3.373 \(\int \frac{1+x^4}{x^3+x^5} \, dx\)

Optimal. Leaf size=18 \[ -\frac{1}{2 x^2}+\log \left (x^2+1\right )-\log (x) \]

[Out]

-1/(2*x^2) - Log[x] + Log[1 + x^2]

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Rubi [A]  time = 0.0298592, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1593, 1252, 894} \[ -\frac{1}{2 x^2}+\log \left (x^2+1\right )-\log (x) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^4)/(x^3 + x^5),x]

[Out]

-1/(2*x^2) - Log[x] + Log[1 + x^2]

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 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^3+x^5} \, dx &=\int \frac{1+x^4}{x^3 \left (1+x^2\right )} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1+x^2}{x^2 (1+x)} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{1}{x}+\frac{2}{1+x}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{2 x^2}-\log (x)+\log \left (1+x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0046253, size = 18, normalized size = 1. \[ -\frac{1}{2 x^2}+\log \left (x^2+1\right )-\log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^4)/(x^3 + x^5),x]

[Out]

-1/(2*x^2) - Log[x] + Log[1 + x^2]

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Maple [A]  time = 0.005, size = 17, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{x}^{2}}}-\ln \left ( x \right ) +\ln \left ({x}^{2}+1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+1)/(x^5+x^3),x)

[Out]

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

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Maxima [A]  time = 1.65857, size = 22, normalized size = 1.22 \begin{align*} -\frac{1}{2 \, x^{2}} + \log \left (x^{2} + 1\right ) - \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)/(x^5+x^3),x, algorithm="maxima")

[Out]

-1/2/x^2 + log(x^2 + 1) - log(x)

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Fricas [A]  time = 1.22791, size = 66, normalized size = 3.67 \begin{align*} \frac{2 \, x^{2} \log \left (x^{2} + 1\right ) - 2 \, x^{2} \log \left (x\right ) - 1}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)/(x^5+x^3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.097069, size = 15, normalized size = 0.83 \begin{align*} - \log{\left (x \right )} + \log{\left (x^{2} + 1 \right )} - \frac{1}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+1)/(x**5+x**3),x)

[Out]

-log(x) + log(x**2 + 1) - 1/(2*x**2)

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Giac [A]  time = 1.12886, size = 31, normalized size = 1.72 \begin{align*} \frac{x^{2} - 1}{2 \, x^{2}} + \log \left (x^{2} + 1\right ) - \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^5+x^3),x, algorithm="giac")

[Out]

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