3.320 \(\int \frac{1}{x^{7/2} \left (1+x^2\right )} \, dx\)

Optimal. Leaf size=108 \[ -\frac{2}{5 x^{5/2}}+\frac{2}{\sqrt{x}}+\frac{\log \left (x-\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}-\frac{\log \left (x+\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{\sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{\sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.140206, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ -\frac{2}{5 x^{5/2}}+\frac{2}{\sqrt{x}}+\frac{\log \left (x-\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}-\frac{\log \left (x+\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{\sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(7/2)*(1 + x^2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 20.9247, size = 99, normalized size = 0.92 \[ \frac{\sqrt{2} \log{\left (- \sqrt{2} \sqrt{x} + x + 1 \right )}}{4} - \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt{x} + x + 1 \right )}}{4} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{2} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{2} + \frac{2}{\sqrt{x}} - \frac{2}{5 x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1)/4 - sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1)/4
+ sqrt(2)*atan(sqrt(2)*sqrt(x) - 1)/2 + sqrt(2)*atan(sqrt(2)*sqrt(x) + 1)/2 + 2
/sqrt(x) - 2/(5*x**(5/2))

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Mathematica [A]  time = 0.0763655, size = 107, normalized size = 0.99 \[ \frac{1}{20} \left (-\frac{8}{x^{5/2}}+\frac{40}{\sqrt{x}}+5 \sqrt{2} \log \left (x-\sqrt{2} \sqrt{x}+1\right )-5 \sqrt{2} \log \left (x+\sqrt{2} \sqrt{x}+1\right )-10 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )+10 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-8/x^(5/2) + 40/Sqrt[x] - 10*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[x]] + 10*Sqrt[2]*A
rcTan[1 + Sqrt[2]*Sqrt[x]] + 5*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[x] + x] - 5*Sqrt[2]*
Log[1 + Sqrt[2]*Sqrt[x] + x])/20

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Maple [A]  time = 0.012, size = 72, normalized size = 0.7 \[ -{\frac{2}{5}{x}^{-{\frac{5}{2}}}}+2\,{\frac{1}{\sqrt{x}}}+{\frac{\sqrt{2}}{2}\arctan \left ( \sqrt{2}\sqrt{x}-1 \right ) }+{\frac{\sqrt{2}}{4}\ln \left ({1 \left ( 1+x-\sqrt{2}\sqrt{x} \right ) \left ( 1+x+\sqrt{2}\sqrt{x} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}}{2}\arctan \left ( 1+\sqrt{2}\sqrt{x} \right ) } \]

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*arctan(2^(1/2)*x^(1/2)-1)*2^(1/2)+1/4*2^(1/2)*ln((1+x
-2^(1/2)*x^(1/2))/(1+x+2^(1/2)*x^(1/2)))+1/2*arctan(1+2^(1/2)*x^(1/2))*2^(1/2)

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Maxima [A]  time = 1.52781, size = 116, normalized size = 1.07 \[ \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{1}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) - \frac{1}{4} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{1}{4} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{2 \,{\left (5 \, x^{2} - 1\right )}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 1/2*sqrt(2)*arctan(-1/2*
sqrt(2)*(sqrt(2) - 2*sqrt(x))) - 1/4*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1) + 1/4*
sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) + 2/5*(5*x^2 - 1)/x^(5/2)

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Fricas [A]  time = 0.251762, size = 178, normalized size = 1.65 \[ -\frac{20 \, \sqrt{2} x^{3} \arctan \left (\frac{1}{\sqrt{2} \sqrt{x} + \sqrt{2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2} + 1}\right ) + 20 \, \sqrt{2} x^{3} \arctan \left (\frac{1}{\sqrt{2} \sqrt{x} + \sqrt{-2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2} - 1}\right ) + 5 \, \sqrt{2} x^{3} \log \left (2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2\right ) - 5 \, \sqrt{2} x^{3} \log \left (-2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2\right ) - 8 \,{\left (5 \, x^{2} - 1\right )} \sqrt{x}}{20 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*(20*sqrt(2)*x^3*arctan(1/(sqrt(2)*sqrt(x) + sqrt(2*sqrt(2)*sqrt(x) + 2*x +
2) + 1)) + 20*sqrt(2)*x^3*arctan(1/(sqrt(2)*sqrt(x) + sqrt(-2*sqrt(2)*sqrt(x) +
2*x + 2) - 1)) + 5*sqrt(2)*x^3*log(2*sqrt(2)*sqrt(x) + 2*x + 2) - 5*sqrt(2)*x^3
*log(-2*sqrt(2)*sqrt(x) + 2*x + 2) - 8*(5*x^2 - 1)*sqrt(x))/x^3

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Sympy [A]  time = 32.9248, size = 105, normalized size = 0.97 \[ \frac{\sqrt{2} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{4} - \frac{\sqrt{2} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{4} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{2} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{2} + \frac{2}{\sqrt{x}} - \frac{2}{5 x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/4 - sqrt(2)*log(4*sqrt(2)*sqrt(x) + 4*
x + 4)/4 + sqrt(2)*atan(sqrt(2)*sqrt(x) - 1)/2 + sqrt(2)*atan(sqrt(2)*sqrt(x) +
1)/2 + 2/sqrt(x) - 2/(5*x**(5/2))

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GIAC/XCAS [A]  time = 0.21297, size = 116, normalized size = 1.07 \[ \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{1}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) - \frac{1}{4} \, \sqrt{2}{\rm ln}\left (\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{1}{4} \, \sqrt{2}{\rm ln}\left (-\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{2 \,{\left (5 \, x^{2} - 1\right )}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 1/2*sqrt(2)*arctan(-1/2*
sqrt(2)*(sqrt(2) - 2*sqrt(x))) - 1/4*sqrt(2)*ln(sqrt(2)*sqrt(x) + x + 1) + 1/4*s
qrt(2)*ln(-sqrt(2)*sqrt(x) + x + 1) + 2/5*(5*x^2 - 1)/x^(5/2)