3.7.56 \(\int \frac {\sqrt {-1+x}}{(1+x^2)^3} \, dx\) [656]

Optimal. Leaf size=272 \[ \frac {\sqrt {-1+x} x}{4 \left (1+x^2\right )^2}-\frac {(1-11 x) \sqrt {-1+x}}{32 \left (1+x^2\right )}-\frac {1}{64} \sqrt {\frac {1}{2} \left (-527+373 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )}-2 \sqrt {-1+x}}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )+\frac {1}{64} \sqrt {\frac {1}{2} \left (-527+373 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )}+2 \sqrt {-1+x}}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )-\frac {1}{128} \sqrt {\frac {1}{2} \left (527+373 \sqrt {2}\right )} \log \left (1-\sqrt {2}-\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {-1+x}-x\right )+\frac {1}{128} \sqrt {\frac {1}{2} \left (527+373 \sqrt {2}\right )} \log \left (1-\sqrt {2}+\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {-1+x}-x\right ) \]

[Out]

1/4*x*(-1+x)^(1/2)/(x^2+1)^2-1/32*(1-11*x)*(-1+x)^(1/2)/(x^2+1)-1/128*arctan((-2*(-1+x)^(1/2)+(-2+2*2^(1/2))^(
1/2))/(2+2*2^(1/2))^(1/2))*(-1054+746*2^(1/2))^(1/2)+1/128*arctan((2*(-1+x)^(1/2)+(-2+2*2^(1/2))^(1/2))/(2+2*2
^(1/2))^(1/2))*(-1054+746*2^(1/2))^(1/2)-1/256*ln(1-x-2^(1/2)-(-1+x)^(1/2)*(-2+2*2^(1/2))^(1/2))*(1054+746*2^(
1/2))^(1/2)+1/256*ln(1-x-2^(1/2)+(-1+x)^(1/2)*(-2+2*2^(1/2))^(1/2))*(1054+746*2^(1/2))^(1/2)

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Rubi [A]
time = 0.24, antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {751, 837, 841, 1183, 648, 632, 210, 642} \begin {gather*} -\frac {1}{64} \sqrt {\frac {1}{2} \left (373 \sqrt {2}-527\right )} \text {ArcTan}\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 )} \text {ArcTan}\left (\frac {2 \sqrt {x-1}+\sqrt {2 \left (\sqrt {2}-1\right )}}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )-\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 ) \end {gather*}

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)) - (Sqrt[(-527 + 373*Sqrt[2])/2]*Ar
cTan[(Sqrt[2*(-1 + Sqrt[2])] - 2*Sqrt[-1 + x])/Sqrt[2*(1 + Sqrt[2])]])/64 + (Sqrt[(-527 + 373*Sqrt[2])/2]*ArcT
an[(Sqrt[2*(-1 + Sqrt[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])/128

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 751

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*(d + e*x)^m*((a + c*x^2)^(p +
1)/(2*a*(p + 1))), x] + Dist[1/(2*a*(p + 1)), Int[(d + e*x)^(m - 1)*(d*(2*p + 3) + e*(m + 2*p + 3)*x)*(a + c*x
^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && (LtQ[m, 1
] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 837

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(
m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] +
Dist[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^
2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g},
x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 841

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 1183

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {\sqrt {-1+x}}{\left (1+x^2\right )^3} \, dx &=\frac {\sqrt {-1+x} x}{4 \left (1+x^2\right )^2}-\frac {1}{4} \int \frac {3-\frac {5 x}{2}}{\sqrt {-1+x} \left (1+x^2\right )^2} \, dx\\ &=\frac {\sqrt {-1+x} x}{4 \left (1+x^2\right )^2}-\frac {(1-11 x) \sqrt {-1+x}}{32 \left (1+x^2\right )}+\frac {1}{16} \int \frac {-\frac {25}{4}+\frac {11 x}{4}}{\sqrt {-1+x} \left (1+x^2\right )} \, dx\\ &=\frac {\sqrt {-1+x} x}{4 \left (1+x^2\right )^2}-\frac {(1-11 x) \sqrt {-1+x}}{32 \left (1+x^2\right )}+\frac {1}{8} \text {Subst}\left (\int \frac {-\frac {7}{2}+\frac {11 x^2}{4}}{2+2 x^2+x^4} \, dx,x,\sqrt {-1+x}\right )\\ &=\frac {\sqrt {-1+x} x}{4 \left (1+x^2\right )^2}-\frac {(1-11 x) \sqrt {-1+x}}{32 \left (1+x^2\right )}+\frac {\text {Subst}\left (\int \frac {-7 \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )}-\left (-\frac {7}{2}-\frac {11}{2 \sqrt {2}}\right ) x}{\sqrt {2}-\sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {-1+x}\right )}{32 \sqrt {-1+\sqrt {2}}}+\frac {\text {Subst}\left (\int \frac {-7 \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )}+\left (-\frac {7}{2}-\frac {11}{2 \sqrt {2}}\right ) x}{\sqrt {2}+\sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {-1+x}\right )}{32 \sqrt {-1+\sqrt {2}}}\\ &=\frac {\sqrt {-1+x} x}{4 \left (1+x^2\right )^2}-\frac {(1-11 x) \sqrt {-1+x}}{32 \left (1+x^2\right )}+\frac {1}{128} \sqrt {219-154 \sqrt {2}} \text {Subst}\left (\int \frac {1}{\sqrt {2}-\sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {-1+x}\right )+\frac {1}{128} \sqrt {219-154 \sqrt {2}} \text {Subst}\left (\int \frac {1}{\sqrt {2}+\sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {-1+x}\right )+\frac {\left (14+11 \sqrt {2}\right ) \text {Subst}\left (\int \frac {-\sqrt {2 \left (-1+\sqrt {2}\right )}+2 x}{\sqrt {2}-\sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {-1+x}\right )}{256 \sqrt {-1+\sqrt {2}}}-\frac {\left (14+11 \sqrt {2}\right ) \text {Subst}\left (\int \frac {\sqrt {2 \left (-1+\sqrt {2}\right )}+2 x}{\sqrt {2}+\sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {-1+x}\right )}{256 \sqrt {-1+\sqrt {2}}}\\ &=\frac {\sqrt {-1+x} x}{4 \left (1+x^2\right )^2}-\frac {(1-11 x) \sqrt {-1+x}}{32 \left (1+x^2\right )}-\frac {1}{256} \sqrt {1054+746 \sqrt {2}} \log \left (1-\sqrt {2}-\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {-1+x}-x\right )+\frac {1}{256} \sqrt {1054+746 \sqrt {2}} \log \left (1-\sqrt {2}+\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {-1+x}-x\right )-\frac {1}{64} \sqrt {219-154 \sqrt {2}} \text {Subst}\left (\int \frac {1}{-2 \left (1+\sqrt {2}\right )-x^2} \, dx,x,-\sqrt {2 \left (-1+\sqrt {2}\right )}+2 \sqrt {-1+x}\right )-\frac {1}{64} \sqrt {219-154 \sqrt {2}} \text {Subst}\left (\int \frac {1}{-2 \left (1+\sqrt {2}\right )-x^2} \, dx,x,\sqrt {2 \left (-1+\sqrt {2}\right )}+2 \sqrt {-1+x}\right )\\ &=\frac {\sqrt {-1+x} x}{4 \left (1+x^2\right )^2}-\frac {(1-11 x) \sqrt {-1+x}}{32 \left (1+x^2\right )}-\frac {1}{64} \sqrt {\frac {1}{2} \left (-527+373 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )}-2 \sqrt {-1+x}}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )+\frac {1}{64} \sqrt {\frac {1}{2} \left (-527+373 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )}+2 \sqrt {-1+x}}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )-\frac {1}{256} \sqrt {1054+746 \sqrt {2}} \log \left (1-\sqrt {2}-\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {-1+x}-x\right )+\frac {1}{256} \sqrt {1054+746 \sqrt {2}} \log \left (1-\sqrt {2}+\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {-1+x}-x\right )\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.20, size = 92, normalized size = 0.34 \begin {gather*} \frac {1}{64} \left (\frac {2 \sqrt {-1+x} \left (-1+19 x-x^2+11 x^3\right )}{\left (1+x^2\right )^2}+\sqrt {-527-23 i} \tan ^{-1}\left (\sqrt {\frac {1}{2}-\frac {i}{2}} \sqrt {-1+x}\right )+\sqrt {-527+23 i} \tan ^{-1}\left (\sqrt {\frac {1}{2}+\frac {i}{2}} \sqrt {-1+x}\right )\right ) \end {gather*}

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 + Sqrt[-527 - 23*I]*ArcTan[Sqrt[1/2 - I/2]*Sqrt[-1 +
x]] + Sqrt[-527 + 23*I]*ArcTan[Sqrt[1/2 + I/2]*Sqrt[-1 + x]])/64

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(553\) vs. \(2(188)=376\).
time = 1.90, size = 554, normalized size = 2.04 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x-1)^(1/2)/(x^2+1)^3,x,method=_RETURNVERBOSE)

[Out]

-1/128*(-4/23*(-759-506*2^(1/2))/(-6-4*2^(1/2))*(x-1)^(3/2)-1/23/(-6-4*2^(1/2))*(-5336-3588*2^(1/2))*(-2+2*2^(
1/2))^(1/2)*(x-1)-2/23*(-2392*2^(1/2)-3036)/(-6-4*2^(1/2))*(x-1)^(1/2)-1/46*(-3312*2^(1/2)-4416)*(-2+2*2^(1/2)
)^(1/2)/(-6-4*2^(1/2)))/(x-1+(x-1)^(1/2)*(-2+2*2^(1/2))^(1/2)+2^(1/2))^2-1/128/(3+2*2^(1/2))*(1/2*(104*2^(1/2)
*(-2+2*2^(1/2))^(1/2)+147*(-2+2*2^(1/2))^(1/2))*ln(x-1+(x-1)^(1/2)*(-2+2*2^(1/2))^(1/2)+2^(1/2))+2*(42*2^(1/2)
+56-1/2*(104*2^(1/2)*(-2+2*2^(1/2))^(1/2)+147*(-2+2*2^(1/2))^(1/2))*(-2+2*2^(1/2))^(1/2))/(2+2*2^(1/2))^(1/2)*
arctan((2*(x-1)^(1/2)+(-2+2*2^(1/2))^(1/2))/(2+2*2^(1/2))^(1/2)))+1/128*(4/23*(-759-506*2^(1/2))/(-6-4*2^(1/2)
)*(x-1)^(3/2)-1/23/(-6-4*2^(1/2))*(-5336-3588*2^(1/2))*(-2+2*2^(1/2))^(1/2)*(x-1)+2/23*(-2392*2^(1/2)-3036)/(-
6-4*2^(1/2))*(x-1)^(1/2)-1/46*(-3312*2^(1/2)-4416)*(-2+2*2^(1/2))^(1/2)/(-6-4*2^(1/2)))/(x-1-(x-1)^(1/2)*(-2+2
*2^(1/2))^(1/2)+2^(1/2))^2+1/128/(3+2*2^(1/2))*(-1/2*(-104*2^(1/2)*(-2+2*2^(1/2))^(1/2)-147*(-2+2*2^(1/2))^(1/
2))*ln(x-1-(x-1)^(1/2)*(-2+2*2^(1/2))^(1/2)+2^(1/2))-2*(42*2^(1/2)+56+1/2*(-104*2^(1/2)*(-2+2*2^(1/2))^(1/2)-1
47*(-2+2*2^(1/2))^(1/2))*(-2+2*2^(1/2))^(1/2))/(2+2*2^(1/2))^(1/2)*arctan((2*(x-1)^(1/2)-(-2+2*2^(1/2))^(1/2))
/(2+2*2^(1/2))^(1/2)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)^(1/2)/(x^2+1)^3,x, algorithm="maxima")

[Out]

integrate(sqrt(x - 1)/(x^2 + 1)^3, x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 436 vs. \(2 (191) = 382\).
time = 2.44, size = 436, normalized size = 1.60 \begin {gather*} -\frac {92 \cdot 278258^{\frac {1}{4}} \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} \sqrt {-393142 \, \sqrt {2} + 556516} \arctan \left (\frac {1}{109810067572} \cdot 278258^{\frac {3}{4}} \sqrt {46} \sqrt {373 \cdot 278258^{\frac {1}{4}} \sqrt {x - 1} {\left (11 \, \sqrt {2} + 14\right )} \sqrt {-393142 \, \sqrt {2} + 556516} + 6399934 \, x + 6399934 \, \sqrt {2} - 6399934} {\left (7 \, \sqrt {2} + 11\right )} \sqrt {-393142 \, \sqrt {2} + 556516} - \frac {1}{6399934} \cdot 278258^{\frac {3}{4}} \sqrt {x - 1} {\left (7 \, \sqrt {2} + 11\right )} \sqrt {-393142 \, \sqrt {2} + 556516} - \sqrt {2} + 1\right ) + 92 \cdot 278258^{\frac {1}{4}} \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} \sqrt {-393142 \, \sqrt {2} + 556516} \arctan \left (\frac {1}{109810067572} \cdot 278258^{\frac {3}{4}} \sqrt {46} \sqrt {-373 \cdot 278258^{\frac {1}{4}} \sqrt {x - 1} {\left (11 \, \sqrt {2} + 14\right )} \sqrt {-393142 \, \sqrt {2} + 556516} + 6399934 \, x + 6399934 \, \sqrt {2} - 6399934} {\left (7 \, \sqrt {2} + 11\right )} \sqrt {-393142 \, \sqrt {2} + 556516} - \frac {1}{6399934} \cdot 278258^{\frac {3}{4}} \sqrt {x - 1} {\left (7 \, \sqrt {2} + 11\right )} \sqrt {-393142 \, \sqrt {2} + 556516} + \sqrt {2} - 1\right ) + 278258^{\frac {1}{4}} {\left (746 \, x^{4} + 1492 \, x^{2} + 527 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} + 746\right )} \sqrt {-393142 \, \sqrt {2} + 556516} \log \left (\frac {373}{46} \cdot 278258^{\frac {1}{4}} \sqrt {x - 1} {\left (11 \, \sqrt {2} + 14\right )} \sqrt {-393142 \, \sqrt {2} + 556516} + 139129 \, x + 139129 \, \sqrt {2} - 139129\right ) - 278258^{\frac {1}{4}} {\left (746 \, x^{4} + 1492 \, x^{2} + 527 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} + 746\right )} \sqrt {-393142 \, \sqrt {2} + 556516} \log \left (-\frac {373}{46} \cdot 278258^{\frac {1}{4}} \sqrt {x - 1} {\left (11 \, \sqrt {2} + 14\right )} \sqrt {-393142 \, \sqrt {2} + 556516} + 139129 \, x + 139129 \, \sqrt {2} - 139129\right ) - 137264 \, {\left (11 \, x^{3} - x^{2} + 19 \, x - 1\right )} \sqrt {x - 1}}{4392448 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)^(1/2)/(x^2+1)^3,x, algorithm="fricas")

[Out]

-1/4392448*(92*278258^(1/4)*sqrt(2)*(x^4 + 2*x^2 + 1)*sqrt(-393142*sqrt(2) + 556516)*arctan(1/109810067572*278
258^(3/4)*sqrt(46)*sqrt(373*278258^(1/4)*sqrt(x - 1)*(11*sqrt(2) + 14)*sqrt(-393142*sqrt(2) + 556516) + 639993
4*x + 6399934*sqrt(2) - 6399934)*(7*sqrt(2) + 11)*sqrt(-393142*sqrt(2) + 556516) - 1/6399934*278258^(3/4)*sqrt
(x - 1)*(7*sqrt(2) + 11)*sqrt(-393142*sqrt(2) + 556516) - sqrt(2) + 1) + 92*278258^(1/4)*sqrt(2)*(x^4 + 2*x^2
+ 1)*sqrt(-393142*sqrt(2) + 556516)*arctan(1/109810067572*278258^(3/4)*sqrt(46)*sqrt(-373*278258^(1/4)*sqrt(x
- 1)*(11*sqrt(2) + 14)*sqrt(-393142*sqrt(2) + 556516) + 6399934*x + 6399934*sqrt(2) - 6399934)*(7*sqrt(2) + 11
)*sqrt(-393142*sqrt(2) + 556516) - 1/6399934*278258^(3/4)*sqrt(x - 1)*(7*sqrt(2) + 11)*sqrt(-393142*sqrt(2) +
556516) + sqrt(2) - 1) + 278258^(1/4)*(746*x^4 + 1492*x^2 + 527*sqrt(2)*(x^4 + 2*x^2 + 1) + 746)*sqrt(-393142*
sqrt(2) + 556516)*log(373/46*278258^(1/4)*sqrt(x - 1)*(11*sqrt(2) + 14)*sqrt(-393142*sqrt(2) + 556516) + 13912
9*x + 139129*sqrt(2) - 139129) - 278258^(1/4)*(746*x^4 + 1492*x^2 + 527*sqrt(2)*(x^4 + 2*x^2 + 1) + 746)*sqrt(
-393142*sqrt(2) + 556516)*log(-373/46*278258^(1/4)*sqrt(x - 1)*(11*sqrt(2) + 14)*sqrt(-393142*sqrt(2) + 556516
) + 139129*x + 139129*sqrt(2) - 139129) - 137264*(11*x^3 - x^2 + 19*x - 1)*sqrt(x - 1))/(x^4 + 2*x^2 + 1)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1185 vs. \(2 (216) = 432\).
time = 163.70, size = 1185, normalized size = 4.36 \begin {gather*} \frac {22 \left (x - 1\right )^{\frac {7}{2}}}{512 x + 64 \left (x - 1\right )^{4} + 256 \left (x - 1\right )^{3} + 512 \left (x - 1\right )^{2} - 256} + \frac {64 \left (x - 1\right )^{\frac {5}{2}}}{512 x + 64 \left (x - 1\right )^{4} + 256 \left (x - 1\right )^{3} + 512 \left (x - 1\right )^{2} - 256} + \frac {100 \left (x - 1\right )^{\frac {3}{2}}}{512 x + 64 \left (x - 1\right )^{4} + 256 \left (x - 1\right )^{3} + 512 \left (x - 1\right )^{2} - 256} + \frac {56 \sqrt {x - 1}}{512 x + 64 \left (x - 1\right )^{4} + 256 \left (x - 1\right )^{3} + 512 \left (x - 1\right )^{2} - 256} + 2 \sqrt {\frac {527}{131072} + \frac {373 \sqrt {2}}{131072}} \log {\left (x - \frac {144 \sqrt {527 + 373 \sqrt {2}} \sqrt {x - 1}}{23} - \frac {19133 \sqrt {2} \sqrt {527 + 373 \sqrt {2}} \sqrt {x - 1}}{8579} + \frac {54 \sqrt {2} \sqrt {527 + 373 \sqrt {2}} \sqrt {393142 \sqrt {2} + 555987} \sqrt {x - 1}}{8579} - \frac {3432348 \sqrt {2} \sqrt {393142 \sqrt {2} + 555987}}{197317} - \frac {1808046973 \sqrt {393142 \sqrt {2} + 555987}}{73599241} + \frac {1907845445314}{73599241} + \frac {3617091686 \sqrt {2}}{197317} \right )} - 2 \sqrt {\frac {527}{131072} + \frac {373 \sqrt {2}}{131072}} \log {\left (x - \frac {54 \sqrt {2} \sqrt {527 + 373 \sqrt {2}} \sqrt {393142 \sqrt {2} + 555987} \sqrt {x - 1}}{8579} + \frac {19133 \sqrt {2} \sqrt {527 + 373 \sqrt {2}} \sqrt {x - 1}}{8579} + \frac {144 \sqrt {527 + 373 \sqrt {2}} \sqrt {x - 1}}{23} - \frac {3432348 \sqrt {2} \sqrt {393142 \sqrt {2} + 555987}}{197317} - \frac {1808046973 \sqrt {393142 \sqrt {2} + 555987}}{73599241} + \frac {1907845445314}{73599241} + \frac {3617091686 \sqrt {2}}{197317} \right )} + 4 \sqrt {- \frac {\sqrt {393142 \sqrt {2} + 555987}}{65536} + \frac {527}{131072} + \frac {1119 \sqrt {2}}{131072}} \operatorname {atan}{\left (\frac {8579 \sqrt {2} \sqrt {x - 1}}{- 161 \sqrt {- 2 \sqrt {393142 \sqrt {2} + 555987} + 527 + 1119 \sqrt {2}} + 18 \sqrt {393142 \sqrt {2} + 555987} \sqrt {- 2 \sqrt {393142 \sqrt {2} + 555987} + 527 + 1119 \sqrt {2}}} - \frac {26856 \sqrt {2} \sqrt {527 + 373 \sqrt {2}}}{- 161 \sqrt {- 2 \sqrt {393142 \sqrt {2} + 555987} + 527 + 1119 \sqrt {2}} + 18 \sqrt {393142 \sqrt {2} + 555987} \sqrt {- 2 \sqrt {393142 \sqrt {2} + 555987} + 527 + 1119 \sqrt {2}}} - \frac {19133 \sqrt {527 + 373 \sqrt {2}}}{- 161 \sqrt {- 2 \sqrt {393142 \sqrt {2} + 555987} + 527 + 1119 \sqrt {2}} + 18 \sqrt {393142 \sqrt {2} + 555987} \sqrt {- 2 \sqrt {393142 \sqrt {2} + 555987} + 527 + 1119 \sqrt {2}}} + \frac {54 \sqrt {527 + 373 \sqrt {2}} \sqrt {393142 \sqrt {2} + 555987}}{- 161 \sqrt {- 2 \sqrt {393142 \sqrt {2} + 555987} + 527 + 1119 \sqrt {2}} + 18 \sqrt {393142 \sqrt {2} + 555987} \sqrt {- 2 \sqrt {393142 \sqrt {2} + 555987} + 527 + 1119 \sqrt {2}}} \right )} + 4 \sqrt {- \frac {\sqrt {393142 \sqrt {2} + 555987}}{65536} + \frac {527}{131072} + \frac {1119 \sqrt {2}}{131072}} \operatorname {atan}{\left (\frac {8579 \sqrt {2} \sqrt {x - 1}}{- 161 \sqrt {- 2 \sqrt {393142 \sqrt {2} + 555987} + 527 + 1119 \sqrt {2}} + 18 \sqrt {393142 \sqrt {2} + 555987} \sqrt {- 2 \sqrt {393142 \sqrt {2} + 555987} + 527 + 1119 \sqrt {2}}} - \frac {54 \sqrt {527 + 373 \sqrt {2}} \sqrt {393142 \sqrt {2} + 555987}}{- 161 \sqrt {- 2 \sqrt {393142 \sqrt {2} + 555987} + 527 + 1119 \sqrt {2}} + 18 \sqrt {393142 \sqrt {2} + 555987} \sqrt {- 2 \sqrt {393142 \sqrt {2} + 555987} + 527 + 1119 \sqrt {2}}} + \frac {19133 \sqrt {527 + 373 \sqrt {2}}}{- 161 \sqrt {- 2 \sqrt {393142 \sqrt {2} + 555987} + 527 + 1119 \sqrt {2}} + 18 \sqrt {393142 \sqrt {2} + 555987} \sqrt {- 2 \sqrt {393142 \sqrt {2} + 555987} + 527 + 1119 \sqrt {2}}} + \frac {26856 \sqrt {2} \sqrt {527 + 373 \sqrt {2}}}{- 161 \sqrt {- 2 \sqrt {393142 \sqrt {2} + 555987} + 527 + 1119 \sqrt {2}} + 18 \sqrt {393142 \sqrt {2} + 555987} \sqrt {- 2 \sqrt {393142 \sqrt {2} + 555987} + 527 + 1119 \sqrt {2}}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)**(1/2)/(x**2+1)**3,x)

[Out]

22*(x - 1)**(7/2)/(512*x + 64*(x - 1)**4 + 256*(x - 1)**3 + 512*(x - 1)**2 - 256) + 64*(x - 1)**(5/2)/(512*x +
64*(x - 1)**4 + 256*(x - 1)**3 + 512*(x - 1)**2 - 256) + 100*(x - 1)**(3/2)/(512*x + 64*(x - 1)**4 + 256*(x -
1)**3 + 512*(x - 1)**2 - 256) + 56*sqrt(x - 1)/(512*x + 64*(x - 1)**4 + 256*(x - 1)**3 + 512*(x - 1)**2 - 256
) + 2*sqrt(527/131072 + 373*sqrt(2)/131072)*log(x - 144*sqrt(527 + 373*sqrt(2))*sqrt(x - 1)/23 - 19133*sqrt(2)
*sqrt(527 + 373*sqrt(2))*sqrt(x - 1)/8579 + 54*sqrt(2)*sqrt(527 + 373*sqrt(2))*sqrt(393142*sqrt(2) + 555987)*s
qrt(x - 1)/8579 - 3432348*sqrt(2)*sqrt(393142*sqrt(2) + 555987)/197317 - 1808046973*sqrt(393142*sqrt(2) + 5559
87)/73599241 + 1907845445314/73599241 + 3617091686*sqrt(2)/197317) - 2*sqrt(527/131072 + 373*sqrt(2)/131072)*l
og(x - 54*sqrt(2)*sqrt(527 + 373*sqrt(2))*sqrt(393142*sqrt(2) + 555987)*sqrt(x - 1)/8579 + 19133*sqrt(2)*sqrt(
527 + 373*sqrt(2))*sqrt(x - 1)/8579 + 144*sqrt(527 + 373*sqrt(2))*sqrt(x - 1)/23 - 3432348*sqrt(2)*sqrt(393142
*sqrt(2) + 555987)/197317 - 1808046973*sqrt(393142*sqrt(2) + 555987)/73599241 + 1907845445314/73599241 + 36170
91686*sqrt(2)/197317) + 4*sqrt(-sqrt(393142*sqrt(2) + 555987)/65536 + 527/131072 + 1119*sqrt(2)/131072)*atan(8
579*sqrt(2)*sqrt(x - 1)/(-161*sqrt(-2*sqrt(393142*sqrt(2) + 555987) + 527 + 1119*sqrt(2)) + 18*sqrt(393142*sqr
t(2) + 555987)*sqrt(-2*sqrt(393142*sqrt(2) + 555987) + 527 + 1119*sqrt(2))) - 26856*sqrt(2)*sqrt(527 + 373*sqr
t(2))/(-161*sqrt(-2*sqrt(393142*sqrt(2) + 555987) + 527 + 1119*sqrt(2)) + 18*sqrt(393142*sqrt(2) + 555987)*sqr
t(-2*sqrt(393142*sqrt(2) + 555987) + 527 + 1119*sqrt(2))) - 19133*sqrt(527 + 373*sqrt(2))/(-161*sqrt(-2*sqrt(3
93142*sqrt(2) + 555987) + 527 + 1119*sqrt(2)) + 18*sqrt(393142*sqrt(2) + 555987)*sqrt(-2*sqrt(393142*sqrt(2) +
555987) + 527 + 1119*sqrt(2))) + 54*sqrt(527 + 373*sqrt(2))*sqrt(393142*sqrt(2) + 555987)/(-161*sqrt(-2*sqrt(
393142*sqrt(2) + 555987) + 527 + 1119*sqrt(2)) + 18*sqrt(393142*sqrt(2) + 555987)*sqrt(-2*sqrt(393142*sqrt(2)
+ 555987) + 527 + 1119*sqrt(2)))) + 4*sqrt(-sqrt(393142*sqrt(2) + 555987)/65536 + 527/131072 + 1119*sqrt(2)/13
1072)*atan(8579*sqrt(2)*sqrt(x - 1)/(-161*sqrt(-2*sqrt(393142*sqrt(2) + 555987) + 527 + 1119*sqrt(2)) + 18*sqr
t(393142*sqrt(2) + 555987)*sqrt(-2*sqrt(393142*sqrt(2) + 555987) + 527 + 1119*sqrt(2))) - 54*sqrt(527 + 373*sq
rt(2))*sqrt(393142*sqrt(2) + 555987)/(-161*sqrt(-2*sqrt(393142*sqrt(2) + 555987) + 527 + 1119*sqrt(2)) + 18*sq
rt(393142*sqrt(2) + 555987)*sqrt(-2*sqrt(393142*sqrt(2) + 555987) + 527 + 1119*sqrt(2))) + 19133*sqrt(527 + 37
3*sqrt(2))/(-161*sqrt(-2*sqrt(393142*sqrt(2) + 555987) + 527 + 1119*sqrt(2)) + 18*sqrt(393142*sqrt(2) + 555987
)*sqrt(-2*sqrt(393142*sqrt(2) + 555987) + 527 + 1119*sqrt(2))) + 26856*sqrt(2)*sqrt(527 + 373*sqrt(2))/(-161*s
qrt(-2*sqrt(393142*sqrt(2) + 555987) + 527 + 1119*sqrt(2)) + 18*sqrt(393142*sqrt(2) + 555987)*sqrt(-2*sqrt(393
142*sqrt(2) + 555987) + 527 + 1119*sqrt(2))))

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Giac [A]
time = 2.06, size = 206, normalized size = 0.76 \begin {gather*} \frac {1}{128} \, \sqrt {746 \, \sqrt {2} - 1054} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {-\sqrt {2} + 2} + 2 \, \sqrt {x - 1}\right )}}{2 \, \sqrt {\sqrt {2} + 2}}\right ) + \frac {1}{128} \, \sqrt {746 \, \sqrt {2} - 1054} \arctan \left (-\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {-\sqrt {2} + 2} - 2 \, \sqrt {x - 1}\right )}}{2 \, \sqrt {\sqrt {2} + 2}}\right ) - \frac {1}{256} \, \sqrt {746 \, \sqrt {2} + 1054} \log \left (2^{\frac {1}{4}} \sqrt {x - 1} \sqrt {-\sqrt {2} + 2} + x + \sqrt {2} - 1\right ) + \frac {1}{256} \, \sqrt {746 \, \sqrt {2} + 1054} \log \left (-2^{\frac {1}{4}} \sqrt {x - 1} \sqrt {-\sqrt {2} + 2} + x + \sqrt {2} - 1\right ) + \frac {11 \, {\left (x - 1\right )}^{\frac {7}{2}} + 32 \, {\left (x - 1\right )}^{\frac {5}{2}} + 50 \, {\left (x - 1\right )}^{\frac {3}{2}} + 28 \, \sqrt {x - 1}}{32 \, {\left ({\left (x - 1\right )}^{2} + 2 \, x\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)^(1/2)/(x^2+1)^3,x, algorithm="giac")

[Out]

1/128*sqrt(746*sqrt(2) - 1054)*arctan(1/2*2^(3/4)*(2^(1/4)*sqrt(-sqrt(2) + 2) + 2*sqrt(x - 1))/sqrt(sqrt(2) +
2)) + 1/128*sqrt(746*sqrt(2) - 1054)*arctan(-1/2*2^(3/4)*(2^(1/4)*sqrt(-sqrt(2) + 2) - 2*sqrt(x - 1))/sqrt(sqr
t(2) + 2)) - 1/256*sqrt(746*sqrt(2) + 1054)*log(2^(1/4)*sqrt(x - 1)*sqrt(-sqrt(2) + 2) + x + sqrt(2) - 1) + 1/
256*sqrt(746*sqrt(2) + 1054)*log(-2^(1/4)*sqrt(x - 1)*sqrt(-sqrt(2) + 2) + x + sqrt(2) - 1) + 1/32*(11*(x - 1)
^(7/2) + 32*(x - 1)^(5/2) + 50*(x - 1)^(3/2) + 28*sqrt(x - 1))/((x - 1)^2 + 2*x)^2

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Mupad [B]
time = 0.12, size = 440, normalized size = 1.62 \begin {gather*} \mathrm {atanh}\left (\frac {275\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}\,\sqrt {x-1}}{64\,\left (28\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}-\frac {207}{4096}\right )}+\frac {275\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}\,\sqrt {x-1}}{64\,\left (28\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}-\frac {207}{4096}\right )}+\frac {373\,\sqrt {2}\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}\,\sqrt {x-1}}{128\,\left (28\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}-\frac {207}{4096}\right )}-\frac {373\,\sqrt {2}\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}\,\sqrt {x-1}}{128\,\left (28\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}-\frac {207}{4096}\right )}\right )\,\left (2\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}+2\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}\right )-\mathrm {atanh}\left (\frac {275\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}\,\sqrt {x-1}}{64\,\left (28\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}+\frac {207}{4096}\right )}-\frac {275\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}\,\sqrt {x-1}}{64\,\left (28\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}+\frac {207}{4096}\right )}+\frac {373\,\sqrt {2}\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}\,\sqrt {x-1}}{128\,\left (28\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}+\frac {207}{4096}\right )}+\frac {373\,\sqrt {2}\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}\,\sqrt {x-1}}{128\,\left (28\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}+\frac {207}{4096}\right )}\right )\,\left (2\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}-2\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}\right )+\frac {\frac {7\,\sqrt {x-1}}{8}+\frac {25\,{\left (x-1\right )}^{3/2}}{16}+{\left (x-1\right )}^{5/2}+\frac {11\,{\left (x-1\right )}^{7/2}}{32}}{8\,x+8\,{\left (x-1\right )}^2+4\,{\left (x-1\right )}^3+{\left (x-1\right )}^4-4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 1)^(1/2)/(x^2 + 1)^3,x)

[Out]

atanh((275*(527/32768 - (373*2^(1/2))/32768)^(1/2)*(x - 1)^(1/2))/(64*(28*(527/32768 - (373*2^(1/2))/32768)^(1
/2)*((373*2^(1/2))/32768 + 527/32768)^(1/2) - 207/4096)) + (275*((373*2^(1/2))/32768 + 527/32768)^(1/2)*(x - 1
)^(1/2))/(64*(28*(527/32768 - (373*2^(1/2))/32768)^(1/2)*((373*2^(1/2))/32768 + 527/32768)^(1/2) - 207/4096))
+ (373*2^(1/2)*(527/32768 - (373*2^(1/2))/32768)^(1/2)*(x - 1)^(1/2))/(128*(28*(527/32768 - (373*2^(1/2))/3276
8)^(1/2)*((373*2^(1/2))/32768 + 527/32768)^(1/2) - 207/4096)) - (373*2^(1/2)*((373*2^(1/2))/32768 + 527/32768)
^(1/2)*(x - 1)^(1/2))/(128*(28*(527/32768 - (373*2^(1/2))/32768)^(1/2)*((373*2^(1/2))/32768 + 527/32768)^(1/2)
- 207/4096)))*(2*(527/32768 - (373*2^(1/2))/32768)^(1/2) + 2*((373*2^(1/2))/32768 + 527/32768)^(1/2)) - atanh
((275*(527/32768 - (373*2^(1/2))/32768)^(1/2)*(x - 1)^(1/2))/(64*(28*(527/32768 - (373*2^(1/2))/32768)^(1/2)*(
(373*2^(1/2))/32768 + 527/32768)^(1/2) + 207/4096)) - (275*((373*2^(1/2))/32768 + 527/32768)^(1/2)*(x - 1)^(1/
2))/(64*(28*(527/32768 - (373*2^(1/2))/32768)^(1/2)*((373*2^(1/2))/32768 + 527/32768)^(1/2) + 207/4096)) + (37
3*2^(1/2)*(527/32768 - (373*2^(1/2))/32768)^(1/2)*(x - 1)^(1/2))/(128*(28*(527/32768 - (373*2^(1/2))/32768)^(1
/2)*((373*2^(1/2))/32768 + 527/32768)^(1/2) + 207/4096)) + (373*2^(1/2)*((373*2^(1/2))/32768 + 527/32768)^(1/2
)*(x - 1)^(1/2))/(128*(28*(527/32768 - (373*2^(1/2))/32768)^(1/2)*((373*2^(1/2))/32768 + 527/32768)^(1/2) + 20
7/4096)))*(2*(527/32768 - (373*2^(1/2))/32768)^(1/2) - 2*((373*2^(1/2))/32768 + 527/32768)^(1/2)) + ((7*(x - 1
)^(1/2))/8 + (25*(x - 1)^(3/2))/16 + (x - 1)^(5/2) + (11*(x - 1)^(7/2))/32)/(8*x + 8*(x - 1)^2 + 4*(x - 1)^3 +
(x - 1)^4 - 4)

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