3.128 \(\int \frac{x}{\log ^3(c (a+b x^2))} \, dx\)

Optimal. Leaf size=73 \[ \frac{\text{li}\left (c \left (b x^2+a\right )\right )}{4 b c}-\frac{a+b x^2}{4 b \log ^2\left (c \left (a+b x^2\right )\right )}-\frac{a+b x^2}{4 b \log \left (c \left (a+b x^2\right )\right )} \]

[Out]

-(a + b*x^2)/(4*b*Log[c*(a + b*x^2)]^2) - (a + b*x^2)/(4*b*Log[c*(a + b*x^2)]) + LogIntegral[c*(a + b*x^2)]/(4
*b*c)

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Rubi [A]  time = 0.0592836, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2454, 2389, 2297, 2298} \[ \frac{\text{li}\left (c \left (b x^2+a\right )\right )}{4 b c}-\frac{a+b x^2}{4 b \log ^2\left (c \left (a+b x^2\right )\right )}-\frac{a+b x^2}{4 b \log \left (c \left (a+b x^2\right )\right )} \]

Antiderivative was successfully verified.

[In]

Int[x/Log[c*(a + b*x^2)]^3,x]

[Out]

-(a + b*x^2)/(4*b*Log[c*(a + b*x^2)]^2) - (a + b*x^2)/(4*b*Log[c*(a + b*x^2)]) + LogIntegral[c*(a + b*x^2)]/(4
*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 2297

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

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

Mathematica [A]  time = 0.0231036, size = 55, normalized size = 0.75 \[ \frac{\frac{\text{li}\left (c \left (b x^2+a\right )\right )}{c}-\frac{\left (a+b x^2\right ) \left (\log \left (c \left (a+b x^2\right )\right )+1\right )}{\log ^2\left (c \left (a+b x^2\right )\right )}}{4 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Log[c*(a + b*x^2)]^3,x]

[Out]

(-(((a + b*x^2)*(1 + Log[c*(a + b*x^2)]))/Log[c*(a + b*x^2)]^2) + LogIntegral[c*(a + b*x^2)]/c)/(4*b)

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Maple [A]  time = 0.068, size = 94, normalized size = 1.3 \begin{align*} -{\frac{{x}^{2}}{4\, \left ( \ln \left ( c \left ( b{x}^{2}+a \right ) \right ) \right ) ^{2}}}-{\frac{a}{4\,b \left ( \ln \left ( c \left ( b{x}^{2}+a \right ) \right ) \right ) ^{2}}}-{\frac{{x}^{2}}{4\,\ln \left ( c \left ( b{x}^{2}+a \right ) \right ) }}-{\frac{a}{4\,b\ln \left ( c \left ( b{x}^{2}+a \right ) \right ) }}-{\frac{{\it Ei} \left ( 1,-\ln \left ( c \left ( b{x}^{2}+a \right ) \right ) \right ) }{4\,bc}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/ln(c*(b*x^2+a))^3,x)

[Out]

-1/4/ln(c*(b*x^2+a))^2*x^2-1/4/b/ln(c*(b*x^2+a))^2*a-1/4/ln(c*(b*x^2+a))*x^2-1/4/b/ln(c*(b*x^2+a))*a-1/4/b/c*E
i(1,-ln(c*(b*x^2+a)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{b x^{2}{\left (\log \left (c\right ) + 1\right )} + a{\left (\log \left (c\right ) + 1\right )} +{\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right )}{4 \,{\left (b \log \left (b x^{2} + a\right )^{2} + 2 \, b \log \left (b x^{2} + a\right ) \log \left (c\right ) + b \log \left (c\right )^{2}\right )}} + \int \frac{x}{2 \,{\left (\log \left (b x^{2} + a\right ) + \log \left (c\right )\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/log(c*(b*x^2+a))^3,x, algorithm="maxima")

[Out]

-1/4*(b*x^2*(log(c) + 1) + a*(log(c) + 1) + (b*x^2 + a)*log(b*x^2 + a))/(b*log(b*x^2 + a)^2 + 2*b*log(b*x^2 +
a)*log(c) + b*log(c)^2) + integrate(1/2*x/(log(b*x^2 + a) + log(c)), x)

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Fricas [A]  time = 1.90631, size = 185, normalized size = 2.53 \begin{align*} -\frac{b c x^{2} - \log \left (b c x^{2} + a c\right )^{2} \logintegral \left (b c x^{2} + a c\right ) + a c +{\left (b c x^{2} + a c\right )} \log \left (b c x^{2} + a c\right )}{4 \, b c \log \left (b c x^{2} + a c\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/log(c*(b*x^2+a))^3,x, algorithm="fricas")

[Out]

-1/4*(b*c*x^2 - log(b*c*x^2 + a*c)^2*log_integral(b*c*x^2 + a*c) + a*c + (b*c*x^2 + a*c)*log(b*c*x^2 + a*c))/(
b*c*log(b*c*x^2 + a*c)^2)

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Sympy [A]  time = 2.9801, size = 70, normalized size = 0.96 \begin{align*} \frac{\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}}{2} + \frac{- a - b x^{2} + \left (- a - b x^{2}\right ) \log{\left (c \left (a + b x^{2}\right ) \right )}}{4 b \log{\left (c \left (a + b x^{2}\right ) \right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/ln(c*(b*x**2+a))**3,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))/2 + (-a - b*x
**2 + (-a - b*x**2)*log(c*(a + b*x**2)))/(4*b*log(c*(a + b*x**2))**2)

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Giac [A]  time = 1.28315, size = 92, normalized size = 1.26 \begin{align*} -\frac{\frac{b c x^{2} + a c}{\log \left ({\left (b x^{2} + a\right )} c\right )} + \frac{b c x^{2} + a c}{\log \left ({\left (b x^{2} + a\right )} c\right )^{2}} -{\rm Ei}\left (\log \left ({\left (b x^{2} + a\right )} c\right )\right )}{4 \, b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/log(c*(b*x^2+a))^3,x, algorithm="giac")

[Out]

-1/4*((b*c*x^2 + a*c)/log((b*x^2 + a)*c) + (b*c*x^2 + a*c)/log((b*x^2 + a)*c)^2 - Ei(log((b*x^2 + a)*c)))/(b*c
)