3.124 \(\int \frac{x}{\log (c (a+b x^2))} \, dx\)

Optimal. Leaf size=20 \[ \frac{\text{li}\left (c \left (b x^2+a\right )\right )}{2 b c} \]

[Out]

LogIntegral[c*(a + b*x^2)]/(2*b*c)

________________________________________________________________________________________

Rubi [A]  time = 0.0271766, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2454, 2389, 2298} \[ \frac{\text{li}\left (c \left (b x^2+a\right )\right )}{2 b c} \]

Antiderivative was successfully verified.

[In]

Int[x/Log[c*(a + b*x^2)],x]

[Out]

LogIntegral[c*(a + b*x^2)]/(2*b*c)

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2298

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, x]

Rubi steps

\begin{align*} \int \frac{x}{\log \left (c \left (a+b x^2\right )\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\log (c (a+b x))} \, dx,x,x^2\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{\log (c x)} \, dx,x,a+b x^2\right )}{2 b}\\ &=\frac{\text{li}\left (c \left (a+b x^2\right )\right )}{2 b c}\\ \end{align*}

Mathematica [A]  time = 0.0135455, size = 20, normalized size = 1. \[ \frac{\text{li}\left (c \left (b x^2+a\right )\right )}{2 b c} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Log[c*(a + b*x^2)],x]

[Out]

LogIntegral[c*(a + b*x^2)]/(2*b*c)

________________________________________________________________________________________

Maple [A]  time = 0.074, size = 23, normalized size = 1.2 \begin{align*} -{\frac{{\it Ei} \left ( 1,-\ln \left ( c \left ( b{x}^{2}+a \right ) \right ) \right ) }{2\,bc}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/ln(c*(b*x^2+a)),x)

[Out]

-1/2/b/c*Ei(1,-ln(c*(b*x^2+a)))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\log \left ({\left (b x^{2} + a\right )} c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/log(c*(b*x^2+a)),x, algorithm="maxima")

[Out]

integrate(x/log((b*x^2 + a)*c), x)

________________________________________________________________________________________

Fricas [A]  time = 1.91236, size = 53, normalized size = 2.65 \begin{align*} \frac{\logintegral \left (b c x^{2} + a c\right )}{2 \, b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/log(c*(b*x^2+a)),x, algorithm="fricas")

[Out]

1/2*log_integral(b*c*x^2 + a*c)/(b*c)

________________________________________________________________________________________

Sympy [A]  time = 2.79422, size = 27, normalized size = 1.35 \begin{align*} \begin{cases} \frac{x^{2}}{2 \log{\left (a c \right )}} & \text{for}\: b = 0 \\0 & \text{for}\: c = 0 \\\frac{\operatorname{Ei}{\left (\log{\left (a c + b c x^{2} \right )} \right )}}{2 b c} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/ln(c*(b*x**2+a)),x)

[Out]

Piecewise((x**2/(2*log(a*c)), Eq(b, 0)), (0, Eq(c, 0)), (Ei(log(a*c + b*c*x**2))/(2*b*c), True))

________________________________________________________________________________________

Giac [A]  time = 1.31621, size = 26, normalized size = 1.3 \begin{align*} \frac{{\rm Ei}\left (\log \left ({\left (b x^{2} + a\right )} c\right )\right )}{2 \, b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/log(c*(b*x^2+a)),x, algorithm="giac")

[Out]

1/2*Ei(log((b*x^2 + a)*c))/(b*c)