3.377 \(\int \frac{1}{(1+x^2) (3+\frac{10 x}{1+x^2})} \, dx\)

Optimal. Leaf size=19 \[ \frac{1}{8} \log (3 x+1)-\frac{1}{8} \log (x+3) \]

[Out]

-Log[3 + x]/8 + Log[1 + 3*x]/8

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Rubi [A]  time = 0.0618751, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6688, 616, 31} \[ \frac{1}{8} \log (3 x+1)-\frac{1}{8} \log (x+3) \]

Antiderivative was successfully verified.

[In]

Int[1/((1 + x^2)*(3 + (10*x)/(1 + x^2))),x]

[Out]

-Log[3 + x]/8 + Log[1 + 3*x]/8

Rule 6688

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

Rule 616

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{\left (1+x^2\right ) \left (3+\frac{10 x}{1+x^2}\right )} \, dx &=\int \frac{1}{3+10 x+3 x^2} \, dx\\ &=\frac{3}{8} \int \frac{1}{1+3 x} \, dx-\frac{3}{8} \int \frac{1}{9+3 x} \, dx\\ &=-\frac{1}{8} \log (3+x)+\frac{1}{8} \log (1+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0031748, size = 19, normalized size = 1. \[ \frac{1}{8} \log (3 x+1)-\frac{1}{8} \log (x+3) \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 + x^2)*(3 + (10*x)/(1 + x^2))),x]

[Out]

-Log[3 + x]/8 + Log[1 + 3*x]/8

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Maple [A]  time = 0.006, size = 16, normalized size = 0.8 \begin{align*} -{\frac{\ln \left ( 3+x \right ) }{8}}+{\frac{\ln \left ( 1+3\,x \right ) }{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2+1)/(3+10*x/(x^2+1)),x)

[Out]

-1/8*ln(3+x)+1/8*ln(1+3*x)

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Maxima [A]  time = 1.08613, size = 20, normalized size = 1.05 \begin{align*} \frac{1}{8} \, \log \left (3 \, x + 1\right ) - \frac{1}{8} \, \log \left (x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+1)/(3+10*x/(x^2+1)),x, algorithm="maxima")

[Out]

1/8*log(3*x + 1) - 1/8*log(x + 3)

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Fricas [A]  time = 1.23929, size = 47, normalized size = 2.47 \begin{align*} \frac{1}{8} \, \log \left (3 \, x + 1\right ) - \frac{1}{8} \, \log \left (x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+1)/(3+10*x/(x^2+1)),x, algorithm="fricas")

[Out]

1/8*log(3*x + 1) - 1/8*log(x + 3)

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Sympy [A]  time = 0.098616, size = 14, normalized size = 0.74 \begin{align*} \frac{\log{\left (x + \frac{1}{3} \right )}}{8} - \frac{\log{\left (x + 3 \right )}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**2+1)/(3+10*x/(x**2+1)),x)

[Out]

log(x + 1/3)/8 - log(x + 3)/8

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Giac [A]  time = 1.15885, size = 23, normalized size = 1.21 \begin{align*} \frac{1}{8} \, \log \left ({\left | 3 \, x + 1 \right |}\right ) - \frac{1}{8} \, \log \left ({\left | x + 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+1)/(3+10*x/(x^2+1)),x, algorithm="giac")

[Out]

1/8*log(abs(3*x + 1)) - 1/8*log(abs(x + 3))