3.48.88 \(\int \frac {x+(-225-i \pi -x-\log (\log (7))) \log (225+i \pi +x+\log (\log (7)))}{(225 x+x^2+x (i \pi +\log (\log (7)))) \log (225+i \pi +x+\log (\log (7)))} \, dx\)

Optimal. Leaf size=21 \[ -1+\log \left (\frac {\log (4) \log (225+i \pi +x+\log (\log (7)))}{x}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.38, antiderivative size = 18, normalized size of antiderivative = 0.86, number of steps used = 8, number of rules used = 7, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {6, 1593, 6742, 2390, 12, 2302, 29} \begin {gather*} -\log (x)+\log (\log (x+i \pi +225+\log (\log (7)))) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x + (-225 - I*Pi - x - Log[Log[7]])*Log[225 + I*Pi + x + Log[Log[7]]])/((225*x + x^2 + x*(I*Pi + Log[Log[
7]]))*Log[225 + I*Pi + x + Log[Log[7]]]),x]

[Out]

-Log[x] + Log[Log[225 + I*Pi + x + Log[Log[7]]]]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 29

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

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 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]

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 {x+(-225-i \pi -x-\log (\log (7))) \log (225+i \pi +x+\log (\log (7)))}{\left (x^2+x (225+i \pi +\log (\log (7)))\right ) \log (225+i \pi +x+\log (\log (7)))} \, dx\\ &=\int \frac {x+(-225-i \pi -x-\log (\log (7))) \log (225+i \pi +x+\log (\log (7)))}{x (225+i \pi +x+\log (\log (7))) \log (225+i \pi +x+\log (\log (7)))} \, dx\\ &=\int \left (-\frac {1}{x}-\frac {i}{(-225 i+\pi -i x-i \log (\log (7))) \log (225+i \pi +x+\log (\log (7)))}\right ) \, dx\\ &=-\log (x)-i \int \frac {1}{(-225 i+\pi -i x-i \log (\log (7))) \log (225+i \pi +x+\log (\log (7)))} \, dx\\ &=-\log (x)-i \operatorname {Subst}\left (\int \frac {225+i \pi +\log (\log (7))}{x \log (x) (-225 i+\pi -i \log (\log (7)))} \, dx,x,225+i \pi +x+\log (\log (7))\right )\\ &=-\log (x)+\operatorname {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,225+i \pi +x+\log (\log (7))\right )\\ &=-\log (x)+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (225+i \pi +x+\log (\log (7)))\right )\\ &=-\log (x)+\log (\log (225+i \pi +x+\log (\log (7))))\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 18, normalized size = 0.86 \begin {gather*} -\log (x)+\log (\log (225+i \pi +x+\log (\log (7)))) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x + (-225 - I*Pi - x - Log[Log[7]])*Log[225 + I*Pi + x + Log[Log[7]]])/((225*x + x^2 + x*(I*Pi + Lo
g[Log[7]]))*Log[225 + I*Pi + x + Log[Log[7]]]),x]

[Out]

-Log[x] + Log[Log[225 + I*Pi + x + Log[Log[7]]]]

________________________________________________________________________________________

fricas [A]  time = 0.55, size = 15, normalized size = 0.71 \begin {gather*} -\log \relax (x) + \log \left (\log \left (x + \log \left (-\log \relax (7)\right ) + 225\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(-log(7))-x-225)*log(log(-log(7))+x+225)+x)/(x*log(-log(7))+x^2+225*x)/log(log(-log(7))+x+225)
,x, algorithm="fricas")

[Out]

-log(x) + log(log(x + log(-log(7)) + 225))

________________________________________________________________________________________

giac [A]  time = 0.12, size = 15, normalized size = 0.71 \begin {gather*} -\log \relax (x) + \log \left (\log \left (x + \log \left (-\log \relax (7)\right ) + 225\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(-log(7))-x-225)*log(log(-log(7))+x+225)+x)/(x*log(-log(7))+x^2+225*x)/log(log(-log(7))+x+225)
,x, algorithm="giac")

[Out]

-log(x) + log(log(x + log(-log(7)) + 225))

________________________________________________________________________________________

maple [A]  time = 0.28, size = 16, normalized size = 0.76

method	result	size
norman	\(-\ln \relax (x )+\ln \left (\ln \left (\ln \left (-\ln \relax (7)\right )+x +225\right )\right )\)	\(16\)
risch	\(-\ln \relax (x )+\ln \left (\ln \left (\ln \left (\ln \relax (7)\right )+i \pi +x +225\right )\right )\)	\(18\)
derivativedivides	\(-\ln \left (\ln \left (\ln \relax (7)\right )+i \pi -\ln \left (-\ln \relax (7)\right )-x \right )+\ln \left (\ln \left (\ln \left (-\ln \relax (7)\right )+x +225\right )\right )\)	\(33\)
default	\(-\ln \left (\ln \left (\ln \relax (7)\right )+i \pi -\ln \left (-\ln \relax (7)\right )-x \right )+\ln \left (\ln \left (\ln \left (-\ln \relax (7)\right )+x +225\right )\right )\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-ln(-ln(7))-x-225)*ln(ln(-ln(7))+x+225)+x)/(x*ln(-ln(7))+x^2+225*x)/ln(ln(-ln(7))+x+225),x,method=_RETUR
NVERBOSE)

[Out]

-ln(x)+ln(ln(ln(-ln(7))+x+225))

________________________________________________________________________________________

maxima [B]  time = 0.50, size = 94, normalized size = 4.48 \begin {gather*} {\left (\frac {\log \left (x + \log \left (-\log \relax (7)\right ) + 225\right )}{\log \left (-\log \relax (7)\right ) + 225} - \frac {\log \relax (x)}{\log \left (-\log \relax (7)\right ) + 225}\right )} \log \left (-\log \relax (7)\right ) + \frac {225 \, \log \left (x + \log \left (-\log \relax (7)\right ) + 225\right )}{\log \left (-\log \relax (7)\right ) + 225} - \frac {225 \, \log \relax (x)}{\log \left (-\log \relax (7)\right ) + 225} - \log \left (x + \log \left (-\log \relax (7)\right ) + 225\right ) + \log \left (\log \left (x + \log \left (-\log \relax (7)\right ) + 225\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(-log(7))-x-225)*log(log(-log(7))+x+225)+x)/(x*log(-log(7))+x^2+225*x)/log(log(-log(7))+x+225)
,x, algorithm="maxima")

[Out]

(log(x + log(-log(7)) + 225)/(log(-log(7)) + 225) - log(x)/(log(-log(7)) + 225))*log(-log(7)) + 225*log(x + lo
g(-log(7)) + 225)/(log(-log(7)) + 225) - 225*log(x)/(log(-log(7)) + 225) - log(x + log(-log(7)) + 225) + log(l
og(x + log(-log(7)) + 225))

________________________________________________________________________________________

mupad [B]  time = 3.62, size = 15, normalized size = 0.71 \begin {gather*} \ln \left (\ln \left (x+\ln \left (-\ln \relax (7)\right )+225\right )\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - log(x + log(-log(7)) + 225)*(x + log(-log(7)) + 225))/(log(x + log(-log(7)) + 225)*(225*x + x*log(-lo
g(7)) + x^2)),x)

[Out]

log(log(x + log(-log(7)) + 225)) - log(x)

________________________________________________________________________________________

sympy [A]  time = 0.25, size = 17, normalized size = 0.81 \begin {gather*} - \log {\relax (x )} + \log {\left (\log {\left (x + \log {\left (\log {\relax (7 )} \right )} + 225 + i \pi \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-ln(-ln(7))-x-225)*ln(ln(-ln(7))+x+225)+x)/(x*ln(-ln(7))+x**2+225*x)/ln(ln(-ln(7))+x+225),x)

[Out]

-log(x) + log(log(x + log(log(7)) + 225 + I*pi))

________________________________________________________________________________________