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

Optimal. Leaf size=229 \[ -\frac{p \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{p \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{\log \left (c (d+e x)^p\right ) \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right )}{2 \sqrt{-f} \sqrt{g}}-\frac{\log \left (c (d+e x)^p\right ) \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 \sqrt{-f} \sqrt{g}} \]

[Out]

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

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Rubi [A]  time = 0.228681, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2409, 2394, 2393, 2391} \[ -\frac{p \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{p \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{\log \left (c (d+e x)^p\right ) \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right )}{2 \sqrt{-f} \sqrt{g}}-\frac{\log \left (c (d+e x)^p\right ) \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 \sqrt{-f} \sqrt{g}} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*x)^p]/(f + g*x^2),x]

[Out]

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

Rule 2409

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))

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

Mathematica [A]  time = 0.110755, size = 178, normalized size = 0.78 \[ \frac{-p \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )+p \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )+\log \left (c (d+e x)^p\right ) \left (\log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right )-\log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )\right )}{2 \sqrt{-f} \sqrt{g}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.533, size = 419, normalized size = 1.8 \begin{align*}{(\ln \left ( \left ( ex+d \right ) ^{p} \right ) -p\ln \left ( ex+d \right ) )\arctan \left ({\frac{2\,g \left ( ex+d \right ) -2\,dg}{2\,e}{\frac{1}{\sqrt{fg}}}} \right ){\frac{1}{\sqrt{fg}}}}+{\frac{p\ln \left ( ex+d \right ) }{2}\ln \left ({ \left ( e\sqrt{-fg}-g \left ( ex+d \right ) +dg \right ) \left ( e\sqrt{-fg}+dg \right ) ^{-1}} \right ){\frac{1}{\sqrt{-fg}}}}-{\frac{p\ln \left ( ex+d \right ) }{2}\ln \left ({ \left ( e\sqrt{-fg}+g \left ( ex+d \right ) -dg \right ) \left ( e\sqrt{-fg}-dg \right ) ^{-1}} \right ){\frac{1}{\sqrt{-fg}}}}+{\frac{p}{2}{\it dilog} \left ({ \left ( e\sqrt{-fg}-g \left ( ex+d \right ) +dg \right ) \left ( e\sqrt{-fg}+dg \right ) ^{-1}} \right ){\frac{1}{\sqrt{-fg}}}}-{\frac{p}{2}{\it dilog} \left ({ \left ( e\sqrt{-fg}+g \left ( ex+d \right ) -dg \right ) \left ( e\sqrt{-fg}-dg \right ) ^{-1}} \right ){\frac{1}{\sqrt{-fg}}}}+{{\frac{i}{2}}\pi \,{\it csgn} \left ( i \left ( ex+d \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( ex+d \right ) ^{p} \right ) \right ) ^{2}\arctan \left ({gx{\frac{1}{\sqrt{fg}}}} \right ){\frac{1}{\sqrt{fg}}}}-{{\frac{i}{2}}\pi \,{\it csgn} \left ( i \left ( ex+d \right ) ^{p} \right ){\it csgn} \left ( ic \left ( ex+d \right ) ^{p} \right ){\it csgn} \left ( ic \right ) \arctan \left ({gx{\frac{1}{\sqrt{fg}}}} \right ){\frac{1}{\sqrt{fg}}}}-{{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( ic \left ( ex+d \right ) ^{p} \right ) \right ) ^{3}\arctan \left ({gx{\frac{1}{\sqrt{fg}}}} \right ){\frac{1}{\sqrt{fg}}}}+{{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( ic \left ( ex+d \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) \arctan \left ({gx{\frac{1}{\sqrt{fg}}}} \right ){\frac{1}{\sqrt{fg}}}}+{\ln \left ( c \right ) \arctan \left ({gx{\frac{1}{\sqrt{fg}}}} \right ){\frac{1}{\sqrt{fg}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(ln((e*x+d)^p)-p*ln(e*x+d))/(f*g)^(1/2)*arctan(1/2*(2*g*(e*x+d)-2*d*g)/e/(f*g)^(1/2))+1/2*p*ln(e*x+d)/(-f*g)^(
1/2)*ln((e*(-f*g)^(1/2)-g*(e*x+d)+d*g)/(e*(-f*g)^(1/2)+d*g))-1/2*p*ln(e*x+d)/(-f*g)^(1/2)*ln((e*(-f*g)^(1/2)+g
*(e*x+d)-d*g)/(e*(-f*g)^(1/2)-d*g))+1/2*p/(-f*g)^(1/2)*dilog((e*(-f*g)^(1/2)-g*(e*x+d)+d*g)/(e*(-f*g)^(1/2)+d*
g))-1/2*p/(-f*g)^(1/2)*dilog((e*(-f*g)^(1/2)+g*(e*x+d)-d*g)/(e*(-f*g)^(1/2)-d*g))+1/2*I/(f*g)^(1/2)*arctan(x*g
/(f*g)^(1/2))*Pi*csgn(I*(e*x+d)^p)*csgn(I*c*(e*x+d)^p)^2-1/2*I/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))*Pi*csgn(I*(
e*x+d)^p)*csgn(I*c*(e*x+d)^p)*csgn(I*c)-1/2*I/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))*Pi*csgn(I*c*(e*x+d)^p)^3+1/2
*I/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))*Pi*csgn(I*c*(e*x+d)^p)^2*csgn(I*c)+1/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2)
)*ln(c)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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