3.645 \(\int \frac{\sqrt{-1+x}}{\left (1+x^2\right )^3} \, dx\)
Optimal. Leaf size=272 \[ -\frac{\sqrt{x-1} (1-11 x)}{32 \left (x^2+1\right )}+\frac{\sqrt{x-1} x}{4 \left (x^2+1\right )^2}-\frac{1}{128} \sqrt{\frac{1}{2} \left (527+373 \sqrt{2}\right )} \log \left (-x-\sqrt{2 \left (\sqrt{2}-1\right )} \sqrt{x-1}-\sqrt{2}+1\right )+\frac{1}{128} \sqrt{\frac{1}{2} \left (527+373 \sqrt{2}\right )} \log \left (-x+\sqrt{2 \left (\sqrt{2}-1\right )} \sqrt{x-1}-\sqrt{2}+1\right )-\frac{1}{64} \sqrt{\frac{1}{2} \left (373 \sqrt{2}-527\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{2}-1\right )}-2 \sqrt{x-1}}{\sqrt{2 \left (1+\sqrt{2}\right )}}\right )+\frac{1}{64} \sqrt{\frac{1}{2} \left (373 \sqrt{2}-527\right )} \tan ^{-1}\left (\frac{2 \sqrt{x-1}+\sqrt{2 \left (\sqrt{2}-1\right )}}{\sqrt{2 \left (1+\sqrt{2}\right )}}\right ) \]
[Out]
(Sqrt[-1 + x]*x)/(4*(1 + x^2)^2) - ((1 - 11*x)*Sqrt[-1 + x])/(32*(1 + x^2)) - (S
qrt[(-527 + 373*Sqrt[2])/2]*ArcTan[(Sqrt[2*(-1 + Sqrt[2])] - 2*Sqrt[-1 + x])/Sqr
t[2*(1 + Sqrt[2])]])/64 + (Sqrt[(-527 + 373*Sqrt[2])/2]*ArcTan[(Sqrt[2*(-1 + Sqr
t[2])] + 2*Sqrt[-1 + x])/Sqrt[2*(1 + Sqrt[2])]])/64 - (Sqrt[(527 + 373*Sqrt[2])/
2]*Log[1 - Sqrt[2] - Sqrt[2*(-1 + Sqrt[2])]*Sqrt[-1 + x] - x])/128 + (Sqrt[(527
+ 373*Sqrt[2])/2]*Log[1 - Sqrt[2] + Sqrt[2*(-1 + Sqrt[2])]*Sqrt[-1 + x] - x])/12
8
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Rubi [A] time = 0.827618, antiderivative size = 272, normalized size of antiderivative = 1.,
number of steps used = 12, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533
\[ -\frac{\sqrt{x-1} (1-11 x)}{32 \left (x^2+1\right )}+\frac{\sqrt{x-1} x}{4 \left (x^2+1\right )^2}-\frac{1}{128} \sqrt{\frac{1}{2} \left (527+373 \sqrt{2}\right )} \log \left (-x-\sqrt{2 \left (\sqrt{2}-1\right )} \sqrt{x-1}-\sqrt{2}+1\right )+\frac{1}{128} \sqrt{\frac{1}{2} \left (527+373 \sqrt{2}\right )} \log \left (-x+\sqrt{2 \left (\sqrt{2}-1\right )} \sqrt{x-1}-\sqrt{2}+1\right )-\frac{1}{64} \sqrt{\frac{1}{2} \left (373 \sqrt{2}-527\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{2}-1\right )}-2 \sqrt{x-1}}{\sqrt{2 \left (1+\sqrt{2}\right )}}\right )+\frac{1}{64} \sqrt{\frac{1}{2} \left (373 \sqrt{2}-527\right )} \tan ^{-1}\left (\frac{2 \sqrt{x-1}+\sqrt{2 \left (\sqrt{2}-1\right )}}{\sqrt{2 \left (1+\sqrt{2}\right )}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[-1 + x]/(1 + x^2)^3,x]
[Out]
(Sqrt[-1 + x]*x)/(4*(1 + x^2)^2) - ((1 - 11*x)*Sqrt[-1 + x])/(32*(1 + x^2)) - (S
qrt[(-527 + 373*Sqrt[2])/2]*ArcTan[(Sqrt[2*(-1 + Sqrt[2])] - 2*Sqrt[-1 + x])/Sqr
t[2*(1 + Sqrt[2])]])/64 + (Sqrt[(-527 + 373*Sqrt[2])/2]*ArcTan[(Sqrt[2*(-1 + Sqr
t[2])] + 2*Sqrt[-1 + x])/Sqrt[2*(1 + Sqrt[2])]])/64 - (Sqrt[(527 + 373*Sqrt[2])/
2]*Log[1 - Sqrt[2] - Sqrt[2*(-1 + Sqrt[2])]*Sqrt[-1 + x] - x])/128 + (Sqrt[(527
+ 373*Sqrt[2])/2]*Log[1 - Sqrt[2] + Sqrt[2*(-1 + Sqrt[2])]*Sqrt[-1 + x] - x])/12
8
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Rubi in Sympy [A] time = 64.7219, size = 374, normalized size = 1.38 \[ \frac{x \sqrt{x - 1}}{4 \left (x^{2} + 1\right )^{2}} - \frac{\left (- \frac{11 x}{2} + \frac{1}{2}\right ) \sqrt{x - 1}}{16 \left (x^{2} + 1\right )} + \frac{\left (\frac{7}{4} + \frac{11 \sqrt{2}}{8}\right ) \log{\left (x - \sqrt{2} \sqrt{-1 + \sqrt{2}} \sqrt{x - 1} - 1 + \sqrt{2} \right )}}{32 \sqrt{-1 + \sqrt{2}}} - \frac{\left (\frac{7}{4} + \frac{11 \sqrt{2}}{8}\right ) \log{\left (x + \sqrt{2} \sqrt{-1 + \sqrt{2}} \sqrt{x - 1} - 1 + \sqrt{2} \right )}}{32 \sqrt{-1 + \sqrt{2}}} - \frac{\sqrt{2} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{2}} \left (\frac{7}{2} + \frac{11 \sqrt{2}}{4}\right )}{2} + \frac{7 \sqrt{2} \sqrt{-1 + \sqrt{2}}}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (\sqrt{x - 1} - \frac{\sqrt{-2 + 2 \sqrt{2}}}{2}\right )}{\sqrt{1 + \sqrt{2}}} \right )}}{32 \sqrt{-1 + \sqrt{2}} \sqrt{1 + \sqrt{2}}} - \frac{\sqrt{2} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{2}} \left (\frac{7}{2} + \frac{11 \sqrt{2}}{4}\right )}{2} + \frac{7 \sqrt{2} \sqrt{-1 + \sqrt{2}}}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (\sqrt{x - 1} + \frac{\sqrt{-2 + 2 \sqrt{2}}}{2}\right )}{\sqrt{1 + \sqrt{2}}} \right )}}{32 \sqrt{-1 + \sqrt{2}} \sqrt{1 + \sqrt{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-1+x)**(1/2)/(x**2+1)**3,x)
[Out]
x*sqrt(x - 1)/(4*(x**2 + 1)**2) - (-11*x/2 + 1/2)*sqrt(x - 1)/(16*(x**2 + 1)) +
(7/4 + 11*sqrt(2)/8)*log(x - sqrt(2)*sqrt(-1 + sqrt(2))*sqrt(x - 1) - 1 + sqrt(2
))/(32*sqrt(-1 + sqrt(2))) - (7/4 + 11*sqrt(2)/8)*log(x + sqrt(2)*sqrt(-1 + sqrt
(2))*sqrt(x - 1) - 1 + sqrt(2))/(32*sqrt(-1 + sqrt(2))) - sqrt(2)*(-sqrt(2)*sqrt
(-1 + sqrt(2))*(7/2 + 11*sqrt(2)/4)/2 + 7*sqrt(2)*sqrt(-1 + sqrt(2))/2)*atan(sqr
t(2)*(sqrt(x - 1) - sqrt(-2 + 2*sqrt(2))/2)/sqrt(1 + sqrt(2)))/(32*sqrt(-1 + sqr
t(2))*sqrt(1 + sqrt(2))) - sqrt(2)*(-sqrt(2)*sqrt(-1 + sqrt(2))*(7/2 + 11*sqrt(2
)/4)/2 + 7*sqrt(2)*sqrt(-1 + sqrt(2))/2)*atan(sqrt(2)*(sqrt(x - 1) + sqrt(-2 + 2
*sqrt(2))/2)/sqrt(1 + sqrt(2)))/(32*sqrt(-1 + sqrt(2))*sqrt(1 + sqrt(2)))
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Mathematica [C] time = 0.155955, size = 90, normalized size = 0.33 \[ \frac{1}{64} \left (\frac{2 \sqrt{x-1} \left (11 x^3-x^2+19 x-1\right )}{\left (x^2+1\right )^2}-(7-18 i) \sqrt{1-i} \tan ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{1-i}}\right )-(7+18 i) \sqrt{1+i} \tan ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{1+i}}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[-1 + x]/(1 + x^2)^3,x]
[Out]
((2*Sqrt[-1 + x]*(-1 + 19*x - x^2 + 11*x^3))/(1 + x^2)^2 - (7 - 18*I)*Sqrt[1 - I
]*ArcTan[Sqrt[-1 + x]/Sqrt[1 - I]] - (7 + 18*I)*Sqrt[1 + I]*ArcTan[Sqrt[-1 + x]/
Sqrt[1 + I]])/64
_______________________________________________________________________________________
Maple [B] time = 0.402, size = 953, normalized size = 3.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-1+x)^(1/2)/(x^2+1)^3,x)
[Out]
1/128*(4/23*(-759-506*2^(1/2))/(-6-4*2^(1/2))*(-1+x)^(3/2)-1/23/(-6-4*2^(1/2))*(
-5336-3588*2^(1/2))*(-2+2*2^(1/2))^(1/2)*(-1+x)+2/23*(-2392*2^(1/2)-3036)/(-6-4*
2^(1/2))*(-1+x)^(1/2)-1/46*(-3312*2^(1/2)-4416)*(-2+2*2^(1/2))^(1/2)/(-6-4*2^(1/
2)))/(x-1-(-1+x)^(1/2)*(-2+2*2^(1/2))^(1/2)+2^(1/2))^2+13/32/(3+2*2^(1/2))*ln(-(
3+2*2^(1/2))*(1-x-2^(1/2)+(-1+x)^(1/2)*(-2+2*2^(1/2))^(1/2)))*2^(1/2)*(-2+2*2^(1
/2))^(1/2)+147/256/(3+2*2^(1/2))*ln(-(3+2*2^(1/2))*(1-x-2^(1/2)+(-1+x)^(1/2)*(-2
+2*2^(1/2))^(1/2)))*(-2+2*2^(1/2))^(1/2)-21/32/(58*2^(1/2)+82)^(1/2)*arctan((2*(
3+2*2^(1/2))*(-1+x)^(1/2)-(3+2*2^(1/2))*(-2+2*2^(1/2))^(1/2))/(58*2^(1/2)+82)^(1
/2))*2^(1/2)-7/8/(58*2^(1/2)+82)^(1/2)*arctan((2*(3+2*2^(1/2))*(-1+x)^(1/2)-(3+2
*2^(1/2))*(-2+2*2^(1/2))^(1/2))/(58*2^(1/2)+82)^(1/2))+13/16/(58*2^(1/2)+82)^(1/
2)*arctan((2*(3+2*2^(1/2))*(-1+x)^(1/2)-(3+2*2^(1/2))*(-2+2*2^(1/2))^(1/2))/(58*
2^(1/2)+82)^(1/2))*(-2+2*2^(1/2))*2^(1/2)+147/128/(58*2^(1/2)+82)^(1/2)*arctan((
2*(3+2*2^(1/2))*(-1+x)^(1/2)-(3+2*2^(1/2))*(-2+2*2^(1/2))^(1/2))/(58*2^(1/2)+82)
^(1/2))*(-2+2*2^(1/2))-1/128*(-4/23*(-759-506*2^(1/2))/(-6-4*2^(1/2))*(-1+x)^(3/
2)-1/23/(-6-4*2^(1/2))*(-5336-3588*2^(1/2))*(-2+2*2^(1/2))^(1/2)*(-1+x)-2/23*(-2
392*2^(1/2)-3036)/(-6-4*2^(1/2))*(-1+x)^(1/2)-1/46*(-3312*2^(1/2)-4416)*(-2+2*2^
(1/2))^(1/2)/(-6-4*2^(1/2)))/(x-1+(-1+x)^(1/2)*(-2+2*2^(1/2))^(1/2)+2^(1/2))^2-1
3/32/(3+2*2^(1/2))*ln((3+2*2^(1/2))*(x-1+(-1+x)^(1/2)*(-2+2*2^(1/2))^(1/2)+2^(1/
2)))*2^(1/2)*(-2+2*2^(1/2))^(1/2)-147/256/(3+2*2^(1/2))*ln((3+2*2^(1/2))*(x-1+(-
1+x)^(1/2)*(-2+2*2^(1/2))^(1/2)+2^(1/2)))*(-2+2*2^(1/2))^(1/2)-21/32/(58*2^(1/2)
+82)^(1/2)*arctan((2*(3+2*2^(1/2))*(-1+x)^(1/2)+(3+2*2^(1/2))*(-2+2*2^(1/2))^(1/
2))/(58*2^(1/2)+82)^(1/2))*2^(1/2)-7/8/(58*2^(1/2)+82)^(1/2)*arctan((2*(3+2*2^(1
/2))*(-1+x)^(1/2)+(3+2*2^(1/2))*(-2+2*2^(1/2))^(1/2))/(58*2^(1/2)+82)^(1/2))+13/
16/(58*2^(1/2)+82)^(1/2)*arctan((2*(3+2*2^(1/2))*(-1+x)^(1/2)+(3+2*2^(1/2))*(-2+
2*2^(1/2))^(1/2))/(58*2^(1/2)+82)^(1/2))*(-2+2*2^(1/2))*2^(1/2)+147/128/(58*2^(1
/2)+82)^(1/2)*arctan((2*(3+2*2^(1/2))*(-1+x)^(1/2)+(3+2*2^(1/2))*(-2+2*2^(1/2))^
(1/2))/(58*2^(1/2)+82)^(1/2))*(-2+2*2^(1/2))
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x - 1}}{{\left (x^{2} + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x - 1)/(x^2 + 1)^3,x, algorithm="maxima")
[Out]
integrate(sqrt(x - 1)/(x^2 + 1)^3, x)
_______________________________________________________________________________________
Fricas [A] time = 0.244523, size = 953, normalized size = 3.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x - 1)/(x^2 + 1)^3,x, algorithm="fricas")
[Out]
1/95488*sqrt(746)*(4*sqrt(746)*(5797*x^3 - 527*x^2 + 373*sqrt(2)*(11*x^3 - x^2 +
19*x - 1) + 10013*x - 527)*sqrt(x - 1)*sqrt((527*sqrt(2) + 746)/(393142*sqrt(2)
+ 555987)) - 92*278258^(1/4)*(x^4 + 2*x^2 + 1)*arctan(373*278258^(1/4)*(18*sqrt
(2) + 25)/(sqrt(746)*sqrt(373)*(373*sqrt(2) + 527)*sqrt((278258^(1/4)*sqrt(746)*
sqrt(x - 1)*(8664282402*sqrt(2) + 12253145681)*sqrt((527*sqrt(2) + 746)/(393142*
sqrt(2) + 555987)) + 218685827213*sqrt(2)*(x - 1) + 373*sqrt(2)*(586289081*sqrt(
2) + 829137970) + 309268462810*x - 309268462810)/(586289081*sqrt(2) + 829137970)
)*sqrt((527*sqrt(2) + 746)/(393142*sqrt(2) + 555987)) + 373*sqrt(746)*sqrt(x - 1
)*(373*sqrt(2) + 527)*sqrt((527*sqrt(2) + 746)/(393142*sqrt(2) + 555987)) + 373*
278258^(1/4)*(7*sqrt(2) + 11))) - 92*278258^(1/4)*(x^4 + 2*x^2 + 1)*arctan(373*2
78258^(1/4)*(18*sqrt(2) + 25)/(sqrt(746)*sqrt(373)*(373*sqrt(2) + 527)*sqrt(-(27
8258^(1/4)*sqrt(746)*sqrt(x - 1)*(8664282402*sqrt(2) + 12253145681)*sqrt((527*sq
rt(2) + 746)/(393142*sqrt(2) + 555987)) - 218685827213*sqrt(2)*(x - 1) - 373*sqr
t(2)*(586289081*sqrt(2) + 829137970) - 309268462810*x + 309268462810)/(586289081
*sqrt(2) + 829137970))*sqrt((527*sqrt(2) + 746)/(393142*sqrt(2) + 555987)) + 373
*sqrt(746)*sqrt(x - 1)*(373*sqrt(2) + 527)*sqrt((527*sqrt(2) + 746)/(393142*sqrt
(2) + 555987)) - 373*278258^(1/4)*(7*sqrt(2) + 11))) - 278258^(1/4)*(527*x^4 + 1
054*x^2 + 373*sqrt(2)*(x^4 + 2*x^2 + 1) + 527)*log(373*(278258^(1/4)*sqrt(746)*s
qrt(x - 1)*(8664282402*sqrt(2) + 12253145681)*sqrt((527*sqrt(2) + 746)/(393142*s
qrt(2) + 555987)) + 218685827213*sqrt(2)*(x - 1) + 373*sqrt(2)*(586289081*sqrt(2
) + 829137970) + 309268462810*x - 309268462810)/(586289081*sqrt(2) + 829137970))
+ 278258^(1/4)*(527*x^4 + 1054*x^2 + 373*sqrt(2)*(x^4 + 2*x^2 + 1) + 527)*log(-
373*(278258^(1/4)*sqrt(746)*sqrt(x - 1)*(8664282402*sqrt(2) + 12253145681)*sqrt(
(527*sqrt(2) + 746)/(393142*sqrt(2) + 555987)) - 218685827213*sqrt(2)*(x - 1) -
373*sqrt(2)*(586289081*sqrt(2) + 829137970) - 309268462810*x + 309268462810)/(58
6289081*sqrt(2) + 829137970)))/((527*x^4 + 1054*x^2 + 373*sqrt(2)*(x^4 + 2*x^2 +
1) + 527)*sqrt((527*sqrt(2) + 746)/(393142*sqrt(2) + 555987)))
_______________________________________________________________________________________
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)**(1/2)/(x**2+1)**3,x)
[Out]
Timed out
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x - 1}}{{\left (x^{2} + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x - 1)/(x^2 + 1)^3,x, algorithm="giac")
[Out]
integrate(sqrt(x - 1)/(x^2 + 1)^3, x)