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