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

Optimal result

Integrand size = 15, antiderivative size = 272 \[ \int \frac {\sqrt {-1+x}}{\left (1+x^2\right )^3} \, dx=\frac {\sqrt {-1+x} x}{4 \left (1+x^2\right )^2}-\frac {(1-11 x) \sqrt {-1+x}}{32 \left (1+x^2\right )}-\frac {1}{64} \sqrt {\frac {1}{2} \left (-527+373 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )}-2 \sqrt {-1+x}}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )+\frac {1}{64} \sqrt {\frac {1}{2} \left (-527+373 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )}+2 \sqrt {-1+x}}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )-\frac {1}{128} \sqrt {\frac {1}{2} \left (527+373 \sqrt {2}\right )} \log \left (1-\sqrt {2}-\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {-1+x}-x\right )+\frac {1}{128} \sqrt {\frac {1}{2} \left (527+373 \sqrt {2}\right )} \log \left (1-\sqrt {2}+\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {-1+x}-x\right ) \]

[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)

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {751, 837, 841, 1183, 648, 632, 210, 642} \[ \int \frac {\sqrt {-1+x}}{\left (1+x^2\right )^3} \, dx=-\frac {1}{64} \sqrt {\frac {1}{2} \left (373 \sqrt {2}-527\right )} \arctan \left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )}-2 \sqrt {x-1}}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )+\frac {1}{64} \sqrt {\frac {1}{2} \left (373 \sqrt {2}-527\right )} \arctan \left (\frac {2 \sqrt {x-1}+\sqrt {2 \left (\sqrt {2}-1\right )}}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )-\frac {\sqrt {x-1} (1-11 x)}{32 \left (x^2+1\right )}+\frac {\sqrt {x-1} x}{4 \left (x^2+1\right )^2}-\frac {1}{128} \sqrt {\frac {1}{2} \left (527+373 \sqrt {2}\right )} \log \left (-x-\sqrt {2 \left (\sqrt {2}-1\right )} \sqrt {x-1}-\sqrt {2}+1\right )+\frac {1}{128} \sqrt {\frac {1}{2} \left (527+373 \sqrt {2}\right )} \log \left (-x+\sqrt {2 \left (\sqrt {2}-1\right )} \sqrt {x-1}-\sqrt {2}+1\right ) \]

[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*} \text {integral}& = \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*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.33 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.34 \[ \int \frac {\sqrt {-1+x}}{\left (1+x^2\right )^3} \, dx=\frac {1}{64} \left (\frac {2 \sqrt {-1+x} \left (-1+19 x-x^2+11 x^3\right )}{\left (1+x^2\right )^2}+\sqrt {-527-23 i} \arctan \left (\sqrt {\frac {1}{2}-\frac {i}{2}} \sqrt {-1+x}\right )+\sqrt {-527+23 i} \arctan \left (\sqrt {\frac {1}{2}+\frac {i}{2}} \sqrt {-1+x}\right )\right ) \]

[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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 2.54 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.63

method	result	size
trager	\(\frac {\left (11 x^{3}-x^{2}+19 x -1\right ) \sqrt {-1+x}}{32 \left (x^{2}+1\right )^{2}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+16 \operatorname {RootOf}\left (128 \textit {\_Z}^{4}-8432 \textit {\_Z}^{2}+139129\right )^{2}-1054\right ) \ln \left (-\frac {3008 \operatorname {RootOf}\left (128 \textit {\_Z}^{4}-8432 \textit {\_Z}^{2}+139129\right )^{4} x \operatorname {RootOf}\left (\textit {\_Z}^{2}+16 \operatorname {RootOf}\left (128 \textit {\_Z}^{4}-8432 \textit {\_Z}^{2}+139129\right )^{2}-1054\right )-185916 \operatorname {RootOf}\left (128 \textit {\_Z}^{4}-8432 \textit {\_Z}^{2}+139129\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+16 \operatorname {RootOf}\left (128 \textit {\_Z}^{4}-8432 \textit {\_Z}^{2}+139129\right )^{2}-1054\right ) x -21620 \operatorname {RootOf}\left (128 \textit {\_Z}^{4}-8432 \textit {\_Z}^{2}+139129\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+16 \operatorname {RootOf}\left (128 \textit {\_Z}^{4}-8432 \textit {\_Z}^{2}+139129\right )^{2}-1054\right )-411792 \sqrt {-1+x}\, \operatorname {RootOf}\left (128 \textit {\_Z}^{4}-8432 \textit {\_Z}^{2}+139129\right )^{2}+2870608 \operatorname {RootOf}\left (\textit {\_Z}^{2}+16 \operatorname {RootOf}\left (128 \textit {\_Z}^{4}-8432 \textit {\_Z}^{2}+139129\right )^{2}-1054\right ) x +686320 \operatorname {RootOf}\left (\textit {\_Z}^{2}+16 \operatorname {RootOf}\left (128 \textit {\_Z}^{4}-8432 \textit {\_Z}^{2}+139129\right )^{2}-1054\right )[phone] \sqrt {-1+x}}{16 \operatorname {RootOf}\left (128 \textit {\_Z}^{4}-8432 \textit {\_Z}^{2}+139129\right )^{2} x -527 x +23}\right )}{128}+\frac {\operatorname {RootOf}\left (128 \textit {\_Z}^{4}-8432 \textit {\_Z}^{2}+139129\right ) \ln \left (-\frac {12032 \operatorname {RootOf}\left (128 \textit {\_Z}^{4}-8432 \textit {\_Z}^{2}+139129\right )^{5} x -841552 \operatorname {RootOf}\left (128 \textit {\_Z}^{4}-8432 \textit {\_Z}^{2}+139129\right )^{3} x +411792 \sqrt {-1+x}\, \operatorname {RootOf}\left (128 \textit {\_Z}^{4}-8432 \textit {\_Z}^{2}+139129\right )^{2}+86480 \operatorname {RootOf}\left (128 \textit {\_Z}^{4}-8432 \textit {\_Z}^{2}+139129\right )^{3}[phone] \operatorname {RootOf}\left (128 \textit {\_Z}^{4}-8432 \textit {\_Z}^{2}+139129\right ) x -12774131 \sqrt {-1+x}-2951590 \operatorname {RootOf}\left (128 \textit {\_Z}^{4}-8432 \textit {\_Z}^{2}+139129\right )}{16 \operatorname {RootOf}\left (128 \textit {\_Z}^{4}-8432 \textit {\_Z}^{2}+139129\right )^{2} x -527 x -23}\right )}{32}\)	\(444\)
risch	\(\frac {\left (11 x^{3}-x^{2}+19 x -1\right ) \sqrt {-1+x}}{32 \left (x^{2}+1\right )^{2}}+\frac {9 \ln \left (-1+x -\sqrt {-1+x}\, \sqrt {-2+2 \sqrt {2}}+\sqrt {2}\right ) \sqrt {2}\, \sqrt {-2+2 \sqrt {2}}}{128}+\frac {25 \ln \left (-1+x -\sqrt {-1+x}\, \sqrt {-2+2 \sqrt {2}}+\sqrt {2}\right ) \sqrt {-2+2 \sqrt {2}}}{256}+\frac {9 \arctan \left (\frac {2 \sqrt {-1+x}-\sqrt {-2+2 \sqrt {2}}}{\sqrt {2+2 \sqrt {2}}}\right ) \left (-2+2 \sqrt {2}\right ) \sqrt {2}}{64 \sqrt {2+2 \sqrt {2}}}+\frac {25 \arctan \left (\frac {2 \sqrt {-1+x}-\sqrt {-2+2 \sqrt {2}}}{\sqrt {2+2 \sqrt {2}}}\right ) \left (-2+2 \sqrt {2}\right )}{128 \sqrt {2+2 \sqrt {2}}}-\frac {7 \arctan \left (\frac {2 \sqrt {-1+x}-\sqrt {-2+2 \sqrt {2}}}{\sqrt {2+2 \sqrt {2}}}\right ) \sqrt {2}}{32 \sqrt {2+2 \sqrt {2}}}-\frac {9 \ln \left (-1+x +\sqrt {-1+x}\, \sqrt {-2+2 \sqrt {2}}+\sqrt {2}\right ) \sqrt {2}\, \sqrt {-2+2 \sqrt {2}}}{128}-\frac {25 \ln \left (-1+x +\sqrt {-1+x}\, \sqrt {-2+2 \sqrt {2}}+\sqrt {2}\right ) \sqrt {-2+2 \sqrt {2}}}{256}+\frac {9 \arctan \left (\frac {2 \sqrt {-1+x}+\sqrt {-2+2 \sqrt {2}}}{\sqrt {2+2 \sqrt {2}}}\right ) \left (-2+2 \sqrt {2}\right ) \sqrt {2}}{64 \sqrt {2+2 \sqrt {2}}}+\frac {25 \arctan \left (\frac {2 \sqrt {-1+x}+\sqrt {-2+2 \sqrt {2}}}{\sqrt {2+2 \sqrt {2}}}\right ) \left (-2+2 \sqrt {2}\right )}{128 \sqrt {2+2 \sqrt {2}}}-\frac {7 \arctan \left (\frac {2 \sqrt {-1+x}+\sqrt {-2+2 \sqrt {2}}}{\sqrt {2+2 \sqrt {2}}}\right ) \sqrt {2}}{32 \sqrt {2+2 \sqrt {2}}}\)	\(451\)
derivativedivides	\(-\frac {-\frac {4 \left (-759-506 \sqrt {2}\right ) \left (-1+x \right )^{\frac {3}{2}}}{23 \left (-6-4 \sqrt {2}\right )}-\frac {\left (-5336-3588 \sqrt {2}\right ) \sqrt {-2+2 \sqrt {2}}\, \left (-1+x \right )}{23 \left (-6-4 \sqrt {2}\right )}-\frac {2 \left (-2392 \sqrt {2}-3036\right ) \sqrt {-1+x}}{23 \left (-6-4 \sqrt {2}\right )}-\frac {\left (-3312 \sqrt {2}-4416\right ) \sqrt {-2+2 \sqrt {2}}}{46 \left (-6-4 \sqrt {2}\right )}}{128 \left (-1+x +\sqrt {-1+x}\, \sqrt {-2+2 \sqrt {2}}+\sqrt {2}\right )^{2}}-\frac {\frac {\left (104 \sqrt {2}\, \sqrt {-2+2 \sqrt {2}}+147 \sqrt {-2+2 \sqrt {2}}\right ) \ln \left (-1+x +\sqrt {-1+x}\, \sqrt {-2+2 \sqrt {2}}+\sqrt {2}\right )}{2}+\frac {2 \left (42 \sqrt {2}+56-\frac {\left (104 \sqrt {2}\, \sqrt {-2+2 \sqrt {2}}+147 \sqrt {-2+2 \sqrt {2}}\right ) \sqrt {-2+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 \sqrt {-1+x}+\sqrt {-2+2 \sqrt {2}}}{\sqrt {2+2 \sqrt {2}}}\right )}{\sqrt {2+2 \sqrt {2}}}}{128 \left (3+2 \sqrt {2}\right )}+\frac {\frac {4 \left (-759-506 \sqrt {2}\right ) \left (-1+x \right )^{\frac {3}{2}}}{23 \left (-6-4 \sqrt {2}\right )}-\frac {\left (-5336-3588 \sqrt {2}\right ) \sqrt {-2+2 \sqrt {2}}\, \left (-1+x \right )}{23 \left (-6-4 \sqrt {2}\right )}+\frac {2 \left (-2392 \sqrt {2}-3036\right ) \sqrt {-1+x}}{23 \left (-6-4 \sqrt {2}\right )}-\frac {\left (-3312 \sqrt {2}-4416\right ) \sqrt {-2+2 \sqrt {2}}}{46 \left (-6-4 \sqrt {2}\right )}}{128 \left (-1+x -\sqrt {-1+x}\, \sqrt {-2+2 \sqrt {2}}+\sqrt {2}\right )^{2}}+\frac {\frac {\left (104 \sqrt {2}\, \sqrt {-2+2 \sqrt {2}}+147 \sqrt {-2+2 \sqrt {2}}\right ) \ln \left (-1+x -\sqrt {-1+x}\, \sqrt {-2+2 \sqrt {2}}+\sqrt {2}\right )}{2}+\frac {2 \left (-42 \sqrt {2}-56+\frac {\left (104 \sqrt {2}\, \sqrt {-2+2 \sqrt {2}}+147 \sqrt {-2+2 \sqrt {2}}\right ) \sqrt {-2+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 \sqrt {-1+x}-\sqrt {-2+2 \sqrt {2}}}{\sqrt {2+2 \sqrt {2}}}\right )}{\sqrt {2+2 \sqrt {2}}}}{384+256 \sqrt {2}}\)	\(554\)
default	\(-\frac {-\frac {4 \left (-759-506 \sqrt {2}\right ) \left (-1+x \right )^{\frac {3}{2}}}{23 \left (-6-4 \sqrt {2}\right )}-\frac {\left (-5336-3588 \sqrt {2}\right ) \sqrt {-2+2 \sqrt {2}}\, \left (-1+x \right )}{23 \left (-6-4 \sqrt {2}\right )}-\frac {2 \left (-2392 \sqrt {2}-3036\right ) \sqrt {-1+x}}{23 \left (-6-4 \sqrt {2}\right )}-\frac {\left (-3312 \sqrt {2}-4416\right ) \sqrt {-2+2 \sqrt {2}}}{46 \left (-6-4 \sqrt {2}\right )}}{128 \left (-1+x +\sqrt {-1+x}\, \sqrt {-2+2 \sqrt {2}}+\sqrt {2}\right )^{2}}-\frac {\frac {\left (104 \sqrt {2}\, \sqrt {-2+2 \sqrt {2}}+147 \sqrt {-2+2 \sqrt {2}}\right ) \ln \left (-1+x +\sqrt {-1+x}\, \sqrt {-2+2 \sqrt {2}}+\sqrt {2}\right )}{2}+\frac {2 \left (42 \sqrt {2}+56-\frac {\left (104 \sqrt {2}\, \sqrt {-2+2 \sqrt {2}}+147 \sqrt {-2+2 \sqrt {2}}\right ) \sqrt {-2+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 \sqrt {-1+x}+\sqrt {-2+2 \sqrt {2}}}{\sqrt {2+2 \sqrt {2}}}\right )}{\sqrt {2+2 \sqrt {2}}}}{128 \left (3+2 \sqrt {2}\right )}+\frac {\frac {4 \left (-759-506 \sqrt {2}\right ) \left (-1+x \right )^{\frac {3}{2}}}{23 \left (-6-4 \sqrt {2}\right )}-\frac {\left (-5336-3588 \sqrt {2}\right ) \sqrt {-2+2 \sqrt {2}}\, \left (-1+x \right )}{23 \left (-6-4 \sqrt {2}\right )}+\frac {2 \left (-2392 \sqrt {2}-3036\right ) \sqrt {-1+x}}{23 \left (-6-4 \sqrt {2}\right )}-\frac {\left (-3312 \sqrt {2}-4416\right ) \sqrt {-2+2 \sqrt {2}}}{46 \left (-6-4 \sqrt {2}\right )}}{128 \left (-1+x -\sqrt {-1+x}\, \sqrt {-2+2 \sqrt {2}}+\sqrt {2}\right )^{2}}+\frac {\frac {\left (104 \sqrt {2}\, \sqrt {-2+2 \sqrt {2}}+147 \sqrt {-2+2 \sqrt {2}}\right ) \ln \left (-1+x -\sqrt {-1+x}\, \sqrt {-2+2 \sqrt {2}}+\sqrt {2}\right )}{2}+\frac {2 \left (-42 \sqrt {2}-56+\frac {\left (104 \sqrt {2}\, \sqrt {-2+2 \sqrt {2}}+147 \sqrt {-2+2 \sqrt {2}}\right ) \sqrt {-2+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 \sqrt {-1+x}-\sqrt {-2+2 \sqrt {2}}}{\sqrt {2+2 \sqrt {2}}}\right )}{\sqrt {2+2 \sqrt {2}}}}{384+256 \sqrt {2}}\)	\(554\)

[In]

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

[Out]

1/32*(11*x^3-x^2+19*x-1)/(x^2+1)^2*(-1+x)^(1/2)+1/128*RootOf(_Z^2+16*RootOf(128*_Z^4-8432*_Z^2+139129)^2-1054)
*ln(-(3008*RootOf(128*_Z^4-8432*_Z^2+139129)^4*x*RootOf(_Z^2+16*RootOf(128*_Z^4-8432*_Z^2+139129)^2-1054)-1859
16*RootOf(128*_Z^4-8432*_Z^2+139129)^2*RootOf(_Z^2+16*RootOf(128*_Z^4-8432*_Z^2+139129)^2-1054)*x-21620*RootOf
(128*_Z^4-8432*_Z^2+139129)^2*RootOf(_Z^2+16*RootOf(128*_Z^4-8432*_Z^2+139129)^2-1054)-411792*(-1+x)^(1/2)*Roo
tOf(128*_Z^4-8432*_Z^2+139129)^2+2870608*RootOf(_Z^2+16*RootOf(128*_Z^4-8432*_Z^2+139129)^2-1054)*x+686320*Roo
tOf(_Z^2+16*RootOf(128*_Z^4-8432*_Z^2+139129)^2-1054)[phone]*(-1+x)^(1/2))/(16*RootOf(128*_Z^4-8432*_Z^2+139
129)^2*x-527*x+23))+1/32*RootOf(128*_Z^4-8432*_Z^2+139129)*ln(-(12032*RootOf(128*_Z^4-8432*_Z^2+139129)^5*x-84
1552*RootOf(128*_Z^4-8432*_Z^2+139129)^3*x+411792*(-1+x)^(1/2)*RootOf(128*_Z^4-8432*_Z^2+139129)^2+86480*RootO
f(128*_Z^4-8432*_Z^2+139129)^3+14706618*RootOf(128*_Z^4-8432*_Z^2+139129)*x-12774131*(-1+x)^(1/2)-2951590*Root
Of(128*_Z^4-8432*_Z^2+139129))/(16*RootOf(128*_Z^4-8432*_Z^2+139129)^2*x-527*x-23))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.84 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.56 \[ \int \frac {\sqrt {-1+x}}{\left (1+x^2\right )^3} \, dx=-\frac {\sqrt {23 i + 527} {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (-\left (18 i - 7\right ) \, \sqrt {23 i + 527} + 373 \, \sqrt {x - 1}\right ) - \sqrt {23 i + 527} {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (\left (18 i - 7\right ) \, \sqrt {23 i + 527} + 373 \, \sqrt {x - 1}\right ) + \sqrt {-23 i + 527} {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (\left (18 i + 7\right ) \, \sqrt {-23 i + 527} + 373 \, \sqrt {x - 1}\right ) - \sqrt {-23 i + 527} {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (-\left (18 i + 7\right ) \, \sqrt {-23 i + 527} + 373 \, \sqrt {x - 1}\right ) - 4 \, {\left (11 \, x^{3} - x^{2} + 19 \, x - 1\right )} \sqrt {x - 1}}{128 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \]

[In]

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

[Out]

-1/128*(sqrt(23*I + 527)*(x^4 + 2*x^2 + 1)*log(-(18*I - 7)*sqrt(23*I + 527) + 373*sqrt(x - 1)) - sqrt(23*I + 5
27)*(x^4 + 2*x^2 + 1)*log((18*I - 7)*sqrt(23*I + 527) + 373*sqrt(x - 1)) + sqrt(-23*I + 527)*(x^4 + 2*x^2 + 1)
*log((18*I + 7)*sqrt(-23*I + 527) + 373*sqrt(x - 1)) - sqrt(-23*I + 527)*(x^4 + 2*x^2 + 1)*log(-(18*I + 7)*sqr
t(-23*I + 527) + 373*sqrt(x - 1)) - 4*(11*x^3 - x^2 + 19*x - 1)*sqrt(x - 1))/(x^4 + 2*x^2 + 1)

Sympy [F]

\[ \int \frac {\sqrt {-1+x}}{\left (1+x^2\right )^3} \, dx=\int \frac {\sqrt {x - 1}}{\left (x^{2} + 1\right )^{3}}\, dx \]

[In]

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

[Out]

Integral(sqrt(x - 1)/(x**2 + 1)**3, x)

Maxima [F]

\[ \int \frac {\sqrt {-1+x}}{\left (1+x^2\right )^3} \, dx=\int { \frac {\sqrt {x - 1}}{{\left (x^{2} + 1\right )}^{3}} \,d x } \]

[In]

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

[Out]

integrate(sqrt(x - 1)/(x^2 + 1)^3, x)

Giac [A] (verification not implemented)

none

Time = 0.68 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {-1+x}}{\left (1+x^2\right )^3} \, dx=\frac {1}{128} \, \sqrt {746 \, \sqrt {2} - 1054} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {-\sqrt {2} + 2} + 2 \, \sqrt {x - 1}\right )}}{2 \, \sqrt {\sqrt {2} + 2}}\right ) + \frac {1}{128} \, \sqrt {746 \, \sqrt {2} - 1054} \arctan \left (-\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {-\sqrt {2} + 2} - 2 \, \sqrt {x - 1}\right )}}{2 \, \sqrt {\sqrt {2} + 2}}\right ) - \frac {1}{256} \, \sqrt {746 \, \sqrt {2} + 1054} \log \left (2^{\frac {1}{4}} \sqrt {x - 1} \sqrt {-\sqrt {2} + 2} + x + \sqrt {2} - 1\right ) + \frac {1}{256} \, \sqrt {746 \, \sqrt {2} + 1054} \log \left (-2^{\frac {1}{4}} \sqrt {x - 1} \sqrt {-\sqrt {2} + 2} + x + \sqrt {2} - 1\right ) + \frac {11 \, {\left (x - 1\right )}^{\frac {7}{2}} + 32 \, {\left (x - 1\right )}^{\frac {5}{2}} + 50 \, {\left (x - 1\right )}^{\frac {3}{2}} + 28 \, \sqrt {x - 1}}{32 \, {\left ({\left (x - 1\right )}^{2} + 2 \, x\right )}^{2}} \]

[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

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.62 \[ \int \frac {\sqrt {-1+x}}{\left (1+x^2\right )^3} \, dx=\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} \]

[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)