3.99.28 \(\int \frac {1}{(22 x+x \log (\sqrt [3]{2} x)) \log (-22-\log (\sqrt [3]{2} x))} \, dx\)

Optimal. Leaf size=14 \[ \log \left (\log \left (-22-\log \left (\sqrt [3]{2} x\right )\right )\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2390, 2302, 29} \begin {gather*} \log \left (\log \left (-\log \left (\sqrt [3]{2} x\right )-22\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((22*x + x*Log[2^(1/3)*x])*Log[-22 - Log[2^(1/3)*x]]),x]

[Out]

Log[Log[-22 - Log[2^(1/3)*x]]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\operatorname {Subst}\left (\int \frac {1}{(22+x) \log (-22-x)} \, dx,x,\log \left (\sqrt [3]{2} x\right )\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,-22-\log \left (\sqrt [3]{2} x\right )\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (-22-\log \left (\sqrt [3]{2} x\right )\right )\right )\\ &=\log \left (\log \left (-22-\log \left (\sqrt [3]{2} x\right )\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (22 x+x \log \left (\sqrt [3]{2} x\right )\right ) \log \left (-22-\log \left (\sqrt [3]{2} x\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[1/((22*x + x*Log[2^(1/3)*x])*Log[-22 - Log[2^(1/3)*x]]),x]

[Out]

Integrate[1/((22*x + x*Log[2^(1/3)*x])*Log[-22 - Log[2^(1/3)*x]]), x]

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fricas [A]  time = 0.90, size = 12, normalized size = 0.86 \begin {gather*} \log \left (\log \left (-\log \left (2^{\frac {1}{3}} x\right ) - 22\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x*log(x*2^(1/3))+22*x)/log(-log(x*2^(1/3))-22),x, algorithm="fricas")

[Out]

log(log(-log(2^(1/3)*x) - 22))

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giac [A]  time = 0.21, size = 17, normalized size = 1.21 \begin {gather*} \log \left (\log \relax (3) - \log \left (-\log \relax (2) - 3 \, \log \relax (x) - 66\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x*log(x*2^(1/3))+22*x)/log(-log(x*2^(1/3))-22),x, algorithm="giac")

[Out]

log(log(3) - log(-log(2) - 3*log(x) - 66))

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maple [A]  time = 0.02, size = 13, normalized size = 0.93

method	result	size
norman	\(\ln \left (\ln \left (-\ln \left (x 2^{\frac {1}{3}}\right )-22\right )\right )\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*ln(x*2^(1/3))+22*x)/ln(-ln(x*2^(1/3))-22),x,method=_RETURNVERBOSE)

[Out]

ln(ln(-ln(x*2^(1/3))-22))

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maxima [A]  time = 0.56, size = 17, normalized size = 1.21 \begin {gather*} \log \left (-\log \relax (3) + \log \left (-\log \relax (2) - 3 \, \log \relax (x) - 66\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x*log(x*2^(1/3))+22*x)/log(-log(x*2^(1/3))-22),x, algorithm="maxima")

[Out]

log(-log(3) + log(-log(2) - 3*log(x) - 66))

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mupad [B]  time = 6.14, size = 12, normalized size = 0.86 \begin {gather*} \ln \left (\ln \left (-\ln \left (2^{1/3}\,x\right )-22\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(log(- log(2^(1/3)*x) - 22)*(22*x + x*log(2^(1/3)*x))),x)

[Out]

log(log(- log(2^(1/3)*x) - 22))

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sympy [A]  time = 0.39, size = 14, normalized size = 1.00 \begin {gather*} \log {\left (\log {\left (- \log {\left (\sqrt [3]{2} x \right )} - 22 \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x*ln(x*2**(1/3))+22*x)/ln(-ln(x*2**(1/3))-22),x)

[Out]

log(log(-log(2**(1/3)*x) - 22))

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