3.336 \(\int \frac{1}{x^{7/2} \left (1+x^2\right )^3} \, dx\)

Optimal. Leaf size=147 \[ -\frac{117}{80 x^{5/2}}+\frac{13}{16 x^{5/2} \left (x^2+1\right )}+\frac{1}{4 x^{5/2} \left (x^2+1\right )^2}+\frac{117}{16 \sqrt{x}}+\frac{117 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}-\frac{117 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}-\frac{117 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}+\frac{117 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{32 \sqrt{2}} \]

[Out]

-117/(80*x^(5/2)) + 117/(16*Sqrt[x]) + 1/(4*x^(5/2)*(1 + x^2)^2) + 13/(16*x^(5/2
)*(1 + x^2)) - (117*ArcTan[1 - Sqrt[2]*Sqrt[x]])/(32*Sqrt[2]) + (117*ArcTan[1 +
Sqrt[2]*Sqrt[x]])/(32*Sqrt[2]) + (117*Log[1 - Sqrt[2]*Sqrt[x] + x])/(64*Sqrt[2])
- (117*Log[1 + Sqrt[2]*Sqrt[x] + x])/(64*Sqrt[2])

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Rubi [A]  time = 0.185712, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.692 \[ -\frac{117}{80 x^{5/2}}+\frac{13}{16 x^{5/2} \left (x^2+1\right )}+\frac{1}{4 x^{5/2} \left (x^2+1\right )^2}+\frac{117}{16 \sqrt{x}}+\frac{117 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}-\frac{117 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}-\frac{117 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}+\frac{117 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{32 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(7/2)*(1 + x^2)^3),x]

[Out]

-117/(80*x^(5/2)) + 117/(16*Sqrt[x]) + 1/(4*x^(5/2)*(1 + x^2)^2) + 13/(16*x^(5/2
)*(1 + x^2)) - (117*ArcTan[1 - Sqrt[2]*Sqrt[x]])/(32*Sqrt[2]) + (117*ArcTan[1 +
Sqrt[2]*Sqrt[x]])/(32*Sqrt[2]) + (117*Log[1 - Sqrt[2]*Sqrt[x] + x])/(64*Sqrt[2])
- (117*Log[1 + Sqrt[2]*Sqrt[x] + x])/(64*Sqrt[2])

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Rubi in Sympy [A]  time = 24.9316, size = 136, normalized size = 0.93 \[ \frac{117 \sqrt{2} \log{\left (- \sqrt{2} \sqrt{x} + x + 1 \right )}}{128} - \frac{117 \sqrt{2} \log{\left (\sqrt{2} \sqrt{x} + x + 1 \right )}}{128} + \frac{117 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{64} + \frac{117 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{64} + \frac{117}{16 \sqrt{x}} - \frac{117}{80 x^{\frac{5}{2}}} + \frac{13}{16 x^{\frac{5}{2}} \left (x^{2} + 1\right )} + \frac{1}{4 x^{\frac{5}{2}} \left (x^{2} + 1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(7/2)/(x**2+1)**3,x)

[Out]

117*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1)/128 - 117*sqrt(2)*log(sqrt(2)*sqrt(x)
+ x + 1)/128 + 117*sqrt(2)*atan(sqrt(2)*sqrt(x) - 1)/64 + 117*sqrt(2)*atan(sqrt(
2)*sqrt(x) + 1)/64 + 117/(16*sqrt(x)) - 117/(80*x**(5/2)) + 13/(16*x**(5/2)*(x**
2 + 1)) + 1/(4*x**(5/2)*(x**2 + 1)**2)

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Mathematica [A]  time = 0.112855, size = 135, normalized size = 0.92 \[ \frac{1}{640} \left (-\frac{256}{x^{5/2}}+\frac{840 x^{3/2}}{x^2+1}+\frac{160 x^{3/2}}{\left (x^2+1\right )^2}+\frac{3840}{\sqrt{x}}+585 \sqrt{2} \log \left (x-\sqrt{2} \sqrt{x}+1\right )-585 \sqrt{2} \log \left (x+\sqrt{2} \sqrt{x}+1\right )-1170 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )+1170 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(7/2)*(1 + x^2)^3),x]

[Out]

(-256/x^(5/2) + 3840/Sqrt[x] + (160*x^(3/2))/(1 + x^2)^2 + (840*x^(3/2))/(1 + x^
2) - 1170*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[x]] + 1170*Sqrt[2]*ArcTan[1 + Sqrt[2]*
Sqrt[x]] + 585*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[x] + x] - 585*Sqrt[2]*Log[1 + Sqrt[2
]*Sqrt[x] + x])/640

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Maple [A]  time = 0.02, size = 92, normalized size = 0.6 \[ -{\frac{2}{5}{x}^{-{\frac{5}{2}}}}+6\,{\frac{1}{\sqrt{x}}}+2\,{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ({\frac{21\,{x}^{7/2}}{32}}+{\frac{25\,{x}^{3/2}}{32}} \right ) }+{\frac{117\,\sqrt{2}}{64}\arctan \left ( \sqrt{2}\sqrt{x}-1 \right ) }+{\frac{117\,\sqrt{2}}{128}\ln \left ({1 \left ( 1+x-\sqrt{2}\sqrt{x} \right ) \left ( 1+x+\sqrt{2}\sqrt{x} \right ) ^{-1}} \right ) }+{\frac{117\,\sqrt{2}}{64}\arctan \left ( 1+\sqrt{2}\sqrt{x} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(7/2)/(x^2+1)^3,x)

[Out]

-2/5/x^(5/2)+6/x^(1/2)+2*(21/32*x^(7/2)+25/32*x^(3/2))/(x^2+1)^2+117/64*arctan(2
^(1/2)*x^(1/2)-1)*2^(1/2)+117/128*2^(1/2)*ln((1+x-2^(1/2)*x^(1/2))/(1+x+2^(1/2)*
x^(1/2)))+117/64*arctan(1+2^(1/2)*x^(1/2))*2^(1/2)

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Maxima [A]  time = 1.50189, size = 144, normalized size = 0.98 \[ \frac{117}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{117}{64} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) - \frac{117}{128} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{117}{128} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{585 \, x^{6} + 1053 \, x^{4} + 416 \, x^{2} - 32}{80 \,{\left (x^{\frac{13}{2}} + 2 \, x^{\frac{9}{2}} + x^{\frac{5}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 1)^3*x^(7/2)),x, algorithm="maxima")

[Out]

117/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 117/64*sqrt(2)*arctan
(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(x))) - 117/128*sqrt(2)*log(sqrt(2)*sqrt(x) + x +
1) + 117/128*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) + 1/80*(585*x^6 + 1053*x^4 +
416*x^2 - 32)/(x^(13/2) + 2*x^(9/2) + x^(5/2))

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Fricas [A]  time = 0.255998, size = 255, normalized size = 1.73 \[ -\frac{2340 \, \sqrt{2}{\left (x^{7} + 2 \, x^{5} + x^{3}\right )} \arctan \left (\frac{1}{\sqrt{2} \sqrt{x} + \sqrt{2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2} + 1}\right ) + 2340 \, \sqrt{2}{\left (x^{7} + 2 \, x^{5} + x^{3}\right )} \arctan \left (\frac{1}{\sqrt{2} \sqrt{x} + \sqrt{-2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2} - 1}\right ) + 585 \, \sqrt{2}{\left (x^{7} + 2 \, x^{5} + x^{3}\right )} \log \left (2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2\right ) - 585 \, \sqrt{2}{\left (x^{7} + 2 \, x^{5} + x^{3}\right )} \log \left (-2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2\right ) - 8 \,{\left (585 \, x^{6} + 1053 \, x^{4} + 416 \, x^{2} - 32\right )} \sqrt{x}}{640 \,{\left (x^{7} + 2 \, x^{5} + x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 1)^3*x^(7/2)),x, algorithm="fricas")

[Out]

-1/640*(2340*sqrt(2)*(x^7 + 2*x^5 + x^3)*arctan(1/(sqrt(2)*sqrt(x) + sqrt(2*sqrt
(2)*sqrt(x) + 2*x + 2) + 1)) + 2340*sqrt(2)*(x^7 + 2*x^5 + x^3)*arctan(1/(sqrt(2
)*sqrt(x) + sqrt(-2*sqrt(2)*sqrt(x) + 2*x + 2) - 1)) + 585*sqrt(2)*(x^7 + 2*x^5
+ x^3)*log(2*sqrt(2)*sqrt(x) + 2*x + 2) - 585*sqrt(2)*(x^7 + 2*x^5 + x^3)*log(-2
*sqrt(2)*sqrt(x) + 2*x + 2) - 8*(585*x^6 + 1053*x^4 + 416*x^2 - 32)*sqrt(x))/(x^
7 + 2*x^5 + x^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(7/2)/(x**2+1)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.210788, size = 143, normalized size = 0.97 \[ \frac{117}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{117}{64} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) - \frac{117}{128} \, \sqrt{2}{\rm ln}\left (\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{117}{128} \, \sqrt{2}{\rm ln}\left (-\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{21 \, x^{\frac{7}{2}} + 25 \, x^{\frac{3}{2}}}{16 \,{\left (x^{2} + 1\right )}^{2}} + \frac{2 \,{\left (15 \, x^{2} - 1\right )}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 1)^3*x^(7/2)),x, algorithm="giac")

[Out]

117/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 117/64*sqrt(2)*arctan
(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(x))) - 117/128*sqrt(2)*ln(sqrt(2)*sqrt(x) + x +
1) + 117/128*sqrt(2)*ln(-sqrt(2)*sqrt(x) + x + 1) + 1/16*(21*x^(7/2) + 25*x^(3/2
))/(x^2 + 1)^2 + 2/5*(15*x^2 - 1)/x^(5/2)