3.354 \(\int \frac {4+x+3 x^2}{x+x^3} \, dx\)

Optimal. Leaf size=17 \[ -\frac {1}{2} \log \left (x^2+1\right )+4 \log (x)+\tan ^{-1}(x) \]

[Out]

arctan(x)+4*ln(x)-1/2*ln(x^2+1)

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Rubi [A]  time = 0.04, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1593, 1802, 635, 203, 260} \[ -\frac {1}{2} \log \left (x^2+1\right )+4 \log (x)+\tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(4 + x + 3*x^2)/(x + x^3),x]

[Out]

ArcTan[x] + 4*Log[x] - Log[1 + x^2]/2

Rule 203

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

Rule 260

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

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

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 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {align*} \int \frac {4+x+3 x^2}{x+x^3} \, dx &=\int \frac {4+x+3 x^2}{x \left (1+x^2\right )} \, dx\\ &=\int \left (\frac {4}{x}+\frac {1-x}{1+x^2}\right ) \, dx\\ &=4 \log (x)+\int \frac {1-x}{1+x^2} \, dx\\ &=4 \log (x)+\int \frac {1}{1+x^2} \, dx-\int \frac {x}{1+x^2} \, dx\\ &=\tan ^{-1}(x)+4 \log (x)-\frac {1}{2} \log \left (1+x^2\right )\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \[ -\frac {1}{2} \log \left (x^2+1\right )+4 \log (x)+\tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[(4 + x + 3*x^2)/(x + x^3),x]

[Out]

ArcTan[x] + 4*Log[x] - Log[1 + x^2]/2

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fricas [A]  time = 0.57, size = 15, normalized size = 0.88 \[ \arctan \relax (x) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) + 4 \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+x+4)/(x^3+x),x, algorithm="fricas")

[Out]

arctan(x) - 1/2*log(x^2 + 1) + 4*log(x)

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giac [A]  time = 0.24, size = 16, normalized size = 0.94 \[ \arctan \relax (x) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) + 4 \, \log \left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+x+4)/(x^3+x),x, algorithm="giac")

[Out]

arctan(x) - 1/2*log(x^2 + 1) + 4*log(abs(x))

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maple [A]  time = 0.00, size = 16, normalized size = 0.94 \[ \arctan \relax (x )+4 \ln \relax (x )-\frac {\ln \left (x^{2}+1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2+x+4)/(x^3+x),x)

[Out]

arctan(x)+4*ln(x)-1/2*ln(x^2+1)

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maxima [A]  time = 1.53, size = 15, normalized size = 0.88 \[ \arctan \relax (x) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) + 4 \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+x+4)/(x^3+x),x, algorithm="maxima")

[Out]

arctan(x) - 1/2*log(x^2 + 1) + 4*log(x)

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mupad [B]  time = 2.28, size = 23, normalized size = 1.35 \[ 4\,\ln \relax (x)+\ln \left (x-\mathrm {i}\right )\,\left (-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )+\ln \left (x+1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 3*x^2 + 4)/(x + x^3),x)

[Out]

4*log(x) - log(x + 1i)*(1/2 - 1i/2) - log(x - 1i)*(1/2 + 1i/2)

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sympy [A]  time = 0.13, size = 15, normalized size = 0.88 \[ 4 \log {\relax (x )} - \frac {\log {\left (x^{2} + 1 \right )}}{2} + \operatorname {atan}{\relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**2+x+4)/(x**3+x),x)

[Out]

4*log(x) - log(x**2 + 1)/2 + atan(x)

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