3.10.64 \(\int \frac {150 x^2+150 x^3+(2 x-75 x^4+(2-75 x^3) \log (x)) \log (x+\log (x))}{(75 x^4+75 x^3 \log (x)) \log (x+\log (x))} \, dx\)

Optimal. Leaf size=20 \[ 2-\frac {1}{75 x^2}-x+\log \left (\log ^2(x+\log (x))\right ) \]

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Rubi [A]  time = 0.46, antiderivative size = 19, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 3, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {2561, 6688, 6684} \begin {gather*} -\frac {1}{75 x^2}-x+2 \log (\log (x+\log (x))) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(150*x^2 + 150*x^3 + (2*x - 75*x^4 + (2 - 75*x^3)*Log[x])*Log[x + Log[x]])/((75*x^4 + 75*x^3*Log[x])*Log[x
+ Log[x]]),x]

[Out]

-1/75*1/x^2 - x + 2*Log[Log[x + Log[x]]]

Rule 2561

Int[(u_.)*((a_.)*(x_)^(m_.) + Log[(c_.)*(x_)^(n_.)]^(q_.)*(b_.)*(x_)^(r_.))^(p_.), x_Symbol] :> Int[u*x^(p*r)*
(a*x^(m - r) + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[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 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {150 x^2+150 x^3+\left (2 x-75 x^4+\left (2-75 x^3\right ) \log (x)\right ) \log (x+\log (x))}{x^3 (75 x+75 \log (x)) \log (x+\log (x))} \, dx\\ &=\int \left (-1+\frac {2}{75 x^3}+\frac {2 (1+x)}{x (x+\log (x)) \log (x+\log (x))}\right ) \, dx\\ &=-\frac {1}{75 x^2}-x+2 \int \frac {1+x}{x (x+\log (x)) \log (x+\log (x))} \, dx\\ &=-\frac {1}{75 x^2}-x+2 \log (\log (x+\log (x)))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 19, normalized size = 0.95 \begin {gather*} -\frac {1}{75 x^2}-x+2 \log (\log (x+\log (x))) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(150*x^2 + 150*x^3 + (2*x - 75*x^4 + (2 - 75*x^3)*Log[x])*Log[x + Log[x]])/((75*x^4 + 75*x^3*Log[x])
*Log[x + Log[x]]),x]

[Out]

-1/75*1/x^2 - x + 2*Log[Log[x + Log[x]]]

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fricas [A]  time = 0.95, size = 23, normalized size = 1.15 \begin {gather*} -\frac {75 \, x^{3} - 150 \, x^{2} \log \left (\log \left (x + \log \relax (x)\right )\right ) + 1}{75 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-75*x^3+2)*log(x)-75*x^4+2*x)*log(x+log(x))+150*x^3+150*x^2)/(75*x^3*log(x)+75*x^4)/log(x+log(x))
,x, algorithm="fricas")

[Out]

-1/75*(75*x^3 - 150*x^2*log(log(x + log(x))) + 1)/x^2

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giac [A]  time = 0.37, size = 17, normalized size = 0.85 \begin {gather*} -x - \frac {1}{75 \, x^{2}} + 2 \, \log \left (\log \left (x + \log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-75*x^3+2)*log(x)-75*x^4+2*x)*log(x+log(x))+150*x^3+150*x^2)/(75*x^3*log(x)+75*x^4)/log(x+log(x))
,x, algorithm="giac")

[Out]

-x - 1/75/x^2 + 2*log(log(x + log(x)))

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maple [A]  time = 0.04, size = 18, normalized size = 0.90

method	result	size
default	\(-x -\frac {1}{75 x^{2}}+2 \ln \left (\ln \left (x +\ln \relax (x )\right )\right )\)	\(18\)
risch	\(-\frac {75 x^{3}+1}{75 x^{2}}+2 \ln \left (\ln \left (x +\ln \relax (x )\right )\right )\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-75*x^3+2)*ln(x)-75*x^4+2*x)*ln(x+ln(x))+150*x^3+150*x^2)/(75*x^3*ln(x)+75*x^4)/ln(x+ln(x)),x,method=_R
ETURNVERBOSE)

[Out]

-x-1/75/x^2+2*ln(ln(x+ln(x)))

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maxima [A]  time = 0.46, size = 21, normalized size = 1.05 \begin {gather*} -\frac {75 \, x^{3} + 1}{75 \, x^{2}} + 2 \, \log \left (\log \left (x + \log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-75*x^3+2)*log(x)-75*x^4+2*x)*log(x+log(x))+150*x^3+150*x^2)/(75*x^3*log(x)+75*x^4)/log(x+log(x))
,x, algorithm="maxima")

[Out]

-1/75*(75*x^3 + 1)/x^2 + 2*log(log(x + log(x)))

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mupad [B]  time = 0.92, size = 17, normalized size = 0.85 \begin {gather*} 2\,\ln \left (\ln \left (x+\ln \relax (x)\right )\right )-x-\frac {1}{75\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((150*x^2 - log(x + log(x))*(75*x^4 - 2*x + log(x)*(75*x^3 - 2)) + 150*x^3)/(log(x + log(x))*(75*x^3*log(x)
+ 75*x^4)),x)

[Out]

2*log(log(x + log(x))) - x - 1/(75*x^2)

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sympy [A]  time = 0.35, size = 17, normalized size = 0.85 \begin {gather*} - x + 2 \log {\left (\log {\left (x + \log {\relax (x )} \right )} \right )} - \frac {1}{75 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-75*x**3+2)*ln(x)-75*x**4+2*x)*ln(x+ln(x))+150*x**3+150*x**2)/(75*x**3*ln(x)+75*x**4)/ln(x+ln(x))
,x)

[Out]

-x + 2*log(log(x + log(x))) - 1/(75*x**2)

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