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

Optimal. Leaf size=111 \[ -\frac{i b m n \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{2 \sqrt{e} \sqrt{f}}+\frac{i b m n \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{2 \sqrt{e} \sqrt{f}}+\frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{\sqrt{e} \sqrt{f}} \]

[Out]

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

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Rubi [A]  time = 0.161518, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {205, 2324, 12, 4848, 2391, 2445} \[ -\frac{i b m n \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{2 \sqrt{e} \sqrt{f}}+\frac{i b m n \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{2 \sqrt{e} \sqrt{f}}+\frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{\sqrt{e} \sqrt{f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rule 2445

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

Rubi steps

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

Mathematica [A]  time = 0.0837419, size = 113, normalized size = 1.02 \[ \frac{b m n \text{PolyLog}\left (2,\frac{\sqrt{f} x}{\sqrt{-e}}\right )-b m n \text{PolyLog}\left (2,\frac{e \sqrt{f} x}{(-e)^{3/2}}\right )+\left (\log \left (\frac{\sqrt{f} x}{\sqrt{-e}}+1\right )-\log \left (\frac{e \sqrt{f} x}{(-e)^{3/2}}+1\right )\right ) \left (-\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )\right )}{2 \sqrt{-e} \sqrt{f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c \left ( d{x}^{m} \right ) ^{n} \right ) }{f{x}^{2}+e}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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