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

Optimal. Leaf size=131 \[ -\frac{9}{10 x^{5/2}}+\frac{1}{2 x^{5/2} \left (x^2+1\right )}+\frac{9}{2 \sqrt{x}}+\frac{9 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{8 \sqrt{2}}-\frac{9 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{8 \sqrt{2}}-\frac{9 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{4 \sqrt{2}}+\frac{9 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{4 \sqrt{2}} \]

[Out]

-9/(10*x^(5/2)) + 9/(2*Sqrt[x]) + 1/(2*x^(5/2)*(1 + x^2)) - (9*ArcTan[1 - Sqrt[2
]*Sqrt[x]])/(4*Sqrt[2]) + (9*ArcTan[1 + Sqrt[2]*Sqrt[x]])/(4*Sqrt[2]) + (9*Log[1
- Sqrt[2]*Sqrt[x] + x])/(8*Sqrt[2]) - (9*Log[1 + Sqrt[2]*Sqrt[x] + x])/(8*Sqrt[
2])

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Rubi [A]  time = 0.162669, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.692 \[ -\frac{9}{10 x^{5/2}}+\frac{1}{2 x^{5/2} \left (x^2+1\right )}+\frac{9}{2 \sqrt{x}}+\frac{9 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{8 \sqrt{2}}-\frac{9 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{8 \sqrt{2}}-\frac{9 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{4 \sqrt{2}}+\frac{9 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{4 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(7/2)*(1 + x^2)^2),x]

[Out]

-9/(10*x^(5/2)) + 9/(2*Sqrt[x]) + 1/(2*x^(5/2)*(1 + x^2)) - (9*ArcTan[1 - Sqrt[2
]*Sqrt[x]])/(4*Sqrt[2]) + (9*ArcTan[1 + Sqrt[2]*Sqrt[x]])/(4*Sqrt[2]) + (9*Log[1
- Sqrt[2]*Sqrt[x] + x])/(8*Sqrt[2]) - (9*Log[1 + Sqrt[2]*Sqrt[x] + x])/(8*Sqrt[
2])

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Rubi in Sympy [A]  time = 24.5787, size = 121, normalized size = 0.92 \[ \frac{9 \sqrt{2} \log{\left (- \sqrt{2} \sqrt{x} + x + 1 \right )}}{16} - \frac{9 \sqrt{2} \log{\left (\sqrt{2} \sqrt{x} + x + 1 \right )}}{16} + \frac{9 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{8} + \frac{9 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{8} + \frac{9}{2 \sqrt{x}} - \frac{9}{10 x^{\frac{5}{2}}} + \frac{1}{2 x^{\frac{5}{2}} \left (x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1)/16 - 9*sqrt(2)*log(sqrt(2)*sqrt(x) + x +
1)/16 + 9*sqrt(2)*atan(sqrt(2)*sqrt(x) - 1)/8 + 9*sqrt(2)*atan(sqrt(2)*sqrt(x)
+ 1)/8 + 9/(2*sqrt(x)) - 9/(10*x**(5/2)) + 1/(2*x**(5/2)*(x**2 + 1))

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Mathematica [A]  time = 0.191439, size = 121, normalized size = 0.92 \[ \frac{1}{80} \left (-\frac{32}{x^{5/2}}+\frac{40 x^{3/2}}{x^2+1}+\frac{320}{\sqrt{x}}+45 \sqrt{2} \log \left (x-\sqrt{2} \sqrt{x}+1\right )-45 \sqrt{2} \log \left (x+\sqrt{2} \sqrt{x}+1\right )-90 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )+90 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-32/x^(5/2) + 320/Sqrt[x] + (40*x^(3/2))/(1 + x^2) - 90*Sqrt[2]*ArcTan[1 - Sqrt
[2]*Sqrt[x]] + 90*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[x]] + 45*Sqrt[2]*Log[1 - Sqrt[
2]*Sqrt[x] + x] - 45*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[x] + x])/80

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Maple [A]  time = 0.02, size = 84, normalized size = 0.6 \[ -{\frac{2}{5}{x}^{-{\frac{5}{2}}}}+4\,{\frac{1}{\sqrt{x}}}+{\frac{1}{2\,{x}^{2}+2}{x}^{{\frac{3}{2}}}}+{\frac{9\,\sqrt{2}}{8}\arctan \left ( 1+\sqrt{2}\sqrt{x} \right ) }+{\frac{9\,\sqrt{2}}{8}\arctan \left ( \sqrt{2}\sqrt{x}-1 \right ) }+{\frac{9\,\sqrt{2}}{16}\ln \left ({1 \left ( 1+x-\sqrt{2}\sqrt{x} \right ) \left ( 1+x+\sqrt{2}\sqrt{x} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(7/2)/(x^2+1)^2,x)

[Out]

-2/5/x^(5/2)+4/x^(1/2)+1/2*x^(3/2)/(x^2+1)+9/8*arctan(1+2^(1/2)*x^(1/2))*2^(1/2)
+9/8*arctan(2^(1/2)*x^(1/2)-1)*2^(1/2)+9/16*2^(1/2)*ln((1+x-2^(1/2)*x^(1/2))/(1+
x+2^(1/2)*x^(1/2)))

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Maxima [A]  time = 1.51088, size = 131, normalized size = 1. \[ \frac{9}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{9}{8} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) - \frac{9}{16} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{9}{16} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{45 \, x^{4} + 36 \, x^{2} - 4}{10 \,{\left (x^{\frac{9}{2}} + x^{\frac{5}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/8*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 9/8*sqrt(2)*arctan(-1/2*
sqrt(2)*(sqrt(2) - 2*sqrt(x))) - 9/16*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1) + 9/1
6*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) + 1/10*(45*x^4 + 36*x^2 - 4)/(x^(9/2) +
x^(5/2))

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Fricas [A]  time = 0.248315, size = 215, normalized size = 1.64 \[ -\frac{180 \, \sqrt{2}{\left (x^{5} + x^{3}\right )} \arctan \left (\frac{1}{\sqrt{2} \sqrt{x} + \sqrt{2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2} + 1}\right ) + 180 \, \sqrt{2}{\left (x^{5} + x^{3}\right )} \arctan \left (\frac{1}{\sqrt{2} \sqrt{x} + \sqrt{-2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2} - 1}\right ) + 45 \, \sqrt{2}{\left (x^{5} + x^{3}\right )} \log \left (2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2\right ) - 45 \, \sqrt{2}{\left (x^{5} + x^{3}\right )} \log \left (-2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2\right ) - 8 \,{\left (45 \, x^{4} + 36 \, x^{2} - 4\right )} \sqrt{x}}{80 \,{\left (x^{5} + x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/80*(180*sqrt(2)*(x^5 + x^3)*arctan(1/(sqrt(2)*sqrt(x) + sqrt(2*sqrt(2)*sqrt(x
) + 2*x + 2) + 1)) + 180*sqrt(2)*(x^5 + x^3)*arctan(1/(sqrt(2)*sqrt(x) + sqrt(-2
*sqrt(2)*sqrt(x) + 2*x + 2) - 1)) + 45*sqrt(2)*(x^5 + x^3)*log(2*sqrt(2)*sqrt(x)
+ 2*x + 2) - 45*sqrt(2)*(x^5 + x^3)*log(-2*sqrt(2)*sqrt(x) + 2*x + 2) - 8*(45*x
^4 + 36*x^2 - 4)*sqrt(x))/(x^5 + x^3)

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Sympy [A]  time = 164.549, size = 384, normalized size = 2.93 \[ \frac{45 \sqrt{2} x^{\frac{9}{2}} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{80 x^{\frac{9}{2}} + 80 x^{\frac{5}{2}}} - \frac{45 \sqrt{2} x^{\frac{9}{2}} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{80 x^{\frac{9}{2}} + 80 x^{\frac{5}{2}}} + \frac{90 \sqrt{2} x^{\frac{9}{2}} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{80 x^{\frac{9}{2}} + 80 x^{\frac{5}{2}}} + \frac{90 \sqrt{2} x^{\frac{9}{2}} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{80 x^{\frac{9}{2}} + 80 x^{\frac{5}{2}}} + \frac{45 \sqrt{2} x^{\frac{5}{2}} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{80 x^{\frac{9}{2}} + 80 x^{\frac{5}{2}}} - \frac{45 \sqrt{2} x^{\frac{5}{2}} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{80 x^{\frac{9}{2}} + 80 x^{\frac{5}{2}}} + \frac{90 \sqrt{2} x^{\frac{5}{2}} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{80 x^{\frac{9}{2}} + 80 x^{\frac{5}{2}}} + \frac{90 \sqrt{2} x^{\frac{5}{2}} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{80 x^{\frac{9}{2}} + 80 x^{\frac{5}{2}}} + \frac{360 x^{4}}{80 x^{\frac{9}{2}} + 80 x^{\frac{5}{2}}} + \frac{288 x^{2}}{80 x^{\frac{9}{2}} + 80 x^{\frac{5}{2}}} - \frac{32}{80 x^{\frac{9}{2}} + 80 x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

45*sqrt(2)*x**(9/2)*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(80*x**(9/2) + 80*x**(5/2)
) - 45*sqrt(2)*x**(9/2)*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(80*x**(9/2) + 80*x**(5
/2)) + 90*sqrt(2)*x**(9/2)*atan(sqrt(2)*sqrt(x) - 1)/(80*x**(9/2) + 80*x**(5/2))
+ 90*sqrt(2)*x**(9/2)*atan(sqrt(2)*sqrt(x) + 1)/(80*x**(9/2) + 80*x**(5/2)) + 4
5*sqrt(2)*x**(5/2)*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(80*x**(9/2) + 80*x**(5/2))
- 45*sqrt(2)*x**(5/2)*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(80*x**(9/2) + 80*x**(5/
2)) + 90*sqrt(2)*x**(5/2)*atan(sqrt(2)*sqrt(x) - 1)/(80*x**(9/2) + 80*x**(5/2))
+ 90*sqrt(2)*x**(5/2)*atan(sqrt(2)*sqrt(x) + 1)/(80*x**(9/2) + 80*x**(5/2)) + 36
0*x**4/(80*x**(9/2) + 80*x**(5/2)) + 288*x**2/(80*x**(9/2) + 80*x**(5/2)) - 32/(
80*x**(9/2) + 80*x**(5/2))

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GIAC/XCAS [A]  time = 0.209794, size = 132, normalized size = 1.01 \[ \frac{9}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{9}{8} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) - \frac{9}{16} \, \sqrt{2}{\rm ln}\left (\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{9}{16} \, \sqrt{2}{\rm ln}\left (-\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{x^{\frac{3}{2}}}{2 \,{\left (x^{2} + 1\right )}} + \frac{2 \,{\left (10 \, x^{2} - 1\right )}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/8*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 9/8*sqrt(2)*arctan(-1/2*
sqrt(2)*(sqrt(2) - 2*sqrt(x))) - 9/16*sqrt(2)*ln(sqrt(2)*sqrt(x) + x + 1) + 9/16
*sqrt(2)*ln(-sqrt(2)*sqrt(x) + x + 1) + 1/2*x^(3/2)/(x^2 + 1) + 2/5*(10*x^2 - 1)
/x^(5/2)