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

Optimal. Leaf size=22 \[ \frac {2 \tan ^{-1}\left (\frac {2 \log (7 x)+1}{\sqrt {3}}\right )}{\sqrt {3}} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {618, 204} \[ \frac {2 \tan ^{-1}\left (\frac {2 \log (7 x)+1}{\sqrt {3}}\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[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 204

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

Rule 618

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

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Mathematica [A]  time = 0.03, size = 22, normalized size = 1.00 \[ \frac {2 \tan ^{-1}\left (\frac {2 \log (7 x)+1}{\sqrt {3}}\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[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]

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fricas [A]  time = 0.42, size = 21, normalized size = 0.95 \[ \frac {2}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \log \left (7 \, x\right ) + \frac {1}{3} \, \sqrt {3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \log \left (7 \, x\right )^{2} + x \log \left (7 \, x\right ) + x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.07, size = 20, normalized size = 0.91 \[ \frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 \ln \left (7 x \right )+1\right ) \sqrt {3}}{3}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \log \left (7 \, x\right )^{2} + x \log \left (7 \, x\right ) + x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.43, size = 19, normalized size = 0.86 \[ \frac {2\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\left (2\,\ln \left (7\,x\right )+1\right )}{3}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.17, size = 22, normalized size = 1.00 \[ \operatorname {RootSum} {\left (3 z^{2} + 1, \left (i \mapsto i \log {\left (\frac {3 i}{2} + \log {\left (7 x \right )} + \frac {1}{2} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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