Optimal. Leaf size=65 \[ -\frac{1}{2} (1-i)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{1-i} \sqrt{x}}{\sqrt{x+1}}\right )-\frac{1}{2} (1+i)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{1+i} \sqrt{x}}{\sqrt{x+1}}\right ) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.126082, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{1}{2} (1-i)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{1-i} \sqrt{x}}{\sqrt{x+1}}\right )-\frac{1}{2} (1+i)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{1+i} \sqrt{x}}{\sqrt{x+1}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]/(Sqrt[1 + x]*(1 + x^2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 19.2993, size = 121, normalized size = 1.86 \[ - \frac{i \left (\sqrt{1 + \sqrt{2}} - \sqrt{- \sqrt{2} + 1}\right ) \operatorname{atanh}{\left (\frac{2 \sqrt{x}}{\sqrt{x + 1} \left (\sqrt{1 + \sqrt{2}} - i \sqrt{-1 + \sqrt{2}}\right )} \right )}}{2} + \frac{i \left (\sqrt{1 + \sqrt{2}} + \sqrt{- \sqrt{2} + 1}\right ) \operatorname{atanh}{\left (\frac{2 \sqrt{x}}{\sqrt{x + 1} \left (\sqrt{1 + \sqrt{2}} + i \sqrt{-1 + \sqrt{2}}\right )} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(1/2)/(x**2+1)/(1+x)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.126005, size = 65, normalized size = 1. \[ \frac{1}{2} \left (\sqrt{2-2 i} \tan ^{-1}\left ((1-i)^{3/2} \sqrt{\frac{x}{2 x+2}}\right )+\sqrt{2+2 i} \tan ^{-1}\left ((1+i)^{3/2} \sqrt{\frac{x}{2 x+2}}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]/(Sqrt[1 + x]*(1 + x^2)),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.19, size = 305, normalized size = 4.7 \[{\frac{ \left ( \sqrt{2}-1+x \right ) \sqrt{2}}{ \left ( -16+12\,\sqrt{2} \right ) \sqrt{1+\sqrt{2}}}\sqrt{{\frac{x \left ( 1+x \right ) }{ \left ( \sqrt{2}-1+x \right ) ^{2}}}} \left ( \sqrt{-2+2\,\sqrt{2}}\arctan \left ({\frac{ \left ( \sqrt{2}+1-x \right ) \left ( -4+3\,\sqrt{2} \right ) \left ( 3+2\,\sqrt{2} \right ) \sqrt{-2+2\,\sqrt{2}} \left ( \sqrt{2}-1+x \right ) }{4\,x \left ( 1+x \right ) }\sqrt{{\frac{x \left ( 1+x \right ) \left ( -4+3\,\sqrt{2} \right ) \left ( 3\,\sqrt{2}+4 \right ) }{ \left ( \sqrt{2}-1+x \right ) ^{2}}}}} \right ) \sqrt{1+\sqrt{2}}\sqrt{2}-2\,\sqrt{-2+2\,\sqrt{2}}\arctan \left ( 1/4\,{\frac{ \left ( \sqrt{2}+1-x \right ) \left ( -4+3\,\sqrt{2} \right ) \left ( 3+2\,\sqrt{2} \right ) \sqrt{-2+2\,\sqrt{2}} \left ( \sqrt{2}-1+x \right ) }{x \left ( 1+x \right ) }\sqrt{{\frac{x \left ( 1+x \right ) \left ( -4+3\,\sqrt{2} \right ) \left ( 3\,\sqrt{2}+4 \right ) }{ \left ( \sqrt{2}-1+x \right ) ^{2}}}}} \right ) \sqrt{1+\sqrt{2}}+4\,{\it Artanh} \left ({\frac{\sqrt{2}}{\sqrt{1+\sqrt{2}}}\sqrt{{\frac{x \left ( 1+x \right ) }{ \left ( \sqrt{2}-1+x \right ) ^{2}}}}} \right ) \sqrt{2}-6\,{\it Artanh} \left ({\frac{\sqrt{2}}{\sqrt{1+\sqrt{2}}}\sqrt{{\frac{x \left ( 1+x \right ) }{ \left ( \sqrt{2}-1+x \right ) ^{2}}}}} \right ) \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{1+x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(1/2)/(x^2+1)/(1+x)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x}}{{\left (x^{2} + 1\right )} \sqrt{x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/((x^2 + 1)*sqrt(x + 1)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.32074, size = 1146, normalized size = 17.63 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/((x^2 + 1)*sqrt(x + 1)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x}}{\sqrt{x + 1} \left (x^{2} + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(1/2)/(x**2+1)/(1+x)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x}}{{\left (x^{2} + 1\right )} \sqrt{x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/((x^2 + 1)*sqrt(x + 1)),x, algorithm="giac")
[Out]