3.85.32 \(\int \frac {6+32 x^2+(-3+581 x+194 x^2+3072 x^3+1024 x^4) \log (3+x)+(2+(192 x+64 x^2) \log (3+x)) \log (\log (3+x))}{(3+x) \log (3+x)} \, dx\)

Optimal. Leaf size=23 \[ -x+x^2+\left (-3-16 x^2-\log (\log (3+x))\right )^2 \]

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Rubi [F]  time = 0.39, 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 {6+32 x^2+\left (-3+581 x+194 x^2+3072 x^3+1024 x^4\right ) \log (3+x)+\left (2+\left (192 x+64 x^2\right ) \log (3+x)\right ) \log (\log (3+x))}{(3+x) \log (3+x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(6 + 32*x^2 + (-3 + 581*x + 194*x^2 + 3072*x^3 + 1024*x^4)*Log[3 + x] + (2 + (192*x + 64*x^2)*Log[3 + x])*
Log[Log[3 + x]])/((3 + x)*Log[3 + x]),x]

[Out]

-x + 97*x^2 + 256*x^4 + 32*ExpIntegralEi[2*Log[3 + x]] + 294*Log[Log[3 + x]] + Log[Log[3 + x]]^2 - 192*LogInte
gral[3 + x] + 64*Defer[Int][x*Log[Log[3 + x]], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1+194 x+1024 x^3+64 x \log (\log (3+x))+\frac {2 \left (3+16 x^2+\log (\log (3+x))\right )}{(3+x) \log (3+x)}\right ) \, dx\\ &=-x+97 x^2+256 x^4+2 \int \frac {3+16 x^2+\log (\log (3+x))}{(3+x) \log (3+x)} \, dx+64 \int x \log (\log (3+x)) \, dx\\ &=-x+97 x^2+256 x^4+2 \int \left (\frac {3+16 x^2}{(3+x) \log (3+x)}+\frac {\log (\log (3+x))}{(3+x) \log (3+x)}\right ) \, dx+64 \int x \log (\log (3+x)) \, dx\\ &=-x+97 x^2+256 x^4+2 \int \frac {3+16 x^2}{(3+x) \log (3+x)} \, dx+2 \int \frac {\log (\log (3+x))}{(3+x) \log (3+x)} \, dx+64 \int x \log (\log (3+x)) \, dx\\ &=-x+97 x^2+256 x^4+\log ^2(\log (3+x))+2 \int \left (-\frac {48}{\log (3+x)}+\frac {16 x}{\log (3+x)}+\frac {147}{(3+x) \log (3+x)}\right ) \, dx+64 \int x \log (\log (3+x)) \, dx\\ &=-x+97 x^2+256 x^4+\log ^2(\log (3+x))+32 \int \frac {x}{\log (3+x)} \, dx+64 \int x \log (\log (3+x)) \, dx-96 \int \frac {1}{\log (3+x)} \, dx+294 \int \frac {1}{(3+x) \log (3+x)} \, dx\\ &=-x+97 x^2+256 x^4+\log ^2(\log (3+x))+32 \int \left (-\frac {3}{\log (3+x)}+\frac {3+x}{\log (3+x)}\right ) \, dx+64 \int x \log (\log (3+x)) \, dx-96 \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,3+x\right )+294 \operatorname {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,3+x\right )\\ &=-x+97 x^2+256 x^4+\log ^2(\log (3+x))-96 \text {li}(3+x)+32 \int \frac {3+x}{\log (3+x)} \, dx+64 \int x \log (\log (3+x)) \, dx-96 \int \frac {1}{\log (3+x)} \, dx+294 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (3+x)\right )\\ &=-x+97 x^2+256 x^4+294 \log (\log (3+x))+\log ^2(\log (3+x))-96 \text {li}(3+x)+32 \operatorname {Subst}\left (\int \frac {x}{\log (x)} \, dx,x,3+x\right )+64 \int x \log (\log (3+x)) \, dx-96 \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,3+x\right )\\ &=-x+97 x^2+256 x^4+294 \log (\log (3+x))+\log ^2(\log (3+x))-192 \text {li}(3+x)+32 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (3+x)\right )+64 \int x \log (\log (3+x)) \, dx\\ &=-x+97 x^2+256 x^4+32 \text {Ei}(2 \log (3+x))+294 \log (\log (3+x))+\log ^2(\log (3+x))-192 \text {li}(3+x)+64 \int x \log (\log (3+x)) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 41, normalized size = 1.78 \begin {gather*} -x+97 x^2+256 x^4+294 \log (\log (3+x))+32 (-3+x) (3+x) \log (\log (3+x))+\log ^2(\log (3+x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6 + 32*x^2 + (-3 + 581*x + 194*x^2 + 3072*x^3 + 1024*x^4)*Log[3 + x] + (2 + (192*x + 64*x^2)*Log[3
+ x])*Log[Log[3 + x]])/((3 + x)*Log[3 + x]),x]

[Out]

-x + 97*x^2 + 256*x^4 + 294*Log[Log[3 + x]] + 32*(-3 + x)*(3 + x)*Log[Log[3 + x]] + Log[Log[3 + x]]^2

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fricas [A]  time = 1.12, size = 35, normalized size = 1.52 \begin {gather*} 256 \, x^{4} + 97 \, x^{2} + 2 \, {\left (16 \, x^{2} + 3\right )} \log \left (\log \left (x + 3\right )\right ) + \log \left (\log \left (x + 3\right )\right )^{2} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((64*x^2+192*x)*log(3+x)+2)*log(log(3+x))+(1024*x^4+3072*x^3+194*x^2+581*x-3)*log(3+x)+32*x^2+6)/(3
+x)/log(3+x),x, algorithm="fricas")

[Out]

256*x^4 + 97*x^2 + 2*(16*x^2 + 3)*log(log(x + 3)) + log(log(x + 3))^2 - x

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giac [A]  time = 0.23, size = 38, normalized size = 1.65 \begin {gather*} 256 \, x^{4} + 32 \, x^{2} \log \left (\log \left (x + 3\right )\right ) + 97 \, x^{2} + \log \left (\log \left (x + 3\right )\right )^{2} - x + 6 \, \log \left (\log \left (x + 3\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((64*x^2+192*x)*log(3+x)+2)*log(log(3+x))+(1024*x^4+3072*x^3+194*x^2+581*x-3)*log(3+x)+32*x^2+6)/(3
+x)/log(3+x),x, algorithm="giac")

[Out]

256*x^4 + 32*x^2*log(log(x + 3)) + 97*x^2 + log(log(x + 3))^2 - x + 6*log(log(x + 3))

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maple [A]  time = 0.37, size = 39, normalized size = 1.70

method	result	size
risch	\(\ln \left (\ln \left (3+x \right )\right )^{2}+32 x^{2} \ln \left (\ln \left (3+x \right )\right )+256 x^{4}+97 x^{2}-x +6 \ln \left (\ln \left (3+x \right )\right )\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((64*x^2+192*x)*ln(3+x)+2)*ln(ln(3+x))+(1024*x^4+3072*x^3+194*x^2+581*x-3)*ln(3+x)+32*x^2+6)/(3+x)/ln(3+x
),x,method=_RETURNVERBOSE)

[Out]

ln(ln(3+x))^2+32*x^2*ln(ln(3+x))+256*x^4+97*x^2-x+6*ln(ln(3+x))

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maxima [A]  time = 0.40, size = 38, normalized size = 1.65 \begin {gather*} 256 \, x^{4} + 32 \, x^{2} \log \left (\log \left (x + 3\right )\right ) + 97 \, x^{2} + \log \left (\log \left (x + 3\right )\right )^{2} - x + 6 \, \log \left (\log \left (x + 3\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((64*x^2+192*x)*log(3+x)+2)*log(log(3+x))+(1024*x^4+3072*x^3+194*x^2+581*x-3)*log(3+x)+32*x^2+6)/(3
+x)/log(3+x),x, algorithm="maxima")

[Out]

256*x^4 + 32*x^2*log(log(x + 3)) + 97*x^2 + log(log(x + 3))^2 - x + 6*log(log(x + 3))

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mupad [B]  time = 5.30, size = 38, normalized size = 1.65 \begin {gather*} 256\,x^4+32\,x^2\,\ln \left (\ln \left (x+3\right )\right )+97\,x^2-x+{\ln \left (\ln \left (x+3\right )\right )}^2+6\,\ln \left (\ln \left (x+3\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x + 3)*(581*x + 194*x^2 + 3072*x^3 + 1024*x^4 - 3) + log(log(x + 3))*(log(x + 3)*(192*x + 64*x^2) + 2
) + 32*x^2 + 6)/(log(x + 3)*(x + 3)),x)

[Out]

6*log(log(x + 3)) - x + log(log(x + 3))^2 + 97*x^2 + 256*x^4 + 32*x^2*log(log(x + 3))

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sympy [A]  time = 0.39, size = 39, normalized size = 1.70 \begin {gather*} 256 x^{4} + 32 x^{2} \log {\left (\log {\left (x + 3 \right )} \right )} + 97 x^{2} - x + \log {\left (\log {\left (x + 3 \right )} \right )}^{2} + 6 \log {\left (\log {\left (x + 3 \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((64*x**2+192*x)*ln(3+x)+2)*ln(ln(3+x))+(1024*x**4+3072*x**3+194*x**2+581*x-3)*ln(3+x)+32*x**2+6)/(
3+x)/ln(3+x),x)

[Out]

256*x**4 + 32*x**2*log(log(x + 3)) + 97*x**2 - x + log(log(x + 3))**2 + 6*log(log(x + 3))

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