3.334 \(\int \frac{1}{x^{3/2} \left (1+x^2\right )^3} \, dx\)

Optimal. Leaf size=138 \[ \frac{9}{16 \sqrt{x} \left (x^2+1\right )}+\frac{1}{4 \sqrt{x} \left (x^2+1\right )^2}-\frac{45}{16 \sqrt{x}}-\frac{45 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}+\frac{45 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}+\frac{45 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}-\frac{45 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{32 \sqrt{2}} \]

[Out]

-45/(16*Sqrt[x]) + 1/(4*Sqrt[x]*(1 + x^2)^2) + 9/(16*Sqrt[x]*(1 + x^2)) + (45*Ar
cTan[1 - Sqrt[2]*Sqrt[x]])/(32*Sqrt[2]) - (45*ArcTan[1 + Sqrt[2]*Sqrt[x]])/(32*S
qrt[2]) - (45*Log[1 - Sqrt[2]*Sqrt[x] + x])/(64*Sqrt[2]) + (45*Log[1 + Sqrt[2]*S
qrt[x] + x])/(64*Sqrt[2])

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Rubi [A]  time = 0.180288, antiderivative size = 138, 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 \[ \frac{9}{16 \sqrt{x} \left (x^2+1\right )}+\frac{1}{4 \sqrt{x} \left (x^2+1\right )^2}-\frac{45}{16 \sqrt{x}}-\frac{45 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}+\frac{45 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}+\frac{45 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}-\frac{45 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{32 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(3/2)*(1 + x^2)^3),x]

[Out]

-45/(16*Sqrt[x]) + 1/(4*Sqrt[x]*(1 + x^2)^2) + 9/(16*Sqrt[x]*(1 + x^2)) + (45*Ar
cTan[1 - Sqrt[2]*Sqrt[x]])/(32*Sqrt[2]) - (45*ArcTan[1 + Sqrt[2]*Sqrt[x]])/(32*S
qrt[2]) - (45*Log[1 - Sqrt[2]*Sqrt[x] + x])/(64*Sqrt[2]) + (45*Log[1 + Sqrt[2]*S
qrt[x] + x])/(64*Sqrt[2])

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Rubi in Sympy [A]  time = 23.3514, size = 128, normalized size = 0.93 \[ - \frac{45 \sqrt{2} \log{\left (- \sqrt{2} \sqrt{x} + x + 1 \right )}}{128} + \frac{45 \sqrt{2} \log{\left (\sqrt{2} \sqrt{x} + x + 1 \right )}}{128} - \frac{45 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{64} - \frac{45 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{64} - \frac{45}{16 \sqrt{x}} + \frac{9}{16 \sqrt{x} \left (x^{2} + 1\right )} + \frac{1}{4 \sqrt{x} \left (x^{2} + 1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(3/2)/(x**2+1)**3,x)

[Out]

-45*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1)/128 + 45*sqrt(2)*log(sqrt(2)*sqrt(x) +
x + 1)/128 - 45*sqrt(2)*atan(sqrt(2)*sqrt(x) - 1)/64 - 45*sqrt(2)*atan(sqrt(2)*
sqrt(x) + 1)/64 - 45/(16*sqrt(x)) + 9/(16*sqrt(x)*(x**2 + 1)) + 1/(4*sqrt(x)*(x*
*2 + 1)**2)

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Mathematica [A]  time = 0.10592, size = 128, normalized size = 0.93 \[ \frac{1}{128} \left (-\frac{104 x^{3/2}}{x^2+1}-\frac{32 x^{3/2}}{\left (x^2+1\right )^2}-\frac{256}{\sqrt{x}}-45 \sqrt{2} \log \left (x-\sqrt{2} \sqrt{x}+1\right )+45 \sqrt{2} \log \left (x+\sqrt{2} \sqrt{x}+1\right )+90 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )-90 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(3/2)*(1 + x^2)^3),x]

[Out]

(-256/Sqrt[x] - (32*x^(3/2))/(1 + x^2)^2 - (104*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])/128

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Maple [A]  time = 0.018, size = 87, normalized size = 0.6 \[ -2\,{\frac{1}{\sqrt{x}}}-2\,{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ({\frac{13\,{x}^{7/2}}{32}}+{\frac{17\,{x}^{3/2}}{32}} \right ) }-{\frac{45\,\sqrt{2}}{64}\arctan \left ( 1+\sqrt{2}\sqrt{x} \right ) }-{\frac{45\,\sqrt{2}}{64}\arctan \left ( \sqrt{2}\sqrt{x}-1 \right ) }-{\frac{45\,\sqrt{2}}{128}\ln \left ({1 \left ( 1+x-\sqrt{2}\sqrt{x} \right ) \left ( 1+x+\sqrt{2}\sqrt{x} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(3/2)/(x^2+1)^3,x)

[Out]

-2/x^(1/2)-2*(13/32*x^(7/2)+17/32*x^(3/2))/(x^2+1)^2-45/64*arctan(1+2^(1/2)*x^(1
/2))*2^(1/2)-45/64*arctan(2^(1/2)*x^(1/2)-1)*2^(1/2)-45/128*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 = 1.49316, size = 138, normalized size = 1. \[ -\frac{45}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) - \frac{45}{64} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) + \frac{45}{128} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{45}{128} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{45 \, x^{4} + 81 \, x^{2} + 32}{16 \,{\left (x^{\frac{9}{2}} + 2 \, x^{\frac{5}{2}} + \sqrt{x}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 1)^3*x^(3/2)),x, algorithm="maxima")

[Out]

-45/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) - 45/64*sqrt(2)*arctan(
-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(x))) + 45/128*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1
) - 45/128*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) - 1/16*(45*x^4 + 81*x^2 + 32)/(
x^(9/2) + 2*x^(5/2) + sqrt(x))

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Fricas [A]  time = 0.254989, size = 235, normalized size = 1.7 \[ \frac{180 \, \sqrt{2}{\left (x^{5} + 2 \, x^{3} + x\right )} \arctan \left (\frac{1}{\sqrt{2} \sqrt{x} + \sqrt{2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2} + 1}\right ) + 180 \, \sqrt{2}{\left (x^{5} + 2 \, x^{3} + x\right )} \arctan \left (\frac{1}{\sqrt{2} \sqrt{x} + \sqrt{-2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2} - 1}\right ) + 45 \, \sqrt{2}{\left (x^{5} + 2 \, x^{3} + x\right )} \log \left (2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2\right ) - 45 \, \sqrt{2}{\left (x^{5} + 2 \, x^{3} + x\right )} \log \left (-2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2\right ) - 8 \,{\left (45 \, x^{4} + 81 \, x^{2} + 32\right )} \sqrt{x}}{128 \,{\left (x^{5} + 2 \, x^{3} + x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 1)^3*x^(3/2)),x, algorithm="fricas")

[Out]

1/128*(180*sqrt(2)*(x^5 + 2*x^3 + x)*arctan(1/(sqrt(2)*sqrt(x) + sqrt(2*sqrt(2)*
sqrt(x) + 2*x + 2) + 1)) + 180*sqrt(2)*(x^5 + 2*x^3 + x)*arctan(1/(sqrt(2)*sqrt(
x) + sqrt(-2*sqrt(2)*sqrt(x) + 2*x + 2) - 1)) + 45*sqrt(2)*(x^5 + 2*x^3 + x)*log
(2*sqrt(2)*sqrt(x) + 2*x + 2) - 45*sqrt(2)*(x^5 + 2*x^3 + x)*log(-2*sqrt(2)*sqrt
(x) + 2*x + 2) - 8*(45*x^4 + 81*x^2 + 32)*sqrt(x))/(x^5 + 2*x^3 + x)

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Sympy [A]  time = 111.671, size = 653, normalized size = 4.73 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(3/2)/(x**2+1)**3,x)

[Out]

-45*sqrt(2)*x**(9/2)*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**(9/2) + 256*x**(5
/2) + 128*sqrt(x)) + 45*sqrt(2)*x**(9/2)*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x
**(9/2) + 256*x**(5/2) + 128*sqrt(x)) - 90*sqrt(2)*x**(9/2)*atan(sqrt(2)*sqrt(x)
- 1)/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(x)) - 90*sqrt(2)*x**(9/2)*atan(sqr
t(2)*sqrt(x) + 1)/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(x)) - 90*sqrt(2)*x**(5
/2)*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(x)
) + 90*sqrt(2)*x**(5/2)*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**(9/2) + 256*x**
(5/2) + 128*sqrt(x)) - 180*sqrt(2)*x**(5/2)*atan(sqrt(2)*sqrt(x) - 1)/(128*x**(9
/2) + 256*x**(5/2) + 128*sqrt(x)) - 180*sqrt(2)*x**(5/2)*atan(sqrt(2)*sqrt(x) +
1)/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(x)) - 45*sqrt(2)*sqrt(x)*log(-4*sqrt(
2)*sqrt(x) + 4*x + 4)/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(x)) + 45*sqrt(2)*s
qrt(x)*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(
x)) - 90*sqrt(2)*sqrt(x)*atan(sqrt(2)*sqrt(x) - 1)/(128*x**(9/2) + 256*x**(5/2)
+ 128*sqrt(x)) - 90*sqrt(2)*sqrt(x)*atan(sqrt(2)*sqrt(x) + 1)/(128*x**(9/2) + 25
6*x**(5/2) + 128*sqrt(x)) - 360*x**4/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(x))
- 648*x**2/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(x)) - 256/(128*x**(9/2) + 25
6*x**(5/2) + 128*sqrt(x))

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GIAC/XCAS [A]  time = 0.211938, size = 134, normalized size = 0.97 \[ -\frac{45}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) - \frac{45}{64} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) + \frac{45}{128} \, \sqrt{2}{\rm ln}\left (\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{45}{128} \, \sqrt{2}{\rm ln}\left (-\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{2}{\sqrt{x}} - \frac{13 \, x^{\frac{7}{2}} + 17 \, x^{\frac{3}{2}}}{16 \,{\left (x^{2} + 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 1)^3*x^(3/2)),x, algorithm="giac")

[Out]

-45/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) - 45/64*sqrt(2)*arctan(
-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(x))) + 45/128*sqrt(2)*ln(sqrt(2)*sqrt(x) + x + 1)
- 45/128*sqrt(2)*ln(-sqrt(2)*sqrt(x) + x + 1) - 2/sqrt(x) - 1/16*(13*x^(7/2) +
17*x^(3/2))/(x^2 + 1)^2