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

Optimal. Leaf size=105 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d} \sqrt {e}}-\frac {i b n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {e}}+\frac {i b n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {e}} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {205, 2324, 12, 4848, 2391} \[ -\frac {i b n \text {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {e}}+\frac {i b n \text {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {e}}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d} \sqrt {e}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/(d + e*x^2),x]

[Out]

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

Rule 12

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

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 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 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]

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 107, normalized size = 1.02 \[ \frac {-\left (\left (\log \left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right )-\log \left (\frac {d \sqrt {e} x}{(-d)^{3/2}}+1\right )\right ) \left (a+b \log \left (c x^n\right )\right )\right )+b n \text {Li}_2\left (\frac {\sqrt {e} x}{\sqrt {-d}}\right )-b n \text {Li}_2\left (\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{2 \sqrt {-d} \sqrt {e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left (c x^{n}\right ) + a}{e x^{2} + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.27, size = 332, normalized size = 3.16 \[ -\frac {i \pi b \arctan \left (\frac {e x}{\sqrt {d e}}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 \sqrt {d e}}+\frac {i \pi b \arctan \left (\frac {e x}{\sqrt {d e}}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 \sqrt {d e}}+\frac {i \pi b \arctan \left (\frac {e x}{\sqrt {d e}}\right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 \sqrt {d e}}-\frac {i \pi b \arctan \left (\frac {e x}{\sqrt {d e}}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2 \sqrt {d e}}-\frac {b n \arctan \left (\frac {e x}{\sqrt {d e}}\right ) \ln \relax (x )}{\sqrt {d e}}+\frac {b n \ln \relax (x ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 \sqrt {-d e}}-\frac {b n \ln \relax (x ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 \sqrt {-d e}}+\frac {b n \dilog \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 \sqrt {-d e}}-\frac {b n \dilog \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 \sqrt {-d e}}+\frac {b \arctan \left (\frac {e x}{\sqrt {d e}}\right ) \ln \relax (c )}{\sqrt {d e}}+\frac {b \arctan \left (\frac {e x}{\sqrt {d e}}\right ) \ln \left (x^{n}\right )}{\sqrt {d e}}+\frac {a \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ b \int \frac {\log \relax (c) + \log \left (x^{n}\right )}{e x^{2} + d}\,{d x} + \frac {a \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{e\,x^2+d} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*x^n))/(d + e*x^2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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