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

Optimal. Leaf size=164 \[ -\frac{i b n \text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{4 d^{3/2} \sqrt{e}}+\frac{i b n \text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{4 d^{3/2} \sqrt{e}}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2} \sqrt{e}}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}-\frac{b n \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e}} \]

[Out]

-(b*n*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*Sqrt[e]) + (x*(a + b*Log[c*x^n]))/(2*d*(d + e*x^2)) + (ArcTan[(S
qrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(2*d^(3/2)*Sqrt[e]) - ((I/4)*b*n*PolyLog[2, ((-I)*Sqrt[e]*x)/Sqrt[d]])/
(d^(3/2)*Sqrt[e]) + ((I/4)*b*n*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*Sqrt[e])

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Rubi [A]  time = 0.0990367, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2323, 205, 2324, 12, 4848, 2391} \[ -\frac{i b n \text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{4 d^{3/2} \sqrt{e}}+\frac{i b n \text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{4 d^{3/2} \sqrt{e}}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2} \sqrt{e}}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}-\frac{b n \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*n*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*Sqrt[e]) + (x*(a + b*Log[c*x^n]))/(2*d*(d + e*x^2)) + (ArcTan[(S
qrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(2*d^(3/2)*Sqrt[e]) - ((I/4)*b*n*PolyLog[2, ((-I)*Sqrt[e]*x)/Sqrt[d]])/
(d^(3/2)*Sqrt[e]) + ((I/4)*b*n*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*Sqrt[e])

Rule 2323

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> -Simp[(x*(d + e*x^2)^(q + 1
)*(a + b*Log[c*x^n]))/(2*d*(q + 1)), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*Log[c*
x^n]), x], x] + Dist[(b*n)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1), x], x]) /; FreeQ[{a, b, c, d, e, n}, x] &&
LtQ[q, -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 2324

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> With[{u = IntHide[1/(d + e*x^2),
x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[u/x, x], x]] /; FreeQ[{a, b, c, d, e, n}, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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{a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^2} \, dx &=\frac{x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}+\frac{\int \frac{a+b \log \left (c x^n\right )}{d+e x^2} \, dx}{2 d}-\frac{(b n) \int \frac{1}{d+e x^2} \, dx}{2 d}\\ &=-\frac{b n \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e}}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2} \sqrt{e}}-\frac{(b n) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e} x} \, dx}{2 d}\\ &=-\frac{b n \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e}}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2} \sqrt{e}}-\frac{(b n) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{2 d^{3/2} \sqrt{e}}\\ &=-\frac{b n \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e}}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2} \sqrt{e}}-\frac{(i b n) \int \frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{4 d^{3/2} \sqrt{e}}+\frac{(i b n) \int \frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{4 d^{3/2} \sqrt{e}}\\ &=-\frac{b n \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e}}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2} \sqrt{e}}-\frac{i b n \text{Li}_2\left (-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{4 d^{3/2} \sqrt{e}}+\frac{i b n \text{Li}_2\left (\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{4 d^{3/2} \sqrt{e}}\\ \end{align*}

Mathematica [A]  time = 0.537206, size = 289, normalized size = 1.76 \[ \frac{1}{4} \left (\frac{b d n \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{(-d)^{5/2} \sqrt{e}}+\frac{b n \text{PolyLog}\left (2,\frac{d \sqrt{e} x}{(-d)^{3/2}}\right )}{(-d)^{3/2} \sqrt{e}}+\frac{\log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{3/2} \sqrt{e}}+\frac{d \log \left (\frac{d \sqrt{e} x}{(-d)^{3/2}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{5/2} \sqrt{e}}+\frac{a+b \log \left (c x^n\right )}{d \left (\sqrt{-d} \sqrt{e}+e x\right )}+\frac{a+b \log \left (c x^n\right )}{d e x+(-d)^{3/2} \sqrt{e}}+\frac{b d n \left (\log (x)-\log \left (\sqrt{-d}-\sqrt{e} x\right )\right )}{(-d)^{5/2} \sqrt{e}}+\frac{b n \left (\log (x)-\log \left (\sqrt{-d}+\sqrt{e} x\right )\right )}{(-d)^{3/2} \sqrt{e}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*Log[c*x^n])/(d*(Sqrt[-d]*Sqrt[e] + e*x)) + (a + b*Log[c*x^n])/((-d)^(3/2)*Sqrt[e] + d*e*x) + (b*d*n*(L
og[x] - Log[Sqrt[-d] - Sqrt[e]*x]))/((-d)^(5/2)*Sqrt[e]) + (b*n*(Log[x] - Log[Sqrt[-d] + Sqrt[e]*x]))/((-d)^(3
/2)*Sqrt[e]) + ((a + b*Log[c*x^n])*Log[1 + (Sqrt[e]*x)/Sqrt[-d]])/((-d)^(3/2)*Sqrt[e]) + (d*(a + b*Log[c*x^n])
*Log[1 + (d*Sqrt[e]*x)/(-d)^(3/2)])/((-d)^(5/2)*Sqrt[e]) + (b*d*n*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]])/((-d)^(5/2
)*Sqrt[e]) + (b*n*PolyLog[2, (d*Sqrt[e]*x)/(-d)^(3/2)])/((-d)^(3/2)*Sqrt[e]))/4

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Maple [C]  time = 0.301, size = 685, normalized size = 4.2 \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))/(e*x^2+d)^2,x)

[Out]

1/2*b*x/d/(e*x^2+d)*ln(x^n)-1/2*b/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*n*ln(x)+1/2*b/d/(d*e)^(1/2)*arctan(x*e
/(d*e)^(1/2))*ln(x^n)-1/2*b*n/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))+1/4*b*n*ln(x)/d/(e*x^2+d)/(-d*e)^(1/2)*ln(
(-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))*x^2*e-1/4*b*n*ln(x)/d/(e*x^2+d)/(-d*e)^(1/2)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1
/2))*x^2*e+1/4*b*n*ln(x)/(e*x^2+d)/(-d*e)^(1/2)*ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-1/4*b*n*ln(x)/(e*x^2+d)/(
-d*e)^(1/2)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+1/4*b*n/d/(-d*e)^(1/2)*dilog((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))
-1/4*b*n/d/(-d*e)^(1/2)*dilog((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-1/4*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*
x/d/(e*x^2+d)-1/4*I*b*Pi*csgn(I*c*x^n)^3/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))-1/4*I*b*Pi*csgn(I*c*x^n)^3*x/d/
(e*x^2+d)+1/4*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*x/d/(e*x^2+d)+1/4*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d/(d*e)^(1
/2)*arctan(x*e/(d*e)^(1/2))+1/4*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)*x/d/(e*x^2+d)+1/4*I*b*Pi*csgn(I*x^n)*csgn(I*c
*x^n)^2/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))-1/4*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d/(d*e)^(1/2)*arc
tan(x*e/(d*e)^(1/2))+1/2*b*ln(c)*x/d/(e*x^2+d)+1/2*b*ln(c)/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))+1/2*a*x/d/(e*
x^2+d)+1/2*a/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))

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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))/(e*x^2+d)^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{b \log \left (c x^{n}\right ) + a}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*log(c*x^n) + a)/(e^2*x^4 + 2*d*e*x^2 + d^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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