3.88.24 \(\int \frac {1+x^2}{-2 x+2 x^3} \, dx\)

Optimal. Leaf size=16 \[ \frac {1}{2} \log \left (\frac {1-x^2}{x}\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.19, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1593, 446, 72} \begin {gather*} \frac {1}{2} \log \left (1-x^2\right )-\frac {\log (x)}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x^2)/(-2*x + 2*x^3),x]

[Out]

-1/2*Log[x] + Log[1 - x^2]/2

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

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]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 19, normalized size = 1.19 \begin {gather*} -\frac {\log (x)}{2}+\frac {1}{2} \log \left (1-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^2)/(-2*x + 2*x^3),x]

[Out]

-1/2*Log[x] + Log[1 - x^2]/2

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fricas [A]  time = 0.52, size = 13, normalized size = 0.81 \begin {gather*} \frac {1}{2} \, \log \left (x^{2} - 1\right ) - \frac {1}{2} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(x^2 - 1) - 1/2*log(x)

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giac [A]  time = 0.17, size = 16, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \, \log \left (x^{2}\right ) + \frac {1}{2} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*log(x^2) + 1/2*log(abs(x^2 - 1))

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maple [A]  time = 0.42, size = 14, normalized size = 0.88

method	result	size
risch	\(-\frac {\ln \relax (x )}{2}+\frac {\ln \left (x^{2}-1\right )}{2}\)	\(14\)
default	\(-\frac {\ln \relax (x )}{2}+\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (x +1\right )}{2}\)	\(18\)
norman	\(-\frac {\ln \relax (x )}{2}+\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (x +1\right )}{2}\)	\(18\)
meijerg	\(-\frac {\ln \relax (x )}{2}-\frac {i \pi }{4}+\frac {\ln \left (-x^{2}+1\right )}{2}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+1)/(2*x^3-2*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.35, size = 17, normalized size = 1.06 \begin {gather*} \frac {1}{2} \, \log \left (x + 1\right ) + \frac {1}{2} \, \log \left (x - 1\right ) - \frac {1}{2} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.07, size = 13, normalized size = 0.81 \begin {gather*} \frac {\ln \left (x^2-1\right )}{2}-\frac {\ln \relax (x)}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2 + 1)/(2*x - 2*x^3),x)

[Out]

log(x^2 - 1)/2 - log(x)/2

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sympy [A]  time = 0.09, size = 12, normalized size = 0.75 \begin {gather*} - \frac {\log {\relax (x )}}{2} + \frac {\log {\left (x^{2} - 1 \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x)/2 + log(x**2 - 1)/2

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