3.341 \(\int \frac{\log (c (d+e x^2)^p)}{x (f+g x^2)} \, dx\)

Optimal. Leaf size=119 \[ -\frac{p \text{PolyLog}\left (2,-\frac{g \left (d+e x^2\right )}{e f-d g}\right )}{2 f}+\frac{p \text{PolyLog}\left (2,\frac{e x^2}{d}+1\right )}{2 f}-\frac{\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac{e \left (f+g x^2\right )}{e f-d g}\right )}{2 f}+\frac{\log \left (-\frac{e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f} \]

[Out]

(Log[-((e*x^2)/d)]*Log[c*(d + e*x^2)^p])/(2*f) - (Log[c*(d + e*x^2)^p]*Log[(e*(f + g*x^2))/(e*f - d*g)])/(2*f)
- (p*PolyLog[2, -((g*(d + e*x^2))/(e*f - d*g))])/(2*f) + (p*PolyLog[2, 1 + (e*x^2)/d])/(2*f)

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Rubi [A]  time = 0.206326, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {2475, 36, 29, 31, 2416, 2394, 2315, 2393, 2391} \[ -\frac{p \text{PolyLog}\left (2,-\frac{g \left (d+e x^2\right )}{e f-d g}\right )}{2 f}+\frac{p \text{PolyLog}\left (2,\frac{e x^2}{d}+1\right )}{2 f}-\frac{\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac{e \left (f+g x^2\right )}{e f-d g}\right )}{2 f}+\frac{\log \left (-\frac{e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*x^2)^p]/(x*(f + g*x^2)),x]

[Out]

(Log[-((e*x^2)/d)]*Log[c*(d + e*x^2)^p])/(2*f) - (Log[c*(d + e*x^2)^p]*Log[(e*(f + g*x^2))/(e*f - d*g)])/(2*f)
- (p*PolyLog[2, -((g*(d + e*x^2))/(e*f - d*g))])/(2*f) + (p*PolyLog[2, 1 + (e*x^2)/d])/(2*f)

Rule 2475

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q,
x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && IntegerQ[r] && IntegerQ[s/n] && Intege
rQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{x \left (f+g x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{x (f+g x)} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{\log \left (c (d+e x)^p\right )}{f x}-\frac{g \log \left (c (d+e x)^p\right )}{f (f+g x)}\right ) \, dx,x,x^2\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^2\right )}{2 f}-\frac{g \operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{f+g x} \, dx,x,x^2\right )}{2 f}\\ &=\frac{\log \left (-\frac{e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f}-\frac{\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac{e \left (f+g x^2\right )}{e f-d g}\right )}{2 f}-\frac{(e p) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx,x,x^2\right )}{2 f}+\frac{(e p) \operatorname{Subst}\left (\int \frac{\log \left (\frac{e (f+g x)}{e f-d g}\right )}{d+e x} \, dx,x,x^2\right )}{2 f}\\ &=\frac{\log \left (-\frac{e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f}-\frac{\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac{e \left (f+g x^2\right )}{e f-d g}\right )}{2 f}+\frac{p \text{Li}_2\left (1+\frac{e x^2}{d}\right )}{2 f}+\frac{p \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{g x}{e f-d g}\right )}{x} \, dx,x,d+e x^2\right )}{2 f}\\ &=\frac{\log \left (-\frac{e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f}-\frac{\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac{e \left (f+g x^2\right )}{e f-d g}\right )}{2 f}-\frac{p \text{Li}_2\left (-\frac{g \left (d+e x^2\right )}{e f-d g}\right )}{2 f}+\frac{p \text{Li}_2\left (1+\frac{e x^2}{d}\right )}{2 f}\\ \end{align*}

Mathematica [A]  time = 0.0356858, size = 92, normalized size = 0.77 \[ \frac{-p \text{PolyLog}\left (2,\frac{g \left (d+e x^2\right )}{d g-e f}\right )+p \text{PolyLog}\left (2,\frac{e x^2}{d}+1\right )+\log \left (c \left (d+e x^2\right )^p\right ) \left (\log \left (-\frac{e x^2}{d}\right )-\log \left (\frac{e \left (f+g x^2\right )}{e f-d g}\right )\right )}{2 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d + e*x^2)^p]/(x*(f + g*x^2)),x]

[Out]

(Log[c*(d + e*x^2)^p]*(Log[-((e*x^2)/d)] - Log[(e*(f + g*x^2))/(e*f - d*g)]) - p*PolyLog[2, (g*(d + e*x^2))/(-
(e*f) + d*g)] + p*PolyLog[2, 1 + (e*x^2)/d])/(2*f)

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Maple [C]  time = 0.665, size = 732, normalized size = 6.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(e*x^2+d)^p)/x/(g*x^2+f),x)

[Out]

-1/2*ln((e*x^2+d)^p)/f*ln(g*x^2+f)+ln((e*x^2+d)^p)/f*ln(x)-p/f*ln(x)*ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-p/f*
ln(x)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-p/f*dilog((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-p/f*dilog((e*x+(-d*e)^(1
/2))/(-d*e)^(1/2))+1/2*p/f*sum(ln(x-_alpha)*ln(g*x^2+f)-ln(x-_alpha)*(ln((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d*g+
e*f,index=1)-x+_alpha)/RootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,index=1))+ln((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d
*g+e*f,index=2)-x+_alpha)/RootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,index=2)))-dilog((RootOf(_Z^2*e*g+2*_Z*_alph
a*e*g-d*g+e*f,index=1)-x+_alpha)/RootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,index=1))-dilog((RootOf(_Z^2*e*g+2*_Z
*_alpha*e*g-d*g+e*f,index=2)-x+_alpha)/RootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,index=2)),_alpha=RootOf(_Z^2*e+
d))+1/2*I*Pi*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)/f*ln(x)-1/2*I*Pi*csgn(I*c*(e*x^2+d)^p)^3/f*ln(x)+1/2*I*Pi*csgn(
I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2/f*ln(x)+1/4*I*Pi*csgn(I*c*(e*x^2+d)^p)^3/f*ln(g*x^2+f)-1/4*I*Pi*csgn(I*
(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2/f*ln(g*x^2+f)-1/4*I*Pi*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)/f*ln(g*x^2+f)-1/
2*I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)/f*ln(x)+1/4*I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^
2+d)^p)*csgn(I*c)/f*ln(g*x^2+f)-1/2*ln(c)/f*ln(g*x^2+f)+1/f*ln(c)*ln(x)

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Maxima [A]  time = 1.72022, size = 189, normalized size = 1.59 \begin{align*} -\frac{1}{2} \, e p{\left (\frac{2 \, \log \left (\frac{e x^{2}}{d} + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-\frac{e x^{2}}{d}\right )}{e f} - \frac{\log \left (g x^{2} + f\right ) \log \left (-\frac{e g x^{2} + e f}{e f - d g} + 1\right ) +{\rm Li}_2\left (\frac{e g x^{2} + e f}{e f - d g}\right )}{e f}\right )} - \frac{1}{2} \,{\left (\frac{\log \left (g x^{2} + f\right )}{f} - \frac{\log \left (x^{2}\right )}{f}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(e*x^2+d)^p)/x/(g*x^2+f),x, algorithm="maxima")

[Out]

-1/2*e*p*((2*log(e*x^2/d + 1)*log(x) + dilog(-e*x^2/d))/(e*f) - (log(g*x^2 + f)*log(-(e*g*x^2 + e*f)/(e*f - d*
g) + 1) + dilog((e*g*x^2 + e*f)/(e*f - d*g)))/(e*f)) - 1/2*(log(g*x^2 + f)/f - log(x^2)/f)*log((e*x^2 + d)^p*c
)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{3} + f x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(e*x^2+d)^p)/x/(g*x^2+f),x, algorithm="fricas")

[Out]

integral(log((e*x^2 + d)^p*c)/(g*x^3 + f*x), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(e*x**2+d)**p)/x/(g*x**2+f),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(e*x^2+d)^p)/x/(g*x^2+f),x, algorithm="giac")

[Out]

integrate(log((e*x^2 + d)^p*c)/((g*x^2 + f)*x), x)