3.331 \(\int \frac{\log (c (a+b x)^n)}{d+e x^2} \, dx\)

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

[Out]

(Log[c*(a + b*x)^n]*Log[(b*(Sqrt[-d] - Sqrt[e]*x))/(b*Sqrt[-d] + a*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) - (Log[c*(a
+ b*x)^n]*Log[(b*(Sqrt[-d] + Sqrt[e]*x))/(b*Sqrt[-d] - a*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) - (n*PolyLog[2, -((S
qrt[e]*(a + b*x))/(b*Sqrt[-d] - a*Sqrt[e]))])/(2*Sqrt[-d]*Sqrt[e]) + (n*PolyLog[2, (Sqrt[e]*(a + b*x))/(b*Sqrt
[-d] + a*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e])

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Rubi [A]  time = 0.164198, 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} \[ -\frac{n \text{PolyLog}\left (2,-\frac{\sqrt{e} (a+b x)}{b \sqrt{-d}-a \sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{n \text{PolyLog}\left (2,\frac{\sqrt{e} (a+b x)}{a \sqrt{e}+b \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\log \left (c (a+b x)^n\right ) \log \left (\frac{b \left (\sqrt{-d}-\sqrt{e} x\right )}{a \sqrt{e}+b \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\log \left (c (a+b x)^n\right ) \log \left (\frac{b \left (\sqrt{-d}+\sqrt{e} x\right )}{b \sqrt{-d}-a \sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b*x)^n]/(d + e*x^2),x]

[Out]

(Log[c*(a + b*x)^n]*Log[(b*(Sqrt[-d] - Sqrt[e]*x))/(b*Sqrt[-d] + a*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) - (Log[c*(a
+ b*x)^n]*Log[(b*(Sqrt[-d] + Sqrt[e]*x))/(b*Sqrt[-d] - a*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) - (n*PolyLog[2, -((S
qrt[e]*(a + b*x))/(b*Sqrt[-d] - a*Sqrt[e]))])/(2*Sqrt[-d]*Sqrt[e]) + (n*PolyLog[2, (Sqrt[e]*(a + b*x))/(b*Sqrt
[-d] + a*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e])

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

Mathematica [A]  time = 0.0907437, size = 178, normalized size = 0.78 \[ \frac{-n \text{PolyLog}\left (2,-\frac{\sqrt{e} (a+b x)}{b \sqrt{-d}-a \sqrt{e}}\right )+n \text{PolyLog}\left (2,\frac{\sqrt{e} (a+b x)}{a \sqrt{e}+b \sqrt{-d}}\right )+\log \left (c (a+b x)^n\right ) \left (\log \left (\frac{b \left (\sqrt{-d}-\sqrt{e} x\right )}{a \sqrt{e}+b \sqrt{-d}}\right )-\log \left (\frac{b \left (\sqrt{-d}+\sqrt{e} x\right )}{b \sqrt{-d}-a \sqrt{e}}\right )\right )}{2 \sqrt{-d} \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b*x)^n]/(d + e*x^2),x]

[Out]

(Log[c*(a + b*x)^n]*(Log[(b*(Sqrt[-d] - Sqrt[e]*x))/(b*Sqrt[-d] + a*Sqrt[e])] - Log[(b*(Sqrt[-d] + Sqrt[e]*x))
/(b*Sqrt[-d] - a*Sqrt[e])]) - n*PolyLog[2, -((Sqrt[e]*(a + b*x))/(b*Sqrt[-d] - a*Sqrt[e]))] + n*PolyLog[2, (Sq
rt[e]*(a + b*x))/(b*Sqrt[-d] + a*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e])

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Maple [C]  time = 0.405, size = 419, normalized size = 1.8 \begin{align*}{(\ln \left ( \left ( bx+a \right ) ^{n} \right ) -n\ln \left ( bx+a \right ) )\arctan \left ({\frac{2\, \left ( bx+a \right ) e-2\,ae}{2\,b}{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{n\ln \left ( bx+a \right ) }{2}\ln \left ({ \left ( b\sqrt{-de}- \left ( bx+a \right ) e+ae \right ) \left ( b\sqrt{-de}+ae \right ) ^{-1}} \right ){\frac{1}{\sqrt{-de}}}}-{\frac{n\ln \left ( bx+a \right ) }{2}\ln \left ({ \left ( b\sqrt{-de}+ \left ( bx+a \right ) e-ae \right ) \left ( b\sqrt{-de}-ae \right ) ^{-1}} \right ){\frac{1}{\sqrt{-de}}}}+{\frac{n}{2}{\it dilog} \left ({ \left ( b\sqrt{-de}- \left ( bx+a \right ) e+ae \right ) \left ( b\sqrt{-de}+ae \right ) ^{-1}} \right ){\frac{1}{\sqrt{-de}}}}-{\frac{n}{2}{\it dilog} \left ({ \left ( b\sqrt{-de}+ \left ( bx+a \right ) e-ae \right ) \left ( b\sqrt{-de}-ae \right ) ^{-1}} \right ){\frac{1}{\sqrt{-de}}}}+{{\frac{i}{2}}\pi \,{\it csgn} \left ( i \left ( bx+a \right ) ^{n} \right ) \left ({\it csgn} \left ( ic \left ( bx+a \right ) ^{n} \right ) \right ) ^{2}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{{\frac{i}{2}}\pi \,{\it csgn} \left ( i \left ( bx+a \right ) ^{n} \right ){\it csgn} \left ( ic \left ( bx+a \right ) ^{n} \right ){\it csgn} \left ( ic \right ) \arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( ic \left ( bx+a \right ) ^{n} \right ) \right ) ^{3}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( ic \left ( bx+a \right ) ^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) \arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\ln \left ( c \right ) \arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(ln((b*x+a)^n)-n*ln(b*x+a))/(d*e)^(1/2)*arctan(1/2*(2*(b*x+a)*e-2*a*e)/b/(d*e)^(1/2))+1/2*n*ln(b*x+a)/(-d*e)^(
1/2)*ln((b*(-d*e)^(1/2)-(b*x+a)*e+a*e)/(b*(-d*e)^(1/2)+a*e))-1/2*n*ln(b*x+a)/(-d*e)^(1/2)*ln((b*(-d*e)^(1/2)+(
b*x+a)*e-a*e)/(b*(-d*e)^(1/2)-a*e))+1/2*n/(-d*e)^(1/2)*dilog((b*(-d*e)^(1/2)-(b*x+a)*e+a*e)/(b*(-d*e)^(1/2)+a*
e))-1/2*n/(-d*e)^(1/2)*dilog((b*(-d*e)^(1/2)+(b*x+a)*e-a*e)/(b*(-d*e)^(1/2)-a*e))+1/2*I/(d*e)^(1/2)*arctan(x*e
/(d*e)^(1/2))*Pi*csgn(I*(b*x+a)^n)*csgn(I*c*(b*x+a)^n)^2-1/2*I/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*Pi*csgn(I*(
b*x+a)^n)*csgn(I*c*(b*x+a)^n)*csgn(I*c)-1/2*I/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*Pi*csgn(I*c*(b*x+a)^n)^3+1/2
*I/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*Pi*csgn(I*c*(b*x+a)^n)^2*csgn(I*c)+1/(d*e)^(1/2)*arctan(x*e/(d*e)^(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*(b*x+a)^n)/(e*x^2+d),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 (b x + a\right )}^{n} c\right )}{e x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(log((b*x + a)^n*c)/(e*x^2 + d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(log(c*(a + b*x)**n)/(d + e*x**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log((b*x + a)^n*c)/(e*x^2 + d), x)