3.35 \(\int \sqrt{1+x^2} \tan ^{-1}(x)^2 \, dx\)

Optimal. Leaf size=121 \[ i \tan ^{-1}(x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(x)}\right )-i \tan ^{-1}(x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(x)}\right )-\text{PolyLog}\left (3,-i e^{i \tan ^{-1}(x)}\right )+\text{PolyLog}\left (3,i e^{i \tan ^{-1}(x)}\right )+\frac{1}{2} x \sqrt{x^2+1} \tan ^{-1}(x)^2-\sqrt{x^2+1} \tan ^{-1}(x)-i \tan ^{-1}\left (e^{i \tan ^{-1}(x)}\right ) \tan ^{-1}(x)^2+\sinh ^{-1}(x) \]

[Out]

ArcSinh[x] - Sqrt[1 + x^2]*ArcTan[x] + (x*Sqrt[1 + x^2]*ArcTan[x]^2)/2 - I*ArcTa
n[E^(I*ArcTan[x])]*ArcTan[x]^2 + I*ArcTan[x]*PolyLog[2, (-I)*E^(I*ArcTan[x])] -
I*ArcTan[x]*PolyLog[2, I*E^(I*ArcTan[x])] - PolyLog[3, (-I)*E^(I*ArcTan[x])] + P
olyLog[3, I*E^(I*ArcTan[x])]

_______________________________________________________________________________________

Rubi [A]  time = 0.178511, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ i \tan ^{-1}(x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(x)}\right )-i \tan ^{-1}(x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(x)}\right )-\text{PolyLog}\left (3,-i e^{i \tan ^{-1}(x)}\right )+\text{PolyLog}\left (3,i e^{i \tan ^{-1}(x)}\right )+\frac{1}{2} x \sqrt{x^2+1} \tan ^{-1}(x)^2-\sqrt{x^2+1} \tan ^{-1}(x)-i \tan ^{-1}\left (e^{i \tan ^{-1}(x)}\right ) \tan ^{-1}(x)^2+\sinh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 + x^2]*ArcTan[x]^2,x]

[Out]

ArcSinh[x] - Sqrt[1 + x^2]*ArcTan[x] + (x*Sqrt[1 + x^2]*ArcTan[x]^2)/2 - I*ArcTa
n[E^(I*ArcTan[x])]*ArcTan[x]^2 + I*ArcTan[x]*PolyLog[2, (-I)*E^(I*ArcTan[x])] -
I*ArcTan[x]*PolyLog[2, I*E^(I*ArcTan[x])] - PolyLog[3, (-I)*E^(I*ArcTan[x])] + P
olyLog[3, I*E^(I*ArcTan[x])]

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x \sqrt{x^{2} + 1} \operatorname{atan}^{2}{\left (x \right )}}{2} - \sqrt{x^{2} + 1} \operatorname{atan}{\left (x \right )} + \operatorname{asinh}{\left (x \right )} + \frac{\int ^{\operatorname{atan}{\left (x \right )}} \frac{x^{2}}{\cos{\left (x \right )}}\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(x**2 + 1)*atan(x)**2/2 - sqrt(x**2 + 1)*atan(x) + asinh(x) + Integral(x**
2/cos(x), (x, atan(x)))/2

_______________________________________________________________________________________

Mathematica [B]  time = 3.28505, size = 405, normalized size = 3.35 \[ \frac{1}{2} \left (2 i \tan ^{-1}(x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(x)}\right )-2 i \tan ^{-1}(x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(x)}\right )-2 \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(x)}\right )+2 \text{PolyLog}\left (3,i e^{i \tan ^{-1}(x)}\right )+\sqrt{x^2+1} \left (x \tan ^{-1}(x)-2\right ) \tan ^{-1}(x)+\tan ^{-1}(x)^2 \log \left (1-i e^{i \tan ^{-1}(x)}\right )-\tan ^{-1}(x)^2 \log \left (1+i e^{i \tan ^{-1}(x)}\right )-\tan ^{-1}(x)^2 \log \left (\left (\frac{1}{2}+\frac{i}{2}\right ) e^{-\frac{1}{2} i \tan ^{-1}(x)} \left (e^{i \tan ^{-1}(x)}-i\right )\right )+\tan ^{-1}(x)^2 \log \left (\frac{1}{2} e^{-\frac{1}{2} i \tan ^{-1}(x)} \left ((1-i) e^{i \tan ^{-1}(x)}+(1+i)\right )\right )+\pi \tan ^{-1}(x) \log \left (\left (-\frac{1}{2}-\frac{i}{2}\right ) e^{-\frac{1}{2} i \tan ^{-1}(x)} \left (e^{i \tan ^{-1}(x)}-i\right )\right )+\pi \tan ^{-1}(x) \log \left (\frac{1}{2} e^{-\frac{1}{2} i \tan ^{-1}(x)} \left ((1-i) e^{i \tan ^{-1}(x)}+(1+i)\right )\right )-\pi \log (2) \tan ^{-1}(x)-\pi \tan ^{-1}(x) \log \left (\sin \left (\frac{1}{4} \left (2 \tan ^{-1}(x)+\pi \right )\right )\right )-\pi \tan ^{-1}(x) \log \left (-\cos \left (\frac{1}{4} \left (2 \tan ^{-1}(x)+\pi \right )\right )\right )+\tan ^{-1}(x)^2 \log \left (\cos \left (\frac{1}{2} \tan ^{-1}(x)\right )-\sin \left (\frac{1}{2} \tan ^{-1}(x)\right )\right )-\tan ^{-1}(x)^2 \log \left (\sin \left (\frac{1}{2} \tan ^{-1}(x)\right )+\cos \left (\frac{1}{2} \tan ^{-1}(x)\right )\right )-2 \log \left (\cos \left (\frac{1}{2} \tan ^{-1}(x)\right )-\sin \left (\frac{1}{2} \tan ^{-1}(x)\right )\right )+2 \log \left (\sin \left (\frac{1}{2} \tan ^{-1}(x)\right )+\cos \left (\frac{1}{2} \tan ^{-1}(x)\right )\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[Sqrt[1 + x^2]*ArcTan[x]^2,x]

[Out]

(Sqrt[1 + x^2]*ArcTan[x]*(-2 + x*ArcTan[x]) - Pi*ArcTan[x]*Log[2] + ArcTan[x]^2*
Log[1 - I*E^(I*ArcTan[x])] - ArcTan[x]^2*Log[1 + I*E^(I*ArcTan[x])] + Pi*ArcTan[
x]*Log[((-1/2 - I/2)*(-I + E^(I*ArcTan[x])))/E^((I/2)*ArcTan[x])] - ArcTan[x]^2*
Log[((1/2 + I/2)*(-I + E^(I*ArcTan[x])))/E^((I/2)*ArcTan[x])] + Pi*ArcTan[x]*Log
[((1 + I) + (1 - I)*E^(I*ArcTan[x]))/(2*E^((I/2)*ArcTan[x]))] + ArcTan[x]^2*Log[
((1 + I) + (1 - I)*E^(I*ArcTan[x]))/(2*E^((I/2)*ArcTan[x]))] - Pi*ArcTan[x]*Log[
-Cos[(Pi + 2*ArcTan[x])/4]] - 2*Log[Cos[ArcTan[x]/2] - Sin[ArcTan[x]/2]] + ArcTa
n[x]^2*Log[Cos[ArcTan[x]/2] - Sin[ArcTan[x]/2]] + 2*Log[Cos[ArcTan[x]/2] + Sin[A
rcTan[x]/2]] - ArcTan[x]^2*Log[Cos[ArcTan[x]/2] + Sin[ArcTan[x]/2]] - Pi*ArcTan[
x]*Log[Sin[(Pi + 2*ArcTan[x])/4]] + (2*I)*ArcTan[x]*PolyLog[2, (-I)*E^(I*ArcTan[
x])] - (2*I)*ArcTan[x]*PolyLog[2, I*E^(I*ArcTan[x])] - 2*PolyLog[3, (-I)*E^(I*Ar
cTan[x])] + 2*PolyLog[3, I*E^(I*ArcTan[x])])/2

_______________________________________________________________________________________

Maple [A]  time = 0.539, size = 171, normalized size = 1.4 \[{\frac{\arctan \left ( x \right ) \left ( x\arctan \left ( x \right ) -2 \right ) }{2}\sqrt{{x}^{2}+1}}+{\frac{ \left ( \arctan \left ( x \right ) \right ) ^{2}}{2}\ln \left ( 1-{i \left ( 1+ix \right ){\frac{1}{\sqrt{{x}^{2}+1}}}} \right ) }-{\frac{ \left ( \arctan \left ( x \right ) \right ) ^{2}}{2}\ln \left ( 1+{i \left ( 1+ix \right ){\frac{1}{\sqrt{{x}^{2}+1}}}} \right ) }-{\it polylog} \left ( 3,{-i \left ( 1+ix \right ){\frac{1}{\sqrt{{x}^{2}+1}}}} \right ) +{\it polylog} \left ( 3,{i \left ( 1+ix \right ){\frac{1}{\sqrt{{x}^{2}+1}}}} \right ) +i\arctan \left ( x \right ){\it polylog} \left ( 2,{-i \left ( 1+ix \right ){\frac{1}{\sqrt{{x}^{2}+1}}}} \right ) -i\arctan \left ( x \right ){\it polylog} \left ( 2,{i \left ( 1+ix \right ){\frac{1}{\sqrt{{x}^{2}+1}}}} \right ) -2\,i\arctan \left ({(1+ix){\frac{1}{\sqrt{{x}^{2}+1}}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(x^2+1)^(1/2)*arctan(x)*(x*arctan(x)-2)+1/2*arctan(x)^2*ln(1-I*(1+I*x)/(x^2+
1)^(1/2))-1/2*arctan(x)^2*ln(1+I*(1+I*x)/(x^2+1)^(1/2))-polylog(3,-I*(1+I*x)/(x^
2+1)^(1/2))+polylog(3,I*(1+I*x)/(x^2+1)^(1/2))+I*arctan(x)*polylog(2,-I*(1+I*x)/
(x^2+1)^(1/2))-I*arctan(x)*polylog(2,I*(1+I*x)/(x^2+1)^(1/2))-2*I*arctan((1+I*x)
/(x^2+1)^(1/2))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{x^{2} + 1} \arctan \left (x\right )^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(x^2 + 1)*arctan(x)^2, x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x^{2} + 1} \operatorname{atan}^{2}{\left (x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(x**2 + 1)*atan(x)**2, x)

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError