3.117 \(\int \frac{x}{(1+x^2) (2+x^2) (3+x^2) (4+x^2)} \, dx\)

Optimal. Leaf size=41 \[ \frac{1}{12} \log \left (x^2+1\right )-\frac{1}{4} \log \left (x^2+2\right )+\frac{1}{4} \log \left (x^2+3\right )-\frac{1}{12} \log \left (x^2+4\right ) \]

[Out]

Log[1 + x^2]/12 - Log[2 + x^2]/4 + Log[3 + x^2]/4 - Log[4 + x^2]/12

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Rubi [A]  time = 0.311424, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {6694, 180} \[ \frac{1}{12} \log \left (x^2+1\right )-\frac{1}{4} \log \left (x^2+2\right )+\frac{1}{4} \log \left (x^2+3\right )-\frac{1}{12} \log \left (x^2+4\right ) \]

Antiderivative was successfully verified.

[In]

Int[x/((1 + x^2)*(2 + x^2)*(3 + x^2)*(4 + x^2)),x]

[Out]

Log[1 + x^2]/12 - Log[2 + x^2]/4 + Log[3 + x^2]/4 - Log[4 + x^2]/12

Rule 6694

Int[(u_)*((c_.) + (d_.)*(v_))^(n_.)*((e_.) + (f_.)*(w_))^(p_.)*((a_.) + (b_.)*(y_))^(m_.)*((g_.) + (h_.)*(z_))
^(q_.), x_Symbol] :> With[{r = DerivativeDivides[y, u, x]}, Dist[r, Subst[Int[(a + b*x)^m*(c + d*x)^n*(e + f*x
)^p*(g + h*x)^q, x], x, y], x] /;  !FalseQ[r]] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q}, x] && EqQ[v, y]
&& EqQ[w, y] && EqQ[z, y]

Rule 180

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_))^(q_), x
_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x], x] /; FreeQ[{a, b, c, d,
e, f, g, h, m, n}, x] && IntegersQ[p, q]

Rubi steps

\begin{align*} \int \frac{x}{\left (1+x^2\right ) \left (2+x^2\right ) \left (3+x^2\right ) \left (4+x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{(1+x) (2+x) (3+x) (4+x)} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{6 (1+x)}-\frac{1}{2 (2+x)}+\frac{1}{2 (3+x)}-\frac{1}{6 (4+x)}\right ) \, dx,x,x^2\right )\\ &=\frac{1}{12} \log \left (1+x^2\right )-\frac{1}{4} \log \left (2+x^2\right )+\frac{1}{4} \log \left (3+x^2\right )-\frac{1}{12} \log \left (4+x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0083798, size = 41, normalized size = 1. \[ \frac{1}{12} \log \left (x^2+1\right )-\frac{1}{4} \log \left (x^2+2\right )+\frac{1}{4} \log \left (x^2+3\right )-\frac{1}{12} \log \left (x^2+4\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/((1 + x^2)*(2 + x^2)*(3 + x^2)*(4 + x^2)),x]

[Out]

Log[1 + x^2]/12 - Log[2 + x^2]/4 + Log[3 + x^2]/4 - Log[4 + x^2]/12

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Maple [A]  time = 0.01, size = 34, normalized size = 0.8 \begin{align*}{\frac{\ln \left ({x}^{2}+1 \right ) }{12}}-{\frac{\ln \left ({x}^{2}+2 \right ) }{4}}+{\frac{\ln \left ({x}^{2}+3 \right ) }{4}}-{\frac{\ln \left ({x}^{2}+4 \right ) }{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^2+1)/(x^2+2)/(x^2+3)/(x^2+4),x)

[Out]

1/12*ln(x^2+1)-1/4*ln(x^2+2)+1/4*ln(x^2+3)-1/12*ln(x^2+4)

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Maxima [A]  time = 0.921042, size = 45, normalized size = 1.1 \begin{align*} -\frac{1}{12} \, \log \left (x^{2} + 4\right ) + \frac{1}{4} \, \log \left (x^{2} + 3\right ) - \frac{1}{4} \, \log \left (x^{2} + 2\right ) + \frac{1}{12} \, \log \left (x^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2+1)/(x^2+2)/(x^2+3)/(x^2+4),x, algorithm="maxima")

[Out]

-1/12*log(x^2 + 4) + 1/4*log(x^2 + 3) - 1/4*log(x^2 + 2) + 1/12*log(x^2 + 1)

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Fricas [A]  time = 2.17329, size = 105, normalized size = 2.56 \begin{align*} -\frac{1}{12} \, \log \left (x^{2} + 4\right ) + \frac{1}{4} \, \log \left (x^{2} + 3\right ) - \frac{1}{4} \, \log \left (x^{2} + 2\right ) + \frac{1}{12} \, \log \left (x^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2+1)/(x^2+2)/(x^2+3)/(x^2+4),x, algorithm="fricas")

[Out]

-1/12*log(x^2 + 4) + 1/4*log(x^2 + 3) - 1/4*log(x^2 + 2) + 1/12*log(x^2 + 1)

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Sympy [A]  time = 0.161699, size = 32, normalized size = 0.78 \begin{align*} \frac{\log{\left (x^{2} + 1 \right )}}{12} - \frac{\log{\left (x^{2} + 2 \right )}}{4} + \frac{\log{\left (x^{2} + 3 \right )}}{4} - \frac{\log{\left (x^{2} + 4 \right )}}{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x**2+1)/(x**2+2)/(x**2+3)/(x**2+4),x)

[Out]

log(x**2 + 1)/12 - log(x**2 + 2)/4 + log(x**2 + 3)/4 - log(x**2 + 4)/12

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Giac [A]  time = 1.0602, size = 45, normalized size = 1.1 \begin{align*} -\frac{1}{12} \, \log \left (x^{2} + 4\right ) + \frac{1}{4} \, \log \left (x^{2} + 3\right ) - \frac{1}{4} \, \log \left (x^{2} + 2\right ) + \frac{1}{12} \, \log \left (x^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2+1)/(x^2+2)/(x^2+3)/(x^2+4),x, algorithm="giac")

[Out]

-1/12*log(x^2 + 4) + 1/4*log(x^2 + 3) - 1/4*log(x^2 + 2) + 1/12*log(x^2 + 1)