3.555 \(\int \frac{1+x^{7/2}}{1-x^2} \, dx\)

Optimal. Leaf size=43 \[ -\frac{2 x^{5/2}}{5}-2 \sqrt{x}-\log \left (1-\sqrt{x}\right )+\frac{1}{2} \log (x+1)+\tan ^{-1}\left (\sqrt{x}\right ) \]

[Out]

-2*Sqrt[x] - (2*x^(5/2))/5 + ArcTan[Sqrt[x]] - Log[1 - Sqrt[x]] + Log[1 + x]/2

_______________________________________________________________________________________

Rubi [A]  time = 0.193995, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ -\frac{2 x^{5/2}}{5}-2 \sqrt{x}-\log \left (1-\sqrt{x}\right )+\frac{1}{2} \log (x+1)+\tan ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^(7/2))/(1 - x^2),x]

[Out]

-2*Sqrt[x] - (2*x^(5/2))/5 + ArcTan[Sqrt[x]] - Log[1 - Sqrt[x]] + Log[1 + x]/2

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.96, size = 41, normalized size = 0.95 \[ - \frac{2 x^{\frac{5}{2}}}{5} - 2 \sqrt{x} - \frac{\log{\left (- x + 1 \right )}}{2} + \frac{\log{\left (x + 1 \right )}}{2} + \operatorname{atan}{\left (\sqrt{x} \right )} + \operatorname{atanh}{\left (\sqrt{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x**(7/2))/(-x**2+1),x)

[Out]

-2*x**(5/2)/5 - 2*sqrt(x) - log(-x + 1)/2 + log(x + 1)/2 + atan(sqrt(x)) + atanh
(sqrt(x))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0187024, size = 43, normalized size = 1. \[ -\frac{2 x^{5/2}}{5}-2 \sqrt{x}-\log \left (1-\sqrt{x}\right )+\frac{1}{2} \log (x+1)+\tan ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x^(7/2))/(1 - x^2),x]

[Out]

-2*Sqrt[x] - (2*x^(5/2))/5 + ArcTan[Sqrt[x]] - Log[1 - Sqrt[x]] + Log[1 + x]/2

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 34, normalized size = 0.8 \[ -{\frac{2}{5}{x}^{{\frac{5}{2}}}}-2\,\sqrt{x}-{\frac{1}{2}\ln \left ( -1+\sqrt{x} \right ) }+{\frac{1}{2}\ln \left ( 1+\sqrt{x} \right ) }+\arctan \left ( \sqrt{x} \right ) +{\it Artanh} \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x^(7/2))/(-x^2+1),x)

[Out]

-2/5*x^(5/2)-2*x^(1/2)-1/2*ln(-1+x^(1/2))+1/2*ln(1+x^(1/2))+arctan(x^(1/2))+arct
anh(x)

_______________________________________________________________________________________

Maxima [A]  time = 0.810667, size = 39, normalized size = 0.91 \[ -\frac{2}{5} \, x^{\frac{5}{2}} - 2 \, \sqrt{x} + \arctan \left (\sqrt{x}\right ) + \frac{1}{2} \, \log \left (x + 1\right ) - \log \left (\sqrt{x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^(7/2) + 1)/(x^2 - 1),x, algorithm="maxima")

[Out]

-2/5*x^(5/2) - 2*sqrt(x) + arctan(sqrt(x)) + 1/2*log(x + 1) - log(sqrt(x) - 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.271261, size = 39, normalized size = 0.91 \[ -\frac{2}{5} \,{\left (x^{2} + 5\right )} \sqrt{x} + \arctan \left (\sqrt{x}\right ) + \frac{1}{2} \, \log \left (x + 1\right ) - \log \left (\sqrt{x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^(7/2) + 1)/(x^2 - 1),x, algorithm="fricas")

[Out]

-2/5*(x^2 + 5)*sqrt(x) + arctan(sqrt(x)) + 1/2*log(x + 1) - log(sqrt(x) - 1)

_______________________________________________________________________________________

Sympy [A]  time = 10.6502, size = 36, normalized size = 0.84 \[ - \frac{2 x^{\frac{5}{2}}}{5} - 2 \sqrt{x} - \log{\left (\sqrt{x} - 1 \right )} + \frac{\log{\left (x + 1 \right )}}{2} + \operatorname{atan}{\left (\sqrt{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x**(7/2))/(-x**2+1),x)

[Out]

-2*x**(5/2)/5 - 2*sqrt(x) - log(sqrt(x) - 1) + log(x + 1)/2 + atan(sqrt(x))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.260682, size = 41, normalized size = 0.95 \[ -\frac{2}{5} \, x^{\frac{5}{2}} - 2 \, \sqrt{x} + \arctan \left (\sqrt{x}\right ) + \frac{1}{2} \,{\rm ln}\left (x + 1\right ) -{\rm ln}\left ({\left | \sqrt{x} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^(7/2) + 1)/(x^2 - 1),x, algorithm="giac")

[Out]

-2/5*x^(5/2) - 2*sqrt(x) + arctan(sqrt(x)) + 1/2*ln(x + 1) - ln(abs(sqrt(x) - 1)
)