3.4.30 \(\int \frac {\log ^2(c (a+b x)^n)}{d+e x^2} \, dx\) [330]

Optimal. Leaf size=347 \[ \frac {\log ^2\left (c (a+b x)^n\right ) \log \left (\frac {b \left (\sqrt {-d}-\sqrt {e} x\right )}{b \sqrt {-d}+a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\log ^2\left (c (a+b x)^n\right ) \log \left (\frac {b \left (\sqrt {-d}+\sqrt {e} x\right )}{b \sqrt {-d}-a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \log \left (c (a+b x)^n\right ) \text {Li}_2\left (-\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}-a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {n \log \left (c (a+b x)^n\right ) \text {Li}_2\left (\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}+a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {n^2 \text {Li}_3\left (-\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}-a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {n^2 \text {Li}_3\left (\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}+a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}} \]

[Out]

1/2*ln(c*(b*x+a)^n)^2*ln(b*((-d)^(1/2)-x*e^(1/2))/(b*(-d)^(1/2)+a*e^(1/2)))/(-d)^(1/2)/e^(1/2)-1/2*ln(c*(b*x+a
)^n)^2*ln(b*((-d)^(1/2)+x*e^(1/2))/(b*(-d)^(1/2)-a*e^(1/2)))/(-d)^(1/2)/e^(1/2)-n*ln(c*(b*x+a)^n)*polylog(2,-(
b*x+a)*e^(1/2)/(b*(-d)^(1/2)-a*e^(1/2)))/(-d)^(1/2)/e^(1/2)+n*ln(c*(b*x+a)^n)*polylog(2,(b*x+a)*e^(1/2)/(b*(-d
)^(1/2)+a*e^(1/2)))/(-d)^(1/2)/e^(1/2)+n^2*polylog(3,-(b*x+a)*e^(1/2)/(b*(-d)^(1/2)-a*e^(1/2)))/(-d)^(1/2)/e^(
1/2)-n^2*polylog(3,(b*x+a)*e^(1/2)/(b*(-d)^(1/2)+a*e^(1/2)))/(-d)^(1/2)/e^(1/2)

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Rubi [A]
time = 0.22, antiderivative size = 347, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2456, 2443, 2481, 2421, 6724} \begin {gather*} -\frac {n \log \left (c (a+b x)^n\right ) \text {PolyLog}\left (2,-\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}-a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {n \log \left (c (a+b x)^n\right ) \text {PolyLog}\left (2,\frac {\sqrt {e} (a+b x)}{a \sqrt {e}+b \sqrt {-d}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {n^2 \text {PolyLog}\left (3,-\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}-a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {n^2 \text {PolyLog}\left (3,\frac {\sqrt {e} (a+b x)}{a \sqrt {e}+b \sqrt {-d}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {\log ^2\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 ^2\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Log[c*(a + b*x)^n]^2*Log[(b*(Sqrt[-d] - Sqrt[e]*x))/(b*Sqrt[-d] + a*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) - (Log[c*
(a + b*x)^n]^2*Log[(b*(Sqrt[-d] + Sqrt[e]*x))/(b*Sqrt[-d] - a*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) - (n*Log[c*(a +
b*x)^n]*PolyLog[2, -((Sqrt[e]*(a + b*x))/(b*Sqrt[-d] - a*Sqrt[e]))])/(Sqrt[-d]*Sqrt[e]) + (n*Log[c*(a + b*x)^n
]*PolyLog[2, (Sqrt[e]*(a + b*x))/(b*Sqrt[-d] + a*Sqrt[e])])/(Sqrt[-d]*Sqrt[e]) + (n^2*PolyLog[3, -((Sqrt[e]*(a
+ b*x))/(b*Sqrt[-d] - a*Sqrt[e]))])/(Sqrt[-d]*Sqrt[e]) - (n^2*PolyLog[3, (Sqrt[e]*(a + b*x))/(b*Sqrt[-d] + a*
Sqrt[e])])/(Sqrt[-d]*Sqrt[e])

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2443

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

Rule 2456

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 2481

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 6724

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

Rubi steps

\begin {align*} \int \frac {\log ^2\left (c (a+b x)^n\right )}{d+e x^2} \, dx &=\int \left (\frac {\sqrt {-d} \log ^2\left (c (a+b x)^n\right )}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} \log ^2\left (c (a+b x)^n\right )}{2 d \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx\\ &=-\frac {\int \frac {\log ^2\left (c (a+b x)^n\right )}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 \sqrt {-d}}-\frac {\int \frac {\log ^2\left (c (a+b x)^n\right )}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 \sqrt {-d}}\\ &=\frac {\log ^2\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 ^2\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 (c (a+b x)^n\right ) \log \left (\frac {b \left (\sqrt {-d}-\sqrt {e} x\right )}{b \sqrt {-d}+a \sqrt {e}}\right )}{a+b x} \, dx}{\sqrt {-d} \sqrt {e}}+\frac {(b n) \int \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 )}{a+b x} \, dx}{\sqrt {-d} \sqrt {e}}\\ &=\frac {\log ^2\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 ^2\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 {Subst}\left (\int \frac {\log \left (c x^n\right ) \log \left (\frac {b \left (\frac {b \sqrt {-d}+a \sqrt {e}}{b}-\frac {\sqrt {e} x}{b}\right )}{b \sqrt {-d}+a \sqrt {e}}\right )}{x} \, dx,x,a+b x\right )}{\sqrt {-d} \sqrt {e}}+\frac {n \text {Subst}\left (\int \frac {\log \left (c x^n\right ) \log \left (\frac {b \left (\frac {b \sqrt {-d}-a \sqrt {e}}{b}+\frac {\sqrt {e} x}{b}\right )}{b \sqrt {-d}-a \sqrt {e}}\right )}{x} \, dx,x,a+b x\right )}{\sqrt {-d} \sqrt {e}}\\ &=\frac {\log ^2\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 ^2\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 \log \left (c (a+b x)^n\right ) \text {Li}_2\left (-\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}-a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {n \log \left (c (a+b x)^n\right ) \text {Li}_2\left (\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}+a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {n^2 \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {\sqrt {e} x}{b \sqrt {-d}-a \sqrt {e}}\right )}{x} \, dx,x,a+b x\right )}{\sqrt {-d} \sqrt {e}}-\frac {n^2 \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {\sqrt {e} x}{b \sqrt {-d}+a \sqrt {e}}\right )}{x} \, dx,x,a+b x\right )}{\sqrt {-d} \sqrt {e}}\\ &=\frac {\log ^2\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 ^2\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 \log \left (c (a+b x)^n\right ) \text {Li}_2\left (-\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}-a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {n \log \left (c (a+b x)^n\right ) \text {Li}_2\left (\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}+a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {n^2 \text {Li}_3\left (-\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}-a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {n^2 \text {Li}_3\left (\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}+a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.08, size = 488, normalized size = 1.41 \begin {gather*} \frac {2 n^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log ^2(a+b x)-4 n \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log (a+b x) \log \left (c (a+b x)^n\right )+2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log ^2\left (c (a+b x)^n\right )-i n^2 \log ^2(a+b x) \log \left (1-\frac {\sqrt {e} (a+b x)}{-i b \sqrt {d}+a \sqrt {e}}\right )+2 i n \log (a+b x) \log \left (c (a+b x)^n\right ) \log \left (1-\frac {\sqrt {e} (a+b x)}{-i b \sqrt {d}+a \sqrt {e}}\right )+i n^2 \log ^2(a+b x) \log \left (1-\frac {\sqrt {e} (a+b x)}{i b \sqrt {d}+a \sqrt {e}}\right )-2 i n \log (a+b x) \log \left (c (a+b x)^n\right ) \log \left (1-\frac {\sqrt {e} (a+b x)}{i b \sqrt {d}+a \sqrt {e}}\right )+2 i n \log \left (c (a+b x)^n\right ) \text {Li}_2\left (\frac {\sqrt {e} (a+b x)}{-i b \sqrt {d}+a \sqrt {e}}\right )-2 i n \log \left (c (a+b x)^n\right ) \text {Li}_2\left (\frac {\sqrt {e} (a+b x)}{i b \sqrt {d}+a \sqrt {e}}\right )-2 i n^2 \text {Li}_3\left (\frac {\sqrt {e} (a+b x)}{-i b \sqrt {d}+a \sqrt {e}}\right )+2 i n^2 \text {Li}_3\left (\frac {\sqrt {e} (a+b x)}{i b \sqrt {d}+a \sqrt {e}}\right )}{2 \sqrt {d} \sqrt {e}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*n^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[a + b*x]^2 - 4*n*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[a + b*x]*Log[c*(a + b*
x)^n] + 2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(a + b*x)^n]^2 - I*n^2*Log[a + b*x]^2*Log[1 - (Sqrt[e]*(a + b*x))/
((-I)*b*Sqrt[d] + a*Sqrt[e])] + (2*I)*n*Log[a + b*x]*Log[c*(a + b*x)^n]*Log[1 - (Sqrt[e]*(a + b*x))/((-I)*b*Sq
rt[d] + a*Sqrt[e])] + I*n^2*Log[a + b*x]^2*Log[1 - (Sqrt[e]*(a + b*x))/(I*b*Sqrt[d] + a*Sqrt[e])] - (2*I)*n*Lo
g[a + b*x]*Log[c*(a + b*x)^n]*Log[1 - (Sqrt[e]*(a + b*x))/(I*b*Sqrt[d] + a*Sqrt[e])] + (2*I)*n*Log[c*(a + b*x)
^n]*PolyLog[2, (Sqrt[e]*(a + b*x))/((-I)*b*Sqrt[d] + a*Sqrt[e])] - (2*I)*n*Log[c*(a + b*x)^n]*PolyLog[2, (Sqrt
[e]*(a + b*x))/(I*b*Sqrt[d] + a*Sqrt[e])] - (2*I)*n^2*PolyLog[3, (Sqrt[e]*(a + b*x))/((-I)*b*Sqrt[d] + a*Sqrt[
e])] + (2*I)*n^2*PolyLog[3, (Sqrt[e]*(a + b*x))/(I*b*Sqrt[d] + a*Sqrt[e])])/(2*Sqrt[d]*Sqrt[e])

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Maple [F]
time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (b x +a \right )^{n}\right )^{2}}{e \,x^{2}+d}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (c \left (a + b x\right )^{n} \right )}^{2}}{d + e x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\ln \left (c\,{\left (a+b\,x\right )}^n\right )}^2}{e\,x^2+d} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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