x)/ln(ln(4+x)),
x,method=_RETURNVERBOSE)

[Out]

7/6*x-16/x^4-32/x^2-1/2*ln(ln(ln(4+x)))

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maxima [B]  time = 0.58, size = 60, normalized size = 1.82 \begin {gather*} \frac {7}{6} \, x + \frac {16 \, {\left (x - 2\right )}}{x^{2}} - \frac {16}{x} - \frac {3 \, x^{2} - 6 \, x + 16}{3 \, x^{3}} + \frac {3 \, x^{3} - 6 \, x^{2} + 16 \, x - 48}{3 \, x^{4}} - \frac {1}{2} \, \log \left (\log \left (\log \left (x + 4\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7*x^6+28*x^5+384*x^3+1536*x^2+384*x+1536)*log(4+x)*log(log(4+x))-3*x^5)/(6*x^6+24*x^5)/log(4+x)/lo
g(log(4+x)),x, algorithm="maxima")

[Out]

7/6*x + 16*(x - 2)/x^2 - 16/x - 1/3*(3*x^2 - 6*x + 16)/x^3 + 1/3*(3*x^3 - 6*x^2 + 16*x - 48)/x^4 - 1/2*log(log
(log(x + 4)))

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mupad [B]  time = 4.21, size = 24, normalized size = 0.73 \begin {gather*} \frac {7\,x}{6}-\frac {\ln \left (\ln \left (\ln \left (x+4\right )\right )\right )}{2}-\frac {32\,x^2+16}{x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x^5 - log(x + 4)*log(log(x + 4))*(384*x + 1536*x^2 + 384*x^3 + 28*x^5 + 7*x^6 + 1536))/(log(x + 4)*log
(log(x + 4))*(24*x^5 + 6*x^6)),x)

[Out]

(7*x)/6 - log(log(log(x + 4)))/2 - (32*x^2 + 16)/x^4

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sympy [A]  time = 0.36, size = 27, normalized size = 0.82 \begin {gather*} \frac {7 x}{6} - \frac {\log {\left (\log {\left (\log {\left (x + 4 \right )} \right )} \right )}}{2} + \frac {- 192 x^{2} - 96}{6 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7*x**6+28*x**5+384*x**3+1536*x**2+384*x+1536)*ln(4+x)*ln(ln(4+x))-3*x**5)/(6*x**6+24*x**5)/ln(4+x)
/ln(ln(4+x)),x)

[Out]

7*x/6 - log(log(log(x + 4)))/2 + (-192*x**2 - 96)/(6*x**4)

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