\(\int \frac {1}{x+x \log (7 x)+x \log ^2(7 x)} \, dx\) [258]

Optimal result

Integrand size = 18, antiderivative size = 22 \[ \int \frac {1}{x+x \log (7 x)+x \log ^2(7 x)} \, dx=\frac {2 \arctan \left (\frac {1+2 \log (7 x)}{\sqrt {3}}\right )}{\sqrt {3}} \]

[Out]

2/3*arctan(1/3*(1+2*ln(7*x))*3^(1/2))*3^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {632, 210} \[ \int \frac {1}{x+x \log (7 x)+x \log ^2(7 x)} \, dx=\frac {2 \arctan \left (\frac {2 \log (7 x)+1}{\sqrt {3}}\right )}{\sqrt {3}} \]

[In]

Int[(x + x*Log[7*x] + x*Log[7*x]^2)^(-1),x]

[Out]

(2*ArcTan[(1 + 2*Log[7*x])/Sqrt[3]])/Sqrt[3]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\log (7 x)\right ) \\ & = -\left (2 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \log (7 x)\right )\right ) \\ & = \frac {2 \tan ^{-1}\left (\frac {1+2 \log (7 x)}{\sqrt {3}}\right )}{\sqrt {3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x+x \log (7 x)+x \log ^2(7 x)} \, dx=\frac {2 \arctan \left (\frac {1+2 \log (7 x)}{\sqrt {3}}\right )}{\sqrt {3}} \]

[In]

Integrate[(x + x*Log[7*x] + x*Log[7*x]^2)^(-1),x]

[Out]

(2*ArcTan[(1 + 2*Log[7*x])/Sqrt[3]])/Sqrt[3]

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

method	result	size
derivativedivides	\(\frac {2 \arctan \left (\frac {\left (1+2 \ln \left (7 x \right )\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}\)	\(20\)
default	\(\frac {2 \arctan \left (\frac {\left (1+2 \ln \left (7 x \right )\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}\)	\(20\)
risch	\(\frac {i \sqrt {3}\, \ln \left (\ln \left (7 x \right )+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{3}-\frac {i \sqrt {3}\, \ln \left (\ln \left (7 x \right )+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{3}\)	\(40\)

[In]

int(1/(x+x*ln(7*x)+x*ln(7*x)^2),x,method=_RETURNVERBOSE)

[Out]

2/3*arctan(1/3*(1+2*ln(7*x))*3^(1/2))*3^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x+x \log (7 x)+x \log ^2(7 x)} \, dx=\frac {2}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \log \left (7 \, x\right ) + \frac {1}{3} \, \sqrt {3}\right ) \]

[In]

integrate(1/(x+x*log(7*x)+x*log(7*x)^2),x, algorithm="fricas")

[Out]

2/3*sqrt(3)*arctan(2/3*sqrt(3)*log(7*x) + 1/3*sqrt(3))

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x+x \log (7 x)+x \log ^2(7 x)} \, dx=\operatorname {RootSum} {\left (3 z^{2} + 1, \left ( i \mapsto i \log {\left (\frac {3 i}{2} + \log {\left (7 x \right )} + \frac {1}{2} \right )} \right )\right )} \]

[In]

integrate(1/(x+x*ln(7*x)+x*ln(7*x)**2),x)

[Out]

RootSum(3*_z**2 + 1, Lambda(_i, _i*log(3*_i/2 + log(7*x) + 1/2)))

Maxima [F]

\[ \int \frac {1}{x+x \log (7 x)+x \log ^2(7 x)} \, dx=\int { \frac {1}{x \log \left (7 \, x\right )^{2} + x \log \left (7 \, x\right ) + x} \,d x } \]

[In]

integrate(1/(x+x*log(7*x)+x*log(7*x)^2),x, algorithm="maxima")

[Out]

integrate(1/(x*log(7*x)^2 + x*log(7*x) + x), x)

Giac [F]

\[ \int \frac {1}{x+x \log (7 x)+x \log ^2(7 x)} \, dx=\int { \frac {1}{x \log \left (7 \, x\right )^{2} + x \log \left (7 \, x\right ) + x} \,d x } \]

[In]

integrate(1/(x+x*log(7*x)+x*log(7*x)^2),x, algorithm="giac")

[Out]

integrate(1/(x*log(7*x)^2 + x*log(7*x) + x), x)

Mupad [B] (verification not implemented)

Time = 1.56 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x+x \log (7 x)+x \log ^2(7 x)} \, dx=\frac {2\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\left (2\,\ln \left (7\,x\right )+1\right )}{3}\right )}{3} \]

[In]

int(1/(x + x*log(7*x) + x*log(7*x)^2),x)

[Out]

(2*3^(1/2)*atan((3^(1/2)*(2*log(7*x) + 1))/3))/3