3.259 \(\int \frac {4 x-\sqrt {1-x^2}}{5+\sqrt {1-x^2}} \, dx\)

Optimal. Leaf size=88 \[ -4 \sqrt {1-x^2}+20 \log \left (\sqrt {1-x^2}+5\right )-\frac {25 \tan ^{-1}\left (\frac {5 x}{2 \sqrt {6} \sqrt {1-x^2}}\right )}{2 \sqrt {6}}-x+5 \sin ^{-1}(x)+\frac {25 \tan ^{-1}\left (\frac {x}{2 \sqrt {6}}\right )}{2 \sqrt {6}} \]

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Rubi [A]  time = 0.24, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6742, 1591, 190, 43, 6740, 203, 402, 216, 377} \[ -4 \sqrt {1-x^2}+20 \log \left (\sqrt {1-x^2}+5\right )-\frac {25 \tan ^{-1}\left (\frac {5 x}{2 \sqrt {6} \sqrt {1-x^2}}\right )}{2 \sqrt {6}}-x+5 \sin ^{-1}(x)+\frac {25 \tan ^{-1}\left (\frac {x}{2 \sqrt {6}}\right )}{2 \sqrt {6}} \]

Antiderivative was successfully verified.

[In]

Int[(4*x - Sqrt[1 - x^2])/(5 + Sqrt[1 - x^2]),x]

[Out]

-x - 4*Sqrt[1 - x^2] + 5*ArcSin[x] + (25*ArcTan[x/(2*Sqrt[6])])/(2*Sqrt[6]) - (25*ArcTan[(5*x)/(2*Sqrt[6]*Sqrt
[1 - x^2])])/(2*Sqrt[6]) + 20*Log[5 + Sqrt[1 - x^2]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 190

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

Rule 203

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

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Dist[b/d, Int[(a + b*x^2)^(p - 1), x], x]
- Dist[(b*c - a*d)/d, Int[(a + b*x^2)^(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d,
0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4])

Rule 1591

Int[((a_.) + (b_.)*(Pq_)^(n_.))^(p_.)*(Qr_), x_Symbol] :> With[{q = Expon[Pq, x], r = Expon[Qr, x]}, Dist[Coef
f[Qr, x, r]/(q*Coeff[Pq, x, q]), Subst[Int[(a + b*x^n)^p, x], x, Pq], x] /; EqQ[r, q - 1] && EqQ[Coeff[Qr, x,
r]*D[Pq, x], q*Coeff[Pq, x, q]*Qr]] /; FreeQ[{a, b, n, p}, x] && PolyQ[Pq, x] && PolyQ[Qr, x]

Rule 6740

Int[(v_)/((a_) + (b_.)*(u_)^(n_.)), x_Symbol] :> Int[ExpandIntegrand[PolynomialInSubst[v, u, x]/(a + b*x^n), x
] /. x -> u, x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && PolynomialInQ[v, u, x]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {4 x-\sqrt {1-x^2}}{5+\sqrt {1-x^2}} \, dx &=\int \left (\frac {4 x}{5+\sqrt {1-x^2}}-\frac {\sqrt {1-x^2}}{5+\sqrt {1-x^2}}\right ) \, dx\\ &=4 \int \frac {x}{5+\sqrt {1-x^2}} \, dx-\int \frac {\sqrt {1-x^2}}{5+\sqrt {1-x^2}} \, dx\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{5+\sqrt {x}} \, dx,x,1-x^2\right )\right )-\int \left (1-\frac {5}{5+\sqrt {1-x^2}}\right ) \, dx\\ &=-x-4 \operatorname {Subst}\left (\int \frac {x}{5+x} \, dx,x,\sqrt {1-x^2}\right )+5 \int \frac {1}{5+\sqrt {1-x^2}} \, dx\\ &=-x-4 \operatorname {Subst}\left (\int \left (1-\frac {5}{5+x}\right ) \, dx,x,\sqrt {1-x^2}\right )+5 \int \left (\frac {5}{24+x^2}-\frac {\sqrt {1-x^2}}{24+x^2}\right ) \, dx\\ &=-x-4 \sqrt {1-x^2}+20 \log \left (5+\sqrt {1-x^2}\right )-5 \int \frac {\sqrt {1-x^2}}{24+x^2} \, dx+25 \int \frac {1}{24+x^2} \, dx\\ &=-x-4 \sqrt {1-x^2}+\frac {25 \tan ^{-1}\left (\frac {x}{2 \sqrt {6}}\right )}{2 \sqrt {6}}+20 \log \left (5+\sqrt {1-x^2}\right )+5 \int \frac {1}{\sqrt {1-x^2}} \, dx-125 \int \frac {1}{\sqrt {1-x^2} \left (24+x^2\right )} \, dx\\ &=-x-4 \sqrt {1-x^2}+5 \sin ^{-1}(x)+\frac {25 \tan ^{-1}\left (\frac {x}{2 \sqrt {6}}\right )}{2 \sqrt {6}}+20 \log \left (5+\sqrt {1-x^2}\right )-125 \operatorname {Subst}\left (\int \frac {1}{24+25 x^2} \, dx,x,\frac {x}{\sqrt {1-x^2}}\right )\\ &=-x-4 \sqrt {1-x^2}+5 \sin ^{-1}(x)+\frac {25 \tan ^{-1}\left (\frac {x}{2 \sqrt {6}}\right )}{2 \sqrt {6}}-\frac {25 \tan ^{-1}\left (\frac {5 x}{2 \sqrt {6} \sqrt {1-x^2}}\right )}{2 \sqrt {6}}+20 \log \left (5+\sqrt {1-x^2}\right )\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 137, normalized size = 1.56 \[ -4 \sqrt {1-x^2}+10 \log \left (x^2+24\right )-10 \log \left (\left (x^2+24\right )^2\right )+10 \log \left (\left (x^2+24\right ) \left (-x^2+10 \sqrt {1-x^2}+26\right )\right )+\frac {25 \tan ^{-1}\left (\frac {4 x^2+409 \sqrt {1-x^2} x+96}{10 \sqrt {6} \left (17 x^2-1\right )}\right )}{2 \sqrt {6}}-x+5 \sin ^{-1}(x)+\frac {25 \tan ^{-1}\left (\frac {x}{2 \sqrt {6}}\right )}{2 \sqrt {6}} \]

Antiderivative was successfully verified.

[In]

Integrate[(4*x - Sqrt[1 - x^2])/(5 + Sqrt[1 - x^2]),x]

[Out]

-x - 4*Sqrt[1 - x^2] + 5*ArcSin[x] + (25*ArcTan[x/(2*Sqrt[6])])/(2*Sqrt[6]) + (25*ArcTan[(96 + 4*x^2 + 409*x*S
qrt[1 - x^2])/(10*Sqrt[6]*(-1 + 17*x^2))])/(2*Sqrt[6]) + 10*Log[24 + x^2] - 10*Log[(24 + x^2)^2] + 10*Log[(24
+ x^2)*(26 - x^2 + 10*Sqrt[1 - x^2])]

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IntegrateAlgebraic [A]  time = 0.32, size = 108, normalized size = 1.23 \[ -4 \sqrt {1-x^2}-20 \log \left (\sqrt {1-x^2}-1\right )+20 \log \left (-x^2+4 \sqrt {1-x^2}-4\right )+10 \tan ^{-1}\left (\frac {x}{\sqrt {1-x^2}-1}\right )-\frac {25 \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {1-x^2}-1}\right )}{\sqrt {6}}-x \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(4*x - Sqrt[1 - x^2])/(5 + Sqrt[1 - x^2]),x]

[Out]

-x - 4*Sqrt[1 - x^2] + 10*ArcTan[x/(-1 + Sqrt[1 - x^2])] - (25*ArcTan[(Sqrt[3/2]*x)/(-1 + Sqrt[1 - x^2])])/Sqr
t[6] - 20*Log[-1 + Sqrt[1 - x^2]] + 20*Log[-4 - x^2 + 4*Sqrt[1 - x^2]]

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fricas [B]  time = 0.59, size = 160, normalized size = 1.82 \[ \frac {25}{12} \, \sqrt {6} \arctan \left (\frac {1}{12} \, \sqrt {6} x\right ) + \frac {25}{12} \, \sqrt {6} \arctan \left (\frac {\sqrt {6} \sqrt {-x^{2} + 1} - \sqrt {6}}{2 \, x}\right ) + \frac {25}{12} \, \sqrt {6} \arctan \left (\frac {\sqrt {6} \sqrt {-x^{2} + 1} - \sqrt {6}}{3 \, x}\right ) - x - 4 \, \sqrt {-x^{2} + 1} - 10 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) + 10 \, \log \left (x^{2} + 24\right ) - 10 \, \log \left (-\frac {x^{2} + 6 \, \sqrt {-x^{2} + 1} - 6}{x^{2}}\right ) + 10 \, \log \left (\frac {x^{2} - 4 \, \sqrt {-x^{2} + 1} + 4}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x-(-x^2+1)^(1/2))/(5+(-x^2+1)^(1/2)),x, algorithm="fricas")

[Out]

25/12*sqrt(6)*arctan(1/12*sqrt(6)*x) + 25/12*sqrt(6)*arctan(1/2*(sqrt(6)*sqrt(-x^2 + 1) - sqrt(6))/x) + 25/12*
sqrt(6)*arctan(1/3*(sqrt(6)*sqrt(-x^2 + 1) - sqrt(6))/x) - x - 4*sqrt(-x^2 + 1) - 10*arctan((sqrt(-x^2 + 1) -
1)/x) + 10*log(x^2 + 24) - 10*log(-(x^2 + 6*sqrt(-x^2 + 1) - 6)/x^2) + 10*log((x^2 - 4*sqrt(-x^2 + 1) + 4)/x^2
)

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giac [B]  time = 0.69, size = 135, normalized size = 1.53 \[ \frac {25}{12} \, \sqrt {6} \arctan \left (\frac {1}{12} \, \sqrt {6} x\right ) - \frac {25}{12} \, \sqrt {6} \arctan \left (-\frac {\sqrt {6} {\left (\sqrt {-x^{2} + 1} - 1\right )}}{3 \, x}\right ) - \frac {25}{12} \, \sqrt {6} \arctan \left (-\frac {\sqrt {6} {\left (\sqrt {-x^{2} + 1} - 1\right )}}{2 \, x}\right ) - x - 4 \, \sqrt {-x^{2} + 1} + 5 \, \arcsin \relax (x) + 10 \, \log \left (x^{2} + 24\right ) - 10 \, \log \left (\frac {3 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + 2\right ) + 10 \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x-(-x^2+1)^(1/2))/(5+(-x^2+1)^(1/2)),x, algorithm="giac")

[Out]

25/12*sqrt(6)*arctan(1/12*sqrt(6)*x) - 25/12*sqrt(6)*arctan(-1/3*sqrt(6)*(sqrt(-x^2 + 1) - 1)/x) - 25/12*sqrt(
6)*arctan(-1/2*sqrt(6)*(sqrt(-x^2 + 1) - 1)/x) - x - 4*sqrt(-x^2 + 1) + 5*arcsin(x) + 10*log(x^2 + 24) - 10*lo
g(3*(sqrt(-x^2 + 1) - 1)^2/x^2 + 2) + 10*log(2*(sqrt(-x^2 + 1) - 1)^2/x^2 + 3)

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maple [A]  time = 0.92, size = 82, normalized size = 0.93

method	result	size
default	\(\frac {25 \arctan \left (\frac {x \sqrt {6}}{12}\right ) \sqrt {6}}{12}+10 \ln \left (x^{2}+24\right )-x +5 \arcsin \relax (x )+\frac {25 \sqrt {6}\, \arctan \left (\frac {5 \sqrt {6}\, \sqrt {-x^{2}+1}\, x}{12 \left (x^{2}-1\right )}\right )}{12}-4 \sqrt {-x^{2}+1}+20 \arctanh \left (\frac {\sqrt {-x^{2}+1}}{5}\right )\)	\(82\)
trager	\(-x -4 \sqrt {-x^{2}+1}+5 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right ) \ln \left (\frac {-25 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right )^{2} \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right ) x -24 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right ) \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right )^{2} x +49 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right ) \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right ) \sqrt {-x^{2}+1}+100 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right )^{2} x -8 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right ) \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right ) x -96 x \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right )^{2}-196 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right ) \sqrt {-x^{2}+1}+196 \sqrt {-x^{2}+1}\, \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right )-5 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right ) \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right )+441 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right ) x +392 x \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right )-834 \sqrt {-x^{2}+1}+20 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right )-20 \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right )+68 x +85}{x \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right )-4 x -5}\right )-5 \ln \left (-\frac {25 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right )^{2} \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right ) x -24 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right ) \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right )^{2} x +49 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right ) \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right ) \sqrt {-x^{2}+1}-100 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right )^{2} x +392 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right ) \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right ) x -96 x \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right )^{2}-196 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right ) \sqrt {-x^{2}+1}+196 \sqrt {-x^{2}+1}\, \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right )-5 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right ) \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right )-1159 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right ) x +1144 x \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right )-734 \sqrt {-x^{2}+1}+20 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right )-20 \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right )-2940 x +75}{x \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right )-4 x -5}\right ) \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right )-5 \ln \left (\frac {24 \sqrt {-x^{2}+1}\, \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right )-96 \sqrt {-x^{2}+1}+25 x}{x \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right )-4 x -5}\right ) \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right )-40 \ln \left (-\frac {25 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right )^{2} \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right ) x -24 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right ) \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right )^{2} x +49 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right ) \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right ) \sqrt {-x^{2}+1}-100 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right )^{2} x +392 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right ) \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right ) x -96 x \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right )^{2}-196 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right ) \sqrt {-x^{2}+1}+196 \sqrt {-x^{2}+1}\, \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right )-5 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right ) \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right )-1159 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right ) x +1144 x \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right )-734 \sqrt {-x^{2}+1}+20 \RootOf \left (\textit {\_Z}^{2}+8 \textit {\_Z} +17\right )-20 \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right )-2940 x +75}{x \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right )-4 x -5}\right )+40 \ln \left (\frac {24 \sqrt {-x^{2}+1}\, \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right )-96 \sqrt {-x^{2}+1}+25 x}{x \RootOf \left (24 \textit {\_Z}^{2}-192 \textit {\_Z} +409\right )-4 x -5}\right )\)	\(1033\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x-(-x^2+1)^(1/2))/(5+(-x^2+1)^(1/2)),x,method=_RETURNVERBOSE)

[Out]

25/12*arctan(1/12*x*6^(1/2))*6^(1/2)+10*ln(x^2+24)-x+5*arcsin(x)+25/12*6^(1/2)*arctan(5/12*6^(1/2)*(-x^2+1)^(1
/2)/(x^2-1)*x)-4*(-x^2+1)^(1/2)+20*arctanh(1/5*(-x^2+1)^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -x - 4 \, \sqrt {-x^{2} + 1} + 5 \, \int \frac {1}{\sqrt {x + 1} \sqrt {-x + 1} + 5}\,{d x} + 20 \, \log \left (\sqrt {-x^{2} + 1} + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x-(-x^2+1)^(1/2))/(5+(-x^2+1)^(1/2)),x, algorithm="maxima")

[Out]

-x - 4*sqrt(-x^2 + 1) + 5*integrate(1/(sqrt(x + 1)*sqrt(-x + 1) + 5), x) + 20*log(sqrt(-x^2 + 1) + 5)

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mupad [B]  time = 0.38, size = 159, normalized size = 1.81 \[ 5\,\mathrm {asin}\relax (x)-x-4\,\sqrt {1-x^2}-\frac {\sqrt {24}\,\ln \left (\frac {\frac {2\,\sqrt {6}\,x}{5}+\sqrt {1-x^2}\,1{}\mathrm {i}+\frac {1}{5}{}\mathrm {i}}{x-\sqrt {6}\,2{}\mathrm {i}}\right )\,\left (125+\sqrt {24}\,100{}\mathrm {i}\right )\,1{}\mathrm {i}}{240}-\frac {\sqrt {24}\,\ln \left (\frac {-\frac {\sqrt {24}\,x}{5}+\sqrt {1-x^2}\,1{}\mathrm {i}+\frac {1}{5}{}\mathrm {i}}{x+\sqrt {24}\,1{}\mathrm {i}}\right )\,\left (-125+\sqrt {24}\,100{}\mathrm {i}\right )\,1{}\mathrm {i}}{240}-\frac {\sqrt {24}\,\ln \left (x-\sqrt {6}\,2{}\mathrm {i}\right )\,\left (25+\sqrt {24}\,20{}\mathrm {i}\right )\,1{}\mathrm {i}}{48}-\frac {\sqrt {24}\,\ln \left (x+\sqrt {24}\,1{}\mathrm {i}\right )\,\left (-25+\sqrt {24}\,20{}\mathrm {i}\right )\,1{}\mathrm {i}}{48} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x - (1 - x^2)^(1/2))/((1 - x^2)^(1/2) + 5),x)

[Out]

5*asin(x) - x - 4*(1 - x^2)^(1/2) - (24^(1/2)*log(((2*6^(1/2)*x)/5 + (1 - x^2)^(1/2)*1i + 1i/5)/(x - 6^(1/2)*2
i))*(24^(1/2)*100i + 125)*1i)/240 - (24^(1/2)*log(((1 - x^2)^(1/2)*1i - (24^(1/2)*x)/5 + 1i/5)/(x + 24^(1/2)*1
i))*(24^(1/2)*100i - 125)*1i)/240 - (24^(1/2)*log(x - 6^(1/2)*2i)*(24^(1/2)*20i + 25)*1i)/48 - (24^(1/2)*log(x
+ 24^(1/2)*1i)*(24^(1/2)*20i - 25)*1i)/48

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {4 x - \sqrt {1 - x^{2}}}{\sqrt {1 - x^{2}} + 5}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x-(-x**2+1)**(1/2))/(5+(-x**2+1)**(1/2)),x)

[Out]

Integral((4*x - sqrt(1 - x**2))/(sqrt(1 - x**2) + 5), x)

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