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

Optimal. Leaf size=102 \[ b e \text {Int}\left (\frac {\tan ^{-1}(c x) \log \left (f+g x^2\right )}{x},x\right )+a d \log (x)+\frac {1}{2} a e \text {Li}_2\left (\frac {g x^2}{f}+1\right )+\frac {1}{2} a e \log \left (-\frac {g x^2}{f}\right ) \log \left (f+g x^2\right )+\frac {1}{2} i b d \text {Li}_2(-i c x)-\frac {1}{2} i b d \text {Li}_2(i c x) \]

[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)

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Rubi [A]  time = 0.28, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 0.27, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx \]

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]

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fricas [A]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b d \arctan \left (c x\right ) + a d + {\left (b e \arctan \left (c x\right ) + a e\right )} \log \left (g x^{2} + f\right )}{x}, x\right ) \]

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)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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maple [A]  time = 3.47, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arctan \left (c x \right )\right ) \left (d +e \ln \left (g \,x^{2}+f \right )\right )}{x}\, dx \]

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)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ a d \log \relax (x) + \frac {1}{2} \, \int \frac {2 \, {\left (b d \arctan \left (c x\right ) + {\left (b e \arctan \left (c x\right ) + a e\right )} \log \left (g x^{2} + f\right )\right )}}{x}\,{d x} \]

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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