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

Optimal. Leaf size=179 \[ -\frac{i b \sqrt{f} m n \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e}}+\frac{i b \sqrt{f} m n \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e}}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x}+\frac{2 \sqrt{f} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{e}}-\frac{b n \log \left (d \left (e+f x^2\right )^m\right )}{x}+\frac{2 b \sqrt{f} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e}} \]

[Out]

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

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Rubi [A]  time = 0.13285, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {2455, 205, 2376, 4848, 2391} \[ -\frac{i b \sqrt{f} m n \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e}}+\frac{i b \sqrt{f} m n \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e}}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x}+\frac{2 \sqrt{f} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{e}}-\frac{b n \log \left (d \left (e+f x^2\right )^m\right )}{x}+\frac{2 b \sqrt{f} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2455

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

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2376

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)^r], x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist
[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] &
& RationalQ[q])) && NeQ[q, -1]

Rule 4848

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[(I*b)/2, Int[Log[1 - I*c*x
]/x, x], x] - Dist[(I*b)/2, Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

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

Mathematica [A]  time = 0.0832571, size = 305, normalized size = 1.7 \[ \frac{-i b \sqrt{f} m n x \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )+i b \sqrt{f} m n x \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )-a \sqrt{e} \log \left (d \left (e+f x^2\right )^m\right )+2 a \sqrt{f} m x \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )-b \sqrt{e} \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+2 b \sqrt{f} m x \log \left (c x^n\right ) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )-b \sqrt{e} n \log \left (d \left (e+f x^2\right )^m\right )+i b \sqrt{f} m n x \log (x) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )-i b \sqrt{f} m n x \log (x) \log \left (1+\frac{i \sqrt{f} x}{\sqrt{e}}\right )+2 b \sqrt{f} m n x \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )-2 b \sqrt{f} m n x \log (x) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e} x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.167, size = 1972, normalized size = 11. \begin{align*} \text{result too large to display} \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^2,x)

[Out]

-1/2*I/x*Pi*ln(d)*b*csgn(I*c)*csgn(I*c*x^n)^2-1/2*I/x*Pi*ln(d)*b*csgn(I*x^n)*csgn(I*c*x^n)^2-I*m*f/(e*f)^(1/2)
*arctan(x*f/(e*f)^(1/2))*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/x*ln(d)*a+2*m*f/(e*f)^(1/2)*arctan(x*f/(e*
f)^(1/2))*b*n+m*f*b*n/(-e*f)^(1/2)*dilog((-f*x+(-e*f)^(1/2))/(-e*f)^(1/2))+1/2*I*Pi*csgn(I*d*(f*x^2+e)^m)^3*b/
x*ln(x^n)-1/4*Pi^2*csgn(I*d*(f*x^2+e)^m)^3/x*b*csgn(I*c)*csgn(I*c*x^n)^2-1/4*Pi^2*csgn(I*d*(f*x^2+e)^m)^3/x*b*
csgn(I*x^n)*csgn(I*c*x^n)^2-1/4*Pi^2*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2/x*b*csgn(I*c*x^n)^3-1/4*Pi^2*csgn(I*(f*
x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2/x*b*csgn(I*c*x^n)^3-1/2*I*Pi*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2/x*a-1/2*I*Pi*
csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2/x*a+1/2*I*Pi*csgn(I*d*(f*x^2+e)^m)^3/x*b*ln(c)+2*m*f*b/(e*f)^(1/2)
*arctan(x*f/(e*f)^(1/2))*ln(x^n)+1/2*I*Pi*csgn(I*d*(f*x^2+e)^m)^3*b*n/x+(-b/x*ln(x^n)-1/2*(-I*b*Pi*csgn(I*c)*c
sgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c*x
^n)^3+2*b*ln(c)+2*b*n+2*a)/x)*ln((f*x^2+e)^m)-2*m*f*b/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*n*ln(x)+m*f*b*n/(-e*
f)^(1/2)*ln(x)*ln((-f*x+(-e*f)^(1/2))/(-e*f)^(1/2))-m*f*b*n/(-e*f)^(1/2)*ln(x)*ln((f*x+(-e*f)^(1/2))/(-e*f)^(1
/2))-m*f*b*n/(-e*f)^(1/2)*dilog((f*x+(-e*f)^(1/2))/(-e*f)^(1/2))+2*m*f/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*b*l
n(c)-ln(d)*b/x*ln(x^n)-1/x*ln(c)*ln(d)*b-1/x*ln(d)*b*n+I*m*f/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*b*Pi*csgn(I*x
^n)*csgn(I*c*x^n)^2+I*m*f/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+1/4*Pi^2*csgn(I*d
)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)/x*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*Pi*csgn(I*d)*csgn(I*
(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)/x*b*ln(c)-1/4*Pi^2*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)/x*b*
csgn(I*c)*csgn(I*c*x^n)^2-I*m*f/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*b*Pi*csgn(I*c*x^n)^3-1/4*Pi^2*csgn(I*(f*x^
2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2/x*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*Pi*csgn(I*d)*csgn(I*(f*x^2+e)^m)
*csgn(I*d*(f*x^2+e)^m)*b/x*ln(x^n)-1/4*Pi^2*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)/x*b*csgn(I*x^n
)*csgn(I*c*x^n)^2-1/4*Pi^2*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2/x*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*Pi*
csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)*b*n/x+1/2*I*Pi*csgn(I*d*(f*x^2+e)^m)^3/x*a+1/4*Pi^2*csgn(I
*d*(f*x^2+e)^m)^3/x*b*csgn(I*c*x^n)^3+2*m*f/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*a+1/2*I/x*Pi*ln(d)*b*csgn(I*c*
x^n)^3-1/2*I*Pi*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2*b/x*ln(x^n)-1/2*I*Pi*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2*b*n/x
-1/2*I*Pi*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2/x*b*ln(c)+1/2*I*Pi*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*
d*(f*x^2+e)^m)/x*a-1/2*I*Pi*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2/x*b*ln(c)+1/2*I/x*Pi*ln(d)*b*csgn(I*c)*csgn(I*x^
n)*csgn(I*c*x^n)+1/4*Pi^2*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2/x*b*csgn(I*c)*csgn(I*c*x^n)^2+1/4*Pi^2*c
sgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2/x*b*csgn(I*c)*csgn(I*c*x^n)^2-1/2*I*Pi*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)
^m)^2*b*n/x+1/4*Pi^2*csgn(I*d*(f*x^2+e)^m)^3/x*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/4*Pi^2*csgn(I*(f*x^2+e)
^m)*csgn(I*d*(f*x^2+e)^m)^2/x*b*csgn(I*x^n)*csgn(I*c*x^n)^2+1/4*Pi^2*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2/x*b*csg
n(I*x^n)*csgn(I*c*x^n)^2+1/4*Pi^2*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)/x*b*csgn(I*c*x^n)^3-1/2*
I*Pi*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2*b/x*ln(x^n)

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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((a+b*log(c*x^n))*log(d*(f*x^2+e)^m)/x^2,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{{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{2}}, 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^2,x, algorithm="fricas")

[Out]

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

[Out]

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