3.1299 \(\int \frac{(a+b \tan ^{-1}(c x)) (d+e \log (f+g x^2))}{x} \, dx\)

Optimal. Leaf size=101 \[ \frac{1}{2} a e \text{PolyLog}\left (2,\frac{g x^2}{f}+1\right )+b e \text{CannotIntegrate}\left (\frac{\tan ^{-1}(c x) \log \left (f+g x^2\right )}{x},x\right )+\frac{1}{2} i b d \text{PolyLog}(2,-i c x)-\frac{1}{2} i b d \text{PolyLog}(2,i c x)+a d \log (x)+\frac{1}{2} a e \log \left (-\frac{g x^2}{f}\right ) \log \left (f+g x^2\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.28075, 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 = {} \[ \int \frac{\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

Rubi steps

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

Mathematica [A]  time = 0.192716, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 1.146, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\arctan \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*arctan(c*x))*(d+e*ln(g*x^2+f))/x,x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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*atan(c*x))*(d+e*ln(g*x**2+f))/x,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arctan \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*arctan(c*x))*(d+e*log(g*x^2+f))/x,x, algorithm="giac")

[Out]

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