3.11 \(\int \frac{x \log \left (1+x^2\right ) \log \left (x+\sqrt{1+x^2}\right )}{\sqrt{1+x^2}} \, dx\)

Optimal. Leaf size=68 \[ x \left (-\log \left (x^2+1\right )\right )+\sqrt{x^2+1} \log \left (x^2+1\right ) \log \left (\sqrt{x^2+1}+x\right )-2 \sqrt{x^2+1} \log \left (\sqrt{x^2+1}+x\right )+4 x-2 \tan ^{-1}(x) \]

[Out]

4*x - 2*ArcTan[x] - x*Log[1 + x^2] - 2*Sqrt[1 + x^2]*Log[x + Sqrt[1 + x^2]] + Sq
rt[1 + x^2]*Log[1 + x^2]*Log[x + Sqrt[1 + x^2]]

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Rubi [A]  time = 0.230531, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276 \[ x \left (-\log \left (x^2+1\right )\right )+\sqrt{x^2+1} \log \left (x^2+1\right ) \log \left (\sqrt{x^2+1}+x\right )-2 \sqrt{x^2+1} \log \left (\sqrt{x^2+1}+x\right )+4 x-2 \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(x*Log[1 + x^2]*Log[x + Sqrt[1 + x^2]])/Sqrt[1 + x^2],x]

[Out]

4*x - 2*ArcTan[x] - x*Log[1 + x^2] - 2*Sqrt[1 + x^2]*Log[x + Sqrt[1 + x^2]] + Sq
rt[1 + x^2]*Log[1 + x^2]*Log[x + Sqrt[1 + x^2]]

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Rubi in Sympy [A]  time = 24.177, size = 65, normalized size = 0.96 \[ - x \log{\left (x^{2} + 1 \right )} + 4 x + \sqrt{x^{2} + 1} \log{\left (x + \sqrt{x^{2} + 1} \right )} \log{\left (x^{2} + 1 \right )} - 2 \sqrt{x^{2} + 1} \log{\left (x + \sqrt{x^{2} + 1} \right )} - 2 \operatorname{atan}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*ln(x**2+1)*ln(x+(x**2+1)**(1/2))/(x**2+1)**(1/2),x)

[Out]

-x*log(x**2 + 1) + 4*x + sqrt(x**2 + 1)*log(x + sqrt(x**2 + 1))*log(x**2 + 1) -
2*sqrt(x**2 + 1)*log(x + sqrt(x**2 + 1)) - 2*atan(x)

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Mathematica [A]  time = 0.060257, size = 64, normalized size = 0.94 \[ -2 \sqrt{x^2+1} \log \left (\sqrt{x^2+1}+x\right )+\log \left (x^2+1\right ) \left (\sqrt{x^2+1} \log \left (\sqrt{x^2+1}+x\right )-x\right )+4 x-2 \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Integrate[(x*Log[1 + x^2]*Log[x + Sqrt[1 + x^2]])/Sqrt[1 + x^2],x]

[Out]

4*x - 2*ArcTan[x] - 2*Sqrt[1 + x^2]*Log[x + Sqrt[1 + x^2]] + Log[1 + x^2]*(-x +
Sqrt[1 + x^2]*Log[x + Sqrt[1 + x^2]])

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Maple [F]  time = 0.026, size = 0, normalized size = 0. \[ \int{x\ln \left ({x}^{2}+1 \right ) \ln \left ( x+\sqrt{{x}^{2}+1} \right ){\frac{1}{\sqrt{{x}^{2}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*ln(x^2+1)*ln(x+(x^2+1)^(1/2))/(x^2+1)^(1/2),x)

[Out]

int(x*ln(x^2+1)*ln(x+(x^2+1)^(1/2))/(x^2+1)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{{\left (2 \, x^{2} -{\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) + 2\right )} \log \left (x + \sqrt{x^{2} + 1}\right )}{\sqrt{x^{2} + 1}} + \int \frac{\log \left (x^{2} + 1\right ) - 2}{x^{2} + \sqrt{x^{2} + 1} x}\,{d x} - \int -\frac{2 \, x^{2} -{\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) + 2}{\sqrt{x^{2} + 1} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*log(x^2 + 1)*log(x + sqrt(x^2 + 1))/sqrt(x^2 + 1),x, algorithm="maxima")

[Out]

-(2*x^2 - (x^2 + 1)*log(x^2 + 1) + 2)*log(x + sqrt(x^2 + 1))/sqrt(x^2 + 1) + int
egrate((log(x^2 + 1) - 2)/(x^2 + sqrt(x^2 + 1)*x), x) - integrate(-(2*x^2 - (x^2
+ 1)*log(x^2 + 1) + 2)/(sqrt(x^2 + 1)*x), x)

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Fricas [A]  time = 0.229495, size = 58, normalized size = 0.85 \[ \sqrt{x^{2} + 1}{\left (\log \left (x^{2} + 1\right ) - 2\right )} \log \left (x + \sqrt{x^{2} + 1}\right ) - x \log \left (x^{2} + 1\right ) + 4 \, x - 2 \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*log(x^2 + 1)*log(x + sqrt(x^2 + 1))/sqrt(x^2 + 1),x, algorithm="fricas")

[Out]

sqrt(x^2 + 1)*(log(x^2 + 1) - 2)*log(x + sqrt(x^2 + 1)) - x*log(x^2 + 1) + 4*x -
2*arctan(x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*ln(x**2+1)*ln(x+(x**2+1)**(1/2))/(x**2+1)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x \log \left (x^{2} + 1\right ) \log \left (x + \sqrt{x^{2} + 1}\right )}{\sqrt{x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*log(x^2 + 1)*log(x + sqrt(x^2 + 1))/sqrt(x^2 + 1),x, algorithm="giac")

[Out]

integrate(x*log(x^2 + 1)*log(x + sqrt(x^2 + 1))/sqrt(x^2 + 1), x)