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

Optimal. Leaf size=10 \[ \frac{1}{x^2+1}+\log (x) \]

[Out]

(1 + x^2)^(-1) + Log[x]

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Rubi [A]  time = 0.0166967, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1252, 894} \[ \frac{1}{x^2+1}+\log (x) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^4)/(x*(1 + x^2)^2),x]

[Out]

(1 + x^2)^(-1) + Log[x]

Rule 1252

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m
- 1)/2)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x] && IntegerQ[(m + 1)/2]

Rule 894

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

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

Mathematica [A]  time = 0.0064212, size = 10, normalized size = 1. \[ \frac{1}{x^2+1}+\log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + x^2)^(-1) + Log[x]

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Maple [A]  time = 0.005, size = 11, normalized size = 1.1 \begin{align*} \left ({x}^{2}+1 \right ) ^{-1}+\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(x^2+1)+ln(x)

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Maxima [A]  time = 0.925533, size = 19, normalized size = 1.9 \begin{align*} \frac{1}{x^{2} + 1} + \frac{1}{2} \, \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(x^2 + 1) + 1/2*log(x^2)

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Fricas [A]  time = 1.91779, size = 46, normalized size = 4.6 \begin{align*} \frac{{\left (x^{2} + 1\right )} \log \left (x\right ) + 1}{x^{2} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x^2 + 1)*log(x) + 1)/(x^2 + 1)

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Sympy [A]  time = 0.09167, size = 8, normalized size = 0.8 \begin{align*} \log{\left (x \right )} + \frac{1}{x^{2} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) + 1/(x**2 + 1)

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Giac [A]  time = 1.06447, size = 19, normalized size = 1.9 \begin{align*} \frac{1}{x^{2} + 1} + \frac{1}{2} \, \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(x^2 + 1) + 1/2*log(x^2)