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

Optimal. Leaf size=113 \[ -\frac{1}{2} m \text{PolyLog}\left (2,-\frac{f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} b m n \text{PolyLog}\left (3,-\frac{f x^2}{e}\right )+\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 b n}-\frac{m \log \left (\frac{f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n} \]

[Out]

((a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m])/(2*b*n) - (m*(a + b*Log[c*x^n])^2*Log[1 + (f*x^2)/e])/(2*b*n) - (m
*(a + b*Log[c*x^n])*PolyLog[2, -((f*x^2)/e)])/2 + (b*m*n*PolyLog[3, -((f*x^2)/e)])/4

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Rubi [A]  time = 0.125981, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2375, 2337, 2374, 6589} \[ -\frac{1}{2} m \text{PolyLog}\left (2,-\frac{f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} b m n \text{PolyLog}\left (3,-\frac{f x^2}{e}\right )+\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 b n}-\frac{m \log \left (\frac{f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m])/(2*b*n) - (m*(a + b*Log[c*x^n])^2*Log[1 + (f*x^2)/e])/(2*b*n) - (m
*(a + b*Log[c*x^n])*PolyLog[2, -((f*x^2)/e)])/2 + (b*m*n*PolyLog[3, -((f*x^2)/e)])/4

Rule 2375

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

Rule 2337

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^(r_)), x_Symbol] :> Si
mp[(f^m*Log[1 + (e*x^r)/d]*(a + b*Log[c*x^n])^p)/(e*r), x] - Dist[(b*f^m*n*p)/(e*r), Int[(Log[1 + (e*x^r)/d]*(
a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] &
& (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(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*Log
[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 6589

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{\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x} \, dx &=\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 b n}-\frac{(f m) \int \frac{x \left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2} \, dx}{b n}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 b n}-\frac{m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x^2}{e}\right )}{2 b n}+m \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x^2}{e}\right )}{x} \, dx\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 b n}-\frac{m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x^2}{e}\right )}{2 b n}-\frac{1}{2} m \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x^2}{e}\right )+\frac{1}{2} (b m n) \int \frac{\text{Li}_2\left (-\frac{f x^2}{e}\right )}{x} \, dx\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 b n}-\frac{m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x^2}{e}\right )}{2 b n}-\frac{1}{2} m \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x^2}{e}\right )+\frac{1}{4} b m n \text{Li}_3\left (-\frac{f x^2}{e}\right )\\ \end{align*}

Mathematica [C]  time = 0.0868114, size = 297, normalized size = 2.63 \[ \frac{1}{2} \left (a m \text{PolyLog}\left (2,\frac{f x^2}{e}+1\right )-2 b m \log \left (c x^n\right ) \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )-2 b m \log \left (c x^n\right ) \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )+2 b m n \text{PolyLog}\left (3,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )+2 b m n \text{PolyLog}\left (3,\frac{i \sqrt{f} x}{\sqrt{e}}\right )+a \log \left (-\frac{f x^2}{e}\right ) \log \left (d \left (e+f x^2\right )^m\right )+2 b \log (x) \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-2 b m \log (x) \log \left (c x^n\right ) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )-2 b m \log (x) \log \left (c x^n\right ) \log \left (1+\frac{i \sqrt{f} x}{\sqrt{e}}\right )-b n \log ^2(x) \log \left (d \left (e+f x^2\right )^m\right )+b m n \log ^2(x) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )+b m n \log ^2(x) \log \left (1+\frac{i \sqrt{f} x}{\sqrt{e}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*m*n*Log[x]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - 2*b*m*Log[x]*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + b*m*
n*Log[x]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - 2*b*m*Log[x]*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - b*n*Log[x
]^2*Log[d*(e + f*x^2)^m] + a*Log[-((f*x^2)/e)]*Log[d*(e + f*x^2)^m] + 2*b*Log[x]*Log[c*x^n]*Log[d*(e + f*x^2)^
m] - 2*b*m*Log[c*x^n]*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]] - 2*b*m*Log[c*x^n]*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]
] + a*m*PolyLog[2, 1 + (f*x^2)/e] + 2*b*m*n*PolyLog[3, ((-I)*Sqrt[f]*x)/Sqrt[e]] + 2*b*m*n*PolyLog[3, (I*Sqrt[
f]*x)/Sqrt[e]])/2

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Maple [F]  time = 0.792, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \ln \left ( d \left ( f{x}^{2}+e \right ) ^{m} \right ) }{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \,{\left (b n \log \left (x\right )^{2} - 2 \, b \log \left (x\right ) \log \left (x^{n}\right ) - 2 \,{\left (b \log \left (c\right ) + a\right )} \log \left (x\right )\right )} \log \left ({\left (f x^{2} + e\right )}^{m}\right ) - \int -\frac{b f m n x^{2} \log \left (x\right )^{2} + b e \log \left (c\right ) \log \left (d\right ) - 2 \,{\left (b f m \log \left (c\right ) + a f m\right )} x^{2} \log \left (x\right ) +{\left (b f \log \left (c\right ) \log \left (d\right ) + a f \log \left (d\right )\right )} x^{2} + a e \log \left (d\right ) -{\left (2 \, b f m x^{2} \log \left (x\right ) - b f x^{2} \log \left (d\right ) - b e \log \left (d\right )\right )} \log \left (x^{n}\right )}{f x^{3} + e x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b*n*log(x)^2 - 2*b*log(x)*log(x^n) - 2*(b*log(c) + a)*log(x))*log((f*x^2 + e)^m) - integrate(-(b*f*m*n*x
^2*log(x)^2 + b*e*log(c)*log(d) - 2*(b*f*m*log(c) + a*f*m)*x^2*log(x) + (b*f*log(c)*log(d) + a*f*log(d))*x^2 +
a*e*log(d) - (2*b*f*m*x^2*log(x) - b*f*x^2*log(d) - b*e*log(d))*log(x^n))/(f*x^3 + e*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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