\(\int \frac {(a+b \arctan (c x)) (d+e \log (f+g x^2))}{x} \, dx\) [1299]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx=a d \log (x)+\frac {1}{2} a e \log \left (-\frac {g x^2}{f}\right ) \log \left (f+g x^2\right )+\frac {1}{2} i b d \operatorname {PolyLog}(2,-i c x)-\frac {1}{2} i b d \operatorname {PolyLog}(2,i c x)+\frac {1}{2} a e \operatorname {PolyLog}\left (2,1+\frac {g x^2}{f}\right )+b e \text {Int}\left (\frac {\arctan (c x) \log \left (f+g x^2\right )}{x},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx=\int \frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx \]

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

Mathematica [N/A]

Not integrable

Time = 0.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx=\int \frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx \]

[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 [N/A] (verified)

Not integrable

Time = 0.71 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

\[\int \frac {\left (a +b \arctan \left (c x \right )\right ) \left (d +e \ln \left (g \,x^{2}+f \right )\right )}{x}d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )}}{x} \,d x } \]

[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 [N/A]

Not integrable

Time = 123.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e \log {\left (f + g x^{2} \right )}\right )}{x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.57 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.79 \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )}}{x} \,d x } \]

[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)

Giac [F(-1)]

Timed out. \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 1.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx=\int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (g\,x^2+f\right )\right )}{x} \,d x \]

[In]

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

[Out]

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