3.33.53 \(\int \frac {-12-4 \log (x)+(3+\log (x))^x (-4+(-12-4 \log (x)) \log (3+\log (x)))}{3+3 x+(1+x) \log (x)+(3+\log (x))^{1+x}} \, dx\) [3253]

Optimal. Leaf size=14 \[ \log \left (\frac {16}{\left (1+x+(3+\log (x))^x\right )^4}\right ) \]

[Out]

ln(16/(x+exp(x*ln(3+ln(x)))+1)^4)

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Rubi [F]
time = 2.31, 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 {-12-4 \log (x)+(3+\log (x))^x (-4+(-12-4 \log (x)) \log (3+\log (x)))}{3+3 x+(1+x) \log (x)+(3+\log (x))^{1+x}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-12 - 4*Log[x] + (3 + Log[x])^x*(-4 + (-12 - 4*Log[x])*Log[3 + Log[x]]))/(3 + 3*x + (1 + x)*Log[x] + (3 +
Log[x])^(1 + x)),x]

[Out]

(-4*ExpIntegralEi[3 + Log[x]])/E^3 - 8*Defer[Int][1/((3 + Log[x])*(1 + x + (3 + Log[x])^x)), x] + 4*Defer[Int]
[x/((3 + Log[x])*(1 + x + (3 + Log[x])^x)), x] - 4*Defer[Int][Log[x]/((3 + Log[x])*(1 + x + (3 + Log[x])^x)),
x] - 4*Defer[Int][Log[3 + Log[x]], x] + 12*Defer[Int][Log[3 + Log[x]]/((3 + Log[x])*(1 + x + (3 + Log[x])^x)),
x] + 12*Defer[Int][(x*Log[3 + Log[x]])/((3 + Log[x])*(1 + x + (3 + Log[x])^x)), x] + 4*Defer[Int][(Log[x]*Log
[3 + Log[x]])/((3 + Log[x])*(1 + x + (3 + Log[x])^x)), x] + 4*Defer[Int][(x*Log[x]*Log[3 + Log[x]])/((3 + Log[
x])*(1 + x + (3 + Log[x])^x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-12-4 \log (x)+(3+\log (x))^x (-4+(-12-4 \log (x)) \log (3+\log (x)))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx\\ &=\int \left (-\frac {4 (1+3 \log (3+\log (x))+\log (x) \log (3+\log (x)))}{3+\log (x)}+\frac {4 (-2+x-\log (x)+3 \log (3+\log (x))+3 x \log (3+\log (x))+\log (x) \log (3+\log (x))+x \log (x) \log (3+\log (x)))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )}\right ) \, dx\\ &=-\left (4 \int \frac {1+3 \log (3+\log (x))+\log (x) \log (3+\log (x))}{3+\log (x)} \, dx\right )+4 \int \frac {-2+x-\log (x)+3 \log (3+\log (x))+3 x \log (3+\log (x))+\log (x) \log (3+\log (x))+x \log (x) \log (3+\log (x))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx\\ &=-\left (4 \int \left (\frac {1}{3+\log (x)}+\log (3+\log (x))\right ) \, dx\right )+4 \int \frac {-2+x+3 (1+x) \log (3+\log (x))+\log (x) (-1+(1+x) \log (3+\log (x)))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx\\ &=-\left (4 \int \frac {1}{3+\log (x)} \, dx\right )-4 \int \log (3+\log (x)) \, dx+4 \int \left (-\frac {2}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )}+\frac {x}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )}-\frac {\log (x)}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )}+\frac {3 \log (3+\log (x))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )}+\frac {3 x \log (3+\log (x))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )}+\frac {\log (x) \log (3+\log (x))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )}+\frac {x \log (x) \log (3+\log (x))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )}\right ) \, dx\\ &=4 \int \frac {x}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx-4 \int \frac {\log (x)}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx-4 \int \log (3+\log (x)) \, dx+4 \int \frac {\log (x) \log (3+\log (x))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx+4 \int \frac {x \log (x) \log (3+\log (x))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx-4 \text {Subst}\left (\int \frac {e^x}{3+x} \, dx,x,\log (x)\right )-8 \int \frac {1}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx+12 \int \frac {\log (3+\log (x))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx+12 \int \frac {x \log (3+\log (x))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx\\ &=-\frac {4 \text {Ei}(3+\log (x))}{e^3}+4 \int \frac {x}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx-4 \int \frac {\log (x)}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx-4 \int \log (3+\log (x)) \, dx+4 \int \frac {\log (x) \log (3+\log (x))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx+4 \int \frac {x \log (x) \log (3+\log (x))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx-8 \int \frac {1}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx+12 \int \frac {\log (3+\log (x))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx+12 \int \frac {x \log (3+\log (x))}{(3+\log (x)) \left (1+x+(3+\log (x))^x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.11, size = 12, normalized size = 0.86 \begin {gather*} -4 \log \left (1+x+(3+\log (x))^x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-12 - 4*Log[x] + (3 + Log[x])^x*(-4 + (-12 - 4*Log[x])*Log[3 + Log[x]]))/(3 + 3*x + (1 + x)*Log[x]
+ (3 + Log[x])^(1 + x)),x]

[Out]

-4*Log[1 + x + (3 + Log[x])^x]

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Maple [A]
time = 0.16, size = 13, normalized size = 0.93

method	result	size
risch	\(-4 \ln \left (\left (3+\ln \left (x \right )\right )^{x}+x +1\right )\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-4*ln(x)-12)*ln(3+ln(x))-4)*exp(x*ln(3+ln(x)))-4*ln(x)-12)/((3+ln(x))*exp(x*ln(3+ln(x)))+ln(x)*(x+1)+3*
x+3),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 12, normalized size = 0.86 \begin {gather*} -4 \, \log \left (x + {\left (\log \left (x\right ) + 3\right )}^{x} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*log(x)-12)*log(3+log(x))-4)*exp(x*log(3+log(x)))-4*log(x)-12)/((3+log(x))*exp(x*log(3+log(x)))
+log(x)*(1+x)+3*x+3),x, algorithm="maxima")

[Out]

-4*log(x + (log(x) + 3)^x + 1)

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Fricas [A]
time = 0.40, size = 12, normalized size = 0.86 \begin {gather*} -4 \, \log \left (x + {\left (\log \left (x\right ) + 3\right )}^{x} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*log(x)-12)*log(3+log(x))-4)*exp(x*log(3+log(x)))-4*log(x)-12)/((3+log(x))*exp(x*log(3+log(x)))
+log(x)*(1+x)+3*x+3),x, algorithm="fricas")

[Out]

-4*log(x + (log(x) + 3)^x + 1)

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Sympy [A]
time = 0.25, size = 17, normalized size = 1.21 \begin {gather*} - 4 \log {\left (x + e^{x \log {\left (\log {\left (x \right )} + 3 \right )}} + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*ln(x)-12)*ln(3+ln(x))-4)*exp(x*ln(3+ln(x)))-4*ln(x)-12)/((3+ln(x))*exp(x*ln(3+ln(x)))+ln(x)*(1
+x)+3*x+3),x)

[Out]

-4*log(x + exp(x*log(log(x) + 3)) + 1)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (14) = 28\).
time = 0.95, size = 67, normalized size = 4.79 \begin {gather*} -\frac {4 \, x e^{3} \log \left (x\right ) \log \left (\log \left (x\right ) + 3\right )}{e^{3} \log \left (x\right ) + 3 \, e^{3}} + 4 \, x \log \left (\log \left (x\right ) + 3\right ) - \frac {12 \, x e^{3} \log \left (\log \left (x\right ) + 3\right )}{e^{3} \log \left (x\right ) + 3 \, e^{3}} - 4 \, \log \left (x + {\left (\log \left (x\right ) + 3\right )}^{x} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*log(x)-12)*log(3+log(x))-4)*exp(x*log(3+log(x)))-4*log(x)-12)/((3+log(x))*exp(x*log(3+log(x)))
+log(x)*(1+x)+3*x+3),x, algorithm="giac")

[Out]

-4*x*e^3*log(x)*log(log(x) + 3)/(e^3*log(x) + 3*e^3) + 4*x*log(log(x) + 3) - 12*x*e^3*log(log(x) + 3)/(e^3*log
(x) + 3*e^3) - 4*log(x + (log(x) + 3)^x + 1)

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Mupad [B]
time = 2.15, size = 12, normalized size = 0.86 \begin {gather*} -4\,\ln \left (x+{\left (\ln \left (x\right )+3\right )}^x+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*log(x) + exp(x*log(log(x) + 3))*(log(log(x) + 3)*(4*log(x) + 12) + 4) + 12)/(3*x + log(x)*(x + 1) + ex
p(x*log(log(x) + 3))*(log(x) + 3) + 3),x)

[Out]

-4*log(x + (log(x) + 3)^x + 1)

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