3.280 \(\int \frac{(a+b \coth ^{-1}(c x)) (d+e \log (f+g x^2))}{x} \, dx\)

Optimal. Leaf size=100 \[ \frac{1}{2} a e \text{PolyLog}\left (2,\frac{g x^2}{f}+1\right )+b e \text{CannotIntegrate}\left (\frac{\coth ^{-1}(c x) \log \left (f+g x^2\right )}{x},x\right )+\frac{1}{2} b d \text{PolyLog}\left (2,-\frac{1}{c x}\right )-\frac{1}{2} b d \text{PolyLog}\left (2,\frac{1}{c x}\right )+a d \log (x)+\frac{1}{2} a e \log \left (-\frac{g x^2}{f}\right ) \log \left (f+g x^2\right ) \]

[Out]

b*e*CannotIntegrate[(ArcCoth[c*x]*Log[f + g*x^2])/x, x] + a*d*Log[x] + (a*e*Log[-((g*x^2)/f)]*Log[f + g*x^2])/
2 + (b*d*PolyLog[2, -(1/(c*x))])/2 - (b*d*PolyLog[2, 1/(c*x)])/2 + (a*e*PolyLog[2, 1 + (g*x^2)/f])/2

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Rubi [A]  time = 0.251519, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

a*d*Log[x] + (a*e*Log[-((g*x^2)/f)]*Log[f + g*x^2])/2 + (b*d*PolyLog[2, -(1/(c*x))])/2 - (b*d*PolyLog[2, 1/(c*
x)])/2 + (a*e*PolyLog[2, 1 + (g*x^2)/f])/2 + b*e*Defer[Int][(ArcCoth[c*x]*Log[f + g*x^2])/x, x]

Rubi steps

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

Mathematica [A]  time = 0.27258, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.74, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm arccoth} \left (cx\right ) \right ) \left ( d+e\ln \left ( g{x}^{2}+f \right ) \right ) }{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*d*arccoth(c*x) + a*d + (b*e*arccoth(c*x) + a*e)*log(g*x^2 + f))/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*acoth(c*x))*(d+e*ln(g*x**2+f))/x,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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