∫ √ ___ 1+x(−1+x2)__ (1+x2)∘ _______ x+√ __

3.30.25 $\int \frac {\sqrt {1+x} (-1+x^2)}{(1+x^2) \sqrt {x+\sqrt {1+x}}} \, dx$

Optimal. Leaf size=337 \[ 8 \text {RootSum}\left [\text {$\#$1}^8+8 \text {$\#$1}^7+12 \text {$\#$1}^6+24 \text {$\#$1}^5+470 \text {$\#$1}^4+120 \text {$\#$1}^3+300 \text {$\#$1}^2+1000 \text {$\#$1}+625\& ,\frac {\text {$\#$1}^5 \log \left (\text {$\#$1}+2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right )+4 \text {$\#$1}^4 \log \left (\text {$\#$1}+2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right )+14 \text {$\#$1}^3 \log \left (\text {$\#$1}+2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right )+20 \text {$\#$1}^2 \log \left (\text {$\#$1}+2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right )+25 \text {$\#$1} \log \left (\text {$\#$1}+2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right )}{\text {$\#$1}^7+7 \text {$\#$1}^6+9 \text {$\#$1}^5+15 \text {$\#$1}^4+235 \text {$\#$1}^3+45 \text {$\#$1}^2+75 \text {$\#$1}+125}\& \right ]+\sqrt {x+1} \sqrt {x+\sqrt {x+1}}-\frac {3}{2} \sqrt {x+\sqrt {x+1}}-\frac {7}{4} \log \left (2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right ) \]

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Rubi [C] time = 1.04, antiderivative size = 420, normalized size of antiderivative = 1.25, number of steps used = 19, number of rules used = 8, integrand size = 33, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.242, Rules used = {6728, 742, 640, 621, 206, 984, 724, 204} \begin {gather*} \sqrt {x+1} \sqrt {x+\sqrt {x+1}}-\frac {3}{2} \sqrt {x+\sqrt {x+1}}-\frac {(1+i) \tan ^{-1}\left (\frac {-2 \left ((-2+2 i)+\sqrt {1-i}\right ) \sqrt {x+1}+4 \sqrt {1-i}+(2-2 i)}{4 \sqrt {(1+i)+(1-i)^{3/2}} \sqrt {x+\sqrt {x+1}}}\right )}{\sqrt {(1+i)+(1-i)^{3/2}}}-\frac {(1+i) \tan ^{-1}\left (\frac {2 \left ((2-2 i)+\sqrt {1-i}\right ) \sqrt {x+1}-4 \sqrt {1-i}+(2-2 i)}{4 \sqrt {(1+i)-(1-i)^{3/2}} \sqrt {x+\sqrt {x+1}}}\right )}{\sqrt {(1+i)-(1-i)^{3/2}}}-\frac {(1-i) \tan ^{-1}\left (\frac {-2 \left ((-2-2 i)+\sqrt {1+i}\right ) \sqrt {x+1}+4 \sqrt {1+i}+(2+2 i)}{4 \sqrt {(1-i)+(1+i)^{3/2}} \sqrt {x+\sqrt {x+1}}}\right )}{\sqrt {(1-i)+(1+i)^{3/2}}}-\frac {(1-i) \tan ^{-1}\left (\frac {2 \left ((2+2 i)+\sqrt {1+i}\right ) \sqrt {x+1}-4 \sqrt {1+i}+(2+2 i)}{4 \sqrt {(1-i)-(1+i)^{3/2}} \sqrt {x+\sqrt {x+1}}}\right )}{\sqrt {(1-i)-(1+i)^{3/2}}}+\frac {7}{4} \tanh ^{-1}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right ) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-3*Sqrt[x + Sqrt[1 + x]])/2 + Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]] - ((1 + I)*ArcTan[((2 - 2*I) + 4*Sqrt[1 - I]

- 2*((-2 + 2*I) + Sqrt[1 - I])*Sqrt[1 + x])/(4*Sqrt[(1 + I) + (1 - I)^(3/2)]*Sqrt[x + Sqrt[1 + x]])])/Sqrt[(1

+ I) + (1 - I)^(3/2)] - ((1 + I)*ArcTan[((2 - 2*I) - 4*Sqrt[1 - I] + 2*((2 - 2*I) + Sqrt[1 - I])*Sqrt[1 + x])/

(4*Sqrt[(1 + I) - (1 - I)^(3/2)]*Sqrt[x + Sqrt[1 + x]])])/Sqrt[(1 + I) - (1 - I)^(3/2)] - ((1 - I)*ArcTan[((2

+ 2*I) + 4*Sqrt[1 + I] - 2*((-2 - 2*I) + Sqrt[1 + I])*Sqrt[1 + x])/(4*Sqrt[(1 - I) + (1 + I)^(3/2)]*Sqrt[x + S

qrt[1 + x]])])/Sqrt[(1 - I) + (1 + I)^(3/2)] - ((1 - I)*ArcTan[((2 + 2*I) - 4*Sqrt[1 + I] + 2*((2 + 2*I) + Sqr

t[1 + I])*Sqrt[1 + x])/(4*Sqrt[(1 - I) - (1 + I)^(3/2)]*Sqrt[x + Sqrt[1 + x]])])/Sqrt[(1 - I) - (1 + I)^(3/2)]

+ (7*ArcTanh[(1 + 2*Sqrt[1 + x])/(2*Sqrt[x + Sqrt[1 + x]])])/4

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /

; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]

/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)

/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +

1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}

, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d

^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,

b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 742

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)

*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2

*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(a + b*x + c*x^2)^p, x], x] /;

FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]

&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,

p, x]

Rule 984

Int[1/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[1/2, Int[1/((a - Rt[-

(a*c), 2]*x)*Sqrt[d + e*x + f*x^2]), x], x] + Dist[1/2, Int[1/((a + Rt[-(a*c), 2]*x)*Sqrt[d + e*x + f*x^2]), x

], x] /; FreeQ[{a, c, d, e, f}, x] && NeQ[e^2 - 4*d*f, 0] && PosQ[-(a*c)]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +

b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {1+x} \left (-1+x^2\right )}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^4 \left (-2+x^2\right )}{\sqrt {-1+x+x^2} \left (2-2 x^2+x^4\right )} \, dx,x,\sqrt {1+x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {x^2}{\sqrt {-1+x+x^2}}-\frac {2 x^2}{\sqrt {-1+x+x^2} \left (2-2 x^2+x^4\right )}\right ) \, dx,x,\sqrt {1+x}\right )\\ &=2 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )-4 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-1+x+x^2} \left (2-2 x^2+x^4\right )} \, dx,x,\sqrt {1+x}\right )\\ &=\sqrt {1+x} \sqrt {x+\sqrt {1+x}}-4 \operatorname {Subst}\left (\int \left (\frac {1-i}{\sqrt {-1+x+x^2} \left ((-2-2 i)+2 x^2\right )}+\frac {1+i}{\sqrt {-1+x+x^2} \left ((-2+2 i)+2 x^2\right )}\right ) \, dx,x,\sqrt {1+x}\right )+\operatorname {Subst}\left (\int \frac {1-\frac {3 x}{2}}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=-\frac {3}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \sqrt {x+\sqrt {1+x}}+\frac {7}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )-(4-4 i) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x+x^2} \left ((-2-2 i)+2 x^2\right )} \, dx,x,\sqrt {1+x}\right )-(4+4 i) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x+x^2} \left ((-2+2 i)+2 x^2\right )} \, dx,x,\sqrt {1+x}\right )\\ &=-\frac {3}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \sqrt {x+\sqrt {1+x}}-(2-2 i) \operatorname {Subst}\left (\int \frac {1}{\left ((-2-2 i)-2 \sqrt {1+i} x\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )-(2-2 i) \operatorname {Subst}\left (\int \frac {1}{\left ((-2-2 i)+2 \sqrt {1+i} x\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )-(2+2 i) \operatorname {Subst}\left (\int \frac {1}{\left ((-2+2 i)-2 \sqrt {1-i} x\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )-(2+2 i) \operatorname {Subst}\left (\int \frac {1}{\left ((-2+2 i)+2 \sqrt {1-i} x\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )+\frac {7}{2} \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1+2 \sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )\\ &=-\frac {3}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \sqrt {x+\sqrt {1+x}}+\frac {7}{4} \tanh ^{-1}\left (\frac {1+2 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right )-(-4-4 i) \operatorname {Subst}\left (\int \frac {1}{(-16-16 i)-16 (1-i)^{3/2}-x^2} \, dx,x,\frac {(2-2 i)+4 \sqrt {1-i}-\left ((-4+4 i)+2 \sqrt {1-i}\right ) \sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )-(-4-4 i) \operatorname {Subst}\left (\int \frac {1}{(-16-16 i)+16 (1-i)^{3/2}-x^2} \, dx,x,\frac {(2-2 i)-4 \sqrt {1-i}-\left ((-4+4 i)-2 \sqrt {1-i}\right ) \sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )-(-4+4 i) \operatorname {Subst}\left (\int \frac {1}{(-16+16 i)-16 (1+i)^{3/2}-x^2} \, dx,x,\frac {(2+2 i)+4 \sqrt {1+i}-\left ((-4-4 i)+2 \sqrt {1+i}\right ) \sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )-(-4+4 i) \operatorname {Subst}\left (\int \frac {1}{(-16+16 i)+16 (1+i)^{3/2}-x^2} \, dx,x,\frac {(2+2 i)-4 \sqrt {1+i}-\left ((-4-4 i)-2 \sqrt {1+i}\right ) \sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )\\ &=-\frac {3}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \sqrt {x+\sqrt {1+x}}-\frac {(1+i) \tan ^{-1}\left (\frac {(2-2 i)+4 \sqrt {1-i}-2 \left ((-2+2 i)+\sqrt {1-i}\right ) \sqrt {1+x}}{4 \sqrt {(1+i)+(1-i)^{3/2}} \sqrt {x+\sqrt {1+x}}}\right )}{\sqrt {(1+i)+(1-i)^{3/2}}}-\frac {(1+i) \tan ^{-1}\left (\frac {(2-2 i)-4 \sqrt {1-i}+2 \left ((2-2 i)+\sqrt {1-i}\right ) \sqrt {1+x}}{4 \sqrt {(1+i)-(1-i)^{3/2}} \sqrt {x+\sqrt {1+x}}}\right )}{\sqrt {(1+i)-(1-i)^{3/2}}}-\frac {(1-i) \tan ^{-1}\left (\frac {(2+2 i)+4 \sqrt {1+i}-2 \left ((-2-2 i)+\sqrt {1+i}\right ) \sqrt {1+x}}{4 \sqrt {(1-i)+(1+i)^{3/2}} \sqrt {x+\sqrt {1+x}}}\right )}{\sqrt {(1-i)+(1+i)^{3/2}}}-\frac {(1-i) \tan ^{-1}\left (\frac {(2+2 i)-4 \sqrt {1+i}+2 \left ((2+2 i)+\sqrt {1+i}\right ) \sqrt {1+x}}{4 \sqrt {(1-i)-(1+i)^{3/2}} \sqrt {x+\sqrt {1+x}}}\right )}{\sqrt {(1-i)-(1+i)^{3/2}}}+\frac {7}{4} \tanh ^{-1}\left (\frac {1+2 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right )\\ \end {align*}

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Mathematica [C] time = 0.28, size = 400, normalized size = 1.19 \begin {gather*} \sqrt {x+1} \sqrt {x+\sqrt {x+1}}-\frac {3}{2} \sqrt {x+\sqrt {x+1}}-\frac {(1+i) \tan ^{-1}\left (\frac {-\left (\left (1-2 \sqrt {1-i}\right ) \sqrt {x+1}\right )+\sqrt {1-i}+2}{2 \sqrt {i+\sqrt {1-i}} \sqrt {x+\sqrt {x+1}}}\right )}{\sqrt {(1+i)+(1-i)^{3/2}}}-\frac {(1-i) \tan ^{-1}\left (\frac {-\left (\left (1-2 \sqrt {1+i}\right ) \sqrt {x+1}\right )+\sqrt {1+i}+2}{2 \sqrt {\sqrt {1+i}-i} \sqrt {x+\sqrt {x+1}}}\right )}{\sqrt {(1-i)+(1+i)^{3/2}}}+\frac {7}{4} \tanh ^{-1}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right )-\frac {(1+i) \tanh ^{-1}\left (\frac {-\left (\left (1+2 \sqrt {1-i}\right ) \sqrt {x+1}\right )-\sqrt {1-i}+2}{2 \sqrt {\sqrt {1-i}-i} \sqrt {x+\sqrt {x+1}}}\right )}{\sqrt {(-1-i)+(1-i)^{3/2}}}-\frac {(1-i) \tanh ^{-1}\left (\frac {-\left (\left (1+2 \sqrt {1+i}\right ) \sqrt {x+1}\right )-\sqrt {1+i}+2}{2 \sqrt {i+\sqrt {1+i}} \sqrt {x+\sqrt {x+1}}}\right )}{\sqrt {(-1+i)+(1+i)^{3/2}}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-3*Sqrt[x + Sqrt[1 + x]])/2 + Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]] - ((1 + I)*ArcTan[(2 + Sqrt[1 - I] - (1 - 2*S

qrt[1 - I])*Sqrt[1 + x])/(2*Sqrt[I + Sqrt[1 - I]]*Sqrt[x + Sqrt[1 + x]])])/Sqrt[(1 + I) + (1 - I)^(3/2)] - ((1

- I)*ArcTan[(2 + Sqrt[1 + I] - (1 - 2*Sqrt[1 + I])*Sqrt[1 + x])/(2*Sqrt[-I + Sqrt[1 + I]]*Sqrt[x + Sqrt[1 + x

]])])/Sqrt[(1 - I) + (1 + I)^(3/2)] + (7*ArcTanh[(1 + 2*Sqrt[1 + x])/(2*Sqrt[x + Sqrt[1 + x]])])/4 - ((1 + I)*

ArcTanh[(2 - Sqrt[1 - I] - (1 + 2*Sqrt[1 - I])*Sqrt[1 + x])/(2*Sqrt[-I + Sqrt[1 - I]]*Sqrt[x + Sqrt[1 + x]])])

/Sqrt[(-1 - I) + (1 - I)^(3/2)] - ((1 - I)*ArcTanh[(2 - Sqrt[1 + I] - (1 + 2*Sqrt[1 + I])*Sqrt[1 + x])/(2*Sqrt

[I + Sqrt[1 + I]]*Sqrt[x + Sqrt[1 + x]])])/Sqrt[(-1 + I) + (1 + I)^(3/2)]

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IntegrateAlgebraic [A] time = 0.00, size = 339, normalized size = 1.01 \begin {gather*} \frac {1}{2} \sqrt {x+\sqrt {1+x}} \left (-3+2 \sqrt {1+x}\right )-\frac {7}{4} \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right )+\text {RootSum}\left [1-8 \text {$\#$1}+40 \text {$\#$1}^2-48 \text {$\#$1}^3+20 \text {$\#$1}^4+8 \text {$\#$1}^5-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right )+2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}-2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2+4 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3-\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^4+2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^5}{-1+10 \text {$\#$1}-18 \text {$\#$1}^2+10 \text {$\#$1}^3+5 \text {$\#$1}^4-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(Sqrt[x + Sqrt[1 + x]]*(-3 + 2*Sqrt[1 + x]))/2 - (7*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]])/4 + Roo

tSum[1 - 8*#1 + 40*#1^2 - 48*#1^3 + 20*#1^4 + 8*#1^5 - 4*#1^6 + #1^8 & , (-Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1

+ x]] - #1] + 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1 - 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]]

- #1]*#1^2 + 4*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^3 - Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]]

- #1]*#1^4 + 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^5)/(-1 + 10*#1 - 18*#1^2 + 10*#1^3 + 5*#1^4

- 3*#1^5 + #1^7) & ]

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fricas [B] time = 13.70, size = 5812, normalized size = 17.25

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*sqrt(5)*sqrt(5*sqrt(56/25*I + 8/25) + 5*sqrt(-56/25*I + 8/25) - 2*sqrt(-3/4*(5*sqrt(56/25*I + 8/25) - 6*I

+ 2)^2 - 3/4*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 1/2*(5*sqrt(56/25*I + 8/25) - 6*I + 2)*(5*sqrt(-56/25*I

+ 8/25) + 6*I - 6) + 20*sqrt(-56/25*I + 8/25) + 24*I - 56) - 4)*log(-1/200*(40*(((11*x - 7)*sqrt(x + 1) + 8*x

- 11)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 30*(3*x - 1)*sqrt(x + 1) - 70*x + 90)*sqrt(x + sqrt(x + 1))*(5*sqr

t(56/25*I + 8/25) - 6*I + 2)^2 + 40*(((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2

- 8*((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) + 40*(3*x - 1)*sqrt(x + 1) - 240*

x - 120)*sqrt(x + sqrt(x + 1))*(5*sqrt(56/25*I + 8/25) - 6*I + 2) + 80*((((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(

5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 30*(3*x - 1)*sqrt(x + 1) - 70*x + 90)*sqrt(x + sqrt(x + 1))*(5*sqrt(56/25

*I + 8/25) - 6*I + 2) - 10*((3*(3*x - 1)*sqrt(x + 1) + 7*x - 9)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 20*(3*x

- 1)*sqrt(x + 1) - 80*x + 60)*sqrt(x + sqrt(x + 1)))*sqrt(-3/4*(5*sqrt(56/25*I + 8/25) - 6*I + 2)^2 - 3/4*(5*s

qrt(-56/25*I + 8/25) + 6*I + 2)^2 - 1/2*(5*sqrt(56/25*I + 8/25) - 6*I + 2)*(5*sqrt(-56/25*I + 8/25) + 6*I - 6)

+ 20*sqrt(-56/25*I + 8/25) + 24*I - 56) - 400*((3*(3*x - 1)*sqrt(x + 1) + 7*x - 9)*(5*sqrt(-56/25*I + 8/25) +

6*I + 2)^2 - 4*((3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 240*(3*x - 1)*sqrt(x +

1) - 560*x - 80)*sqrt(x + sqrt(x + 1)) + ((40*sqrt(5)*(11*x + 8)*sqrt(x + 1) - (2*sqrt(5)*(33*x + 19)*sqrt(x

+ 1) + sqrt(5)*(31*x^2 + 88*x + 35))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) + 20*sqrt(5)*(7*x^2 + 36*x + 15))*(5*

sqrt(56/25*I + 8/25) - 6*I + 2)^2 + 20*(2*sqrt(5)*(11*x + 8)*sqrt(x + 1) + sqrt(5)*(7*x^2 + 36*x + 15))*(5*sqr

t(-56/25*I + 8/25) + 6*I + 2)^2 - 3200*sqrt(5)*(6*x + 13)*sqrt(x + 1) - ((2*sqrt(5)*(33*x + 19)*sqrt(x + 1) +

sqrt(5)*(31*x^2 + 88*x + 35))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 80*sqrt(5)*(x + 3)*sqrt(x + 1) - 8*(2*sq

rt(5)*(33*x + 19)*sqrt(x + 1) + sqrt(5)*(31*x^2 + 88*x + 35))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) + 40*sqrt(5)

*(13*x^2 + 24*x - 15))*(5*sqrt(56/25*I + 8/25) - 6*I + 2) + 40*(2*sqrt(5)*(x + 3)*sqrt(x + 1) - sqrt(5)*(13*x^

2 + 24*x - 15))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 3200*sqrt(5)*(x^2 + 8*x + 5) - 2*(400*sqrt(5)*(9*x + 7)*

sqrt(x + 1) - (40*sqrt(5)*(11*x + 8)*sqrt(x + 1) - (2*sqrt(5)*(33*x + 19)*sqrt(x + 1) + sqrt(5)*(31*x^2 + 88*x

+ 35))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) + 20*sqrt(5)*(7*x^2 + 36*x + 15))*(5*sqrt(56/25*I + 8/25) - 6*I +

2) - 20*(2*sqrt(5)*(11*x + 8)*sqrt(x + 1) + sqrt(5)*(7*x^2 + 36*x + 15))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) +

600*sqrt(5)*(x^2 + 8*x + 5))*sqrt(-3/4*(5*sqrt(56/25*I + 8/25) - 6*I + 2)^2 - 3/4*(5*sqrt(-56/25*I + 8/25) +

6*I + 2)^2 - 1/2*(5*sqrt(56/25*I + 8/25) - 6*I + 2)*(5*sqrt(-56/25*I + 8/25) + 6*I - 6) + 20*sqrt(-56/25*I + 8

/25) + 24*I - 56))*sqrt(5*sqrt(56/25*I + 8/25) + 5*sqrt(-56/25*I + 8/25) - 2*sqrt(-3/4*(5*sqrt(56/25*I + 8/25)

- 6*I + 2)^2 - 3/4*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 1/2*(5*sqrt(56/25*I + 8/25) - 6*I + 2)*(5*sqrt(-56

/25*I + 8/25) + 6*I - 6) + 20*sqrt(-56/25*I + 8/25) + 24*I - 56) - 4))/(x^2 + 1)) - 1/20*sqrt(5)*sqrt(5*sqrt(5

6/25*I + 8/25) + 5*sqrt(-56/25*I + 8/25) - 2*sqrt(-3/4*(5*sqrt(56/25*I + 8/25) - 6*I + 2)^2 - 3/4*(5*sqrt(-56/

25*I + 8/25) + 6*I + 2)^2 - 1/2*(5*sqrt(56/25*I + 8/25) - 6*I + 2)*(5*sqrt(-56/25*I + 8/25) + 6*I - 6) + 20*sq

rt(-56/25*I + 8/25) + 24*I - 56) - 4)*log(-1/200*(40*(((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(5*sqrt(-56/25*I + 8

/25) + 6*I + 2) - 30*(3*x - 1)*sqrt(x + 1) - 70*x + 90)*sqrt(x + sqrt(x + 1))*(5*sqrt(56/25*I + 8/25) - 6*I +

2)^2 + 40*(((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 8*((11*x - 7)*sqrt(x +

1) + 8*x - 11)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) + 40*(3*x - 1)*sqrt(x + 1) - 240*x - 120)*sqrt(x + sqrt(x +

1))*(5*sqrt(56/25*I + 8/25) - 6*I + 2) + 80*((((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(5*sqrt(-56/25*I + 8/25) +

6*I + 2) - 30*(3*x - 1)*sqrt(x + 1) - 70*x + 90)*sqrt(x + sqrt(x + 1))*(5*sqrt(56/25*I + 8/25) - 6*I + 2) - 10

*((3*(3*x - 1)*sqrt(x + 1) + 7*x - 9)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 20*(3*x - 1)*sqrt(x + 1) - 80*x +

60)*sqrt(x + sqrt(x + 1)))*sqrt(-3/4*(5*sqrt(56/25*I + 8/25) - 6*I + 2)^2 - 3/4*(5*sqrt(-56/25*I + 8/25) + 6*I

+ 2)^2 - 1/2*(5*sqrt(56/25*I + 8/25) - 6*I + 2)*(5*sqrt(-56/25*I + 8/25) + 6*I - 6) + 20*sqrt(-56/25*I + 8/25

) + 24*I - 56) - 400*((3*(3*x - 1)*sqrt(x + 1) + 7*x - 9)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 4*((3*x - 1)

*sqrt(x + 1) - 6*x - 3)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 240*(3*x - 1)*sqrt(x + 1) - 560*x - 80)*sqrt(x +

sqrt(x + 1)) - ((40*sqrt(5)*(11*x + 8)*sqrt(x + 1) - (2*sqrt(5)*(33*x + 19)*sqrt(x + 1) + sqrt(5)*(31*x^2 + 8

8*x + 35))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) + 20*sqrt(5)*(7*x^2 + 36*x + 15))*(5*sqrt(56/25*I + 8/25) - 6*I

+ 2)^2 + 20*(2*sqrt(5)*(11*x + 8)*sqrt(x + 1) + sqrt(5)*(7*x^2 + 36*x + 15))*(5*sqrt(-56/25*I + 8/25) + 6*I +

2)^2 - 3200*sqrt(5)*(6*x + 13)*sqrt(x + 1) - ((2*sqrt(5)*(33*x + 19)*sqrt(x + 1) + sqrt(5)*(31*x^2 + 88*x + 3

5))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 80*sqrt(5)*(x + 3)*sqrt(x + 1) - 8*(2*sqrt(5)*(33*x + 19)*sqrt(x +

1) + sqrt(5)*(31*x^2 + 88*x + 35))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) + 40*sqrt(5)*(13*x^2 + 24*x - 15))*(5*

sqrt(56/25*I + 8/25) - 6*I + 2) + 40*(2*sqrt(5)*(x + 3)*sqrt(x + 1) - sqrt(5)*(13*x^2 + 24*x - 15))*(5*sqrt(-5

6/25*I + 8/25) + 6*I + 2) - 3200*sqrt(5)*(x^2 + 8*x + 5) - 2*(400*sqrt(5)*(9*x + 7)*sqrt(x + 1) - (40*sqrt(5)*

(11*x + 8)*sqrt(x + 1) - (2*sqrt(5)*(33*x + 19)*sqrt(x + 1) + sqrt(5)*(31*x^2 + 88*x + 35))*(5*sqrt(-56/25*I +

8/25) + 6*I + 2) + 20*sqrt(5)*(7*x^2 + 36*x + 15))*(5*sqrt(56/25*I + 8/25) - 6*I + 2) - 20*(2*sqrt(5)*(11*x +

8)*sqrt(x + 1) + sqrt(5)*(7*x^2 + 36*x + 15))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) + 600*sqrt(5)*(x^2 + 8*x +

5))*sqrt(-3/4*(5*sqrt(56/25*I + 8/25) - 6*I + 2)^2 - 3/4*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 1/2*(5*sqrt(5

6/25*I + 8/25) - 6*I + 2)*(5*sqrt(-56/25*I + 8/25) + 6*I - 6) + 20*sqrt(-56/25*I + 8/25) + 24*I - 56))*sqrt(5*

sqrt(56/25*I + 8/25) + 5*sqrt(-56/25*I + 8/25) - 2*sqrt(-3/4*(5*sqrt(56/25*I + 8/25) - 6*I + 2)^2 - 3/4*(5*sqr

t(-56/25*I + 8/25) + 6*I + 2)^2 - 1/2*(5*sqrt(56/25*I + 8/25) - 6*I + 2)*(5*sqrt(-56/25*I + 8/25) + 6*I - 6) +

20*sqrt(-56/25*I + 8/25) + 24*I - 56) - 4))/(x^2 + 1)) + 1/20*sqrt(5)*sqrt(5*sqrt(56/25*I + 8/25) + 5*sqrt(-5

6/25*I + 8/25) + 2*sqrt(-3/4*(5*sqrt(56/25*I + 8/25) - 6*I + 2)^2 - 3/4*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2

- 1/2*(5*sqrt(56/25*I + 8/25) - 6*I + 2)*(5*sqrt(-56/25*I + 8/25) + 6*I - 6) + 20*sqrt(-56/25*I + 8/25) + 24*I

- 56) - 4)*log(-1/200*(40*(((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 30*(3*x

- 1)*sqrt(x + 1) - 70*x + 90)*sqrt(x + sqrt(x + 1))*(5*sqrt(56/25*I + 8/25) - 6*I + 2)^2 + 40*(((11*x - 7)*sqr

t(x + 1) + 8*x - 11)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 8*((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(5*sqrt(-56

/25*I + 8/25) + 6*I + 2) + 40*(3*x - 1)*sqrt(x + 1) - 240*x - 120)*sqrt(x + sqrt(x + 1))*(5*sqrt(56/25*I + 8/2

5) - 6*I + 2) - 80*((((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 30*(3*x - 1)*sq

rt(x + 1) - 70*x + 90)*sqrt(x + sqrt(x + 1))*(5*sqrt(56/25*I + 8/25) - 6*I + 2) - 10*((3*(3*x - 1)*sqrt(x + 1)

+ 7*x - 9)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 20*(3*x - 1)*sqrt(x + 1) - 80*x + 60)*sqrt(x + sqrt(x + 1)))

*sqrt(-3/4*(5*sqrt(56/25*I + 8/25) - 6*I + 2)^2 - 3/4*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 1/2*(5*sqrt(56/2

5*I + 8/25) - 6*I + 2)*(5*sqrt(-56/25*I + 8/25) + 6*I - 6) + 20*sqrt(-56/25*I + 8/25) + 24*I - 56) - 400*((3*(

3*x - 1)*sqrt(x + 1) + 7*x - 9)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 4*((3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5

*sqrt(-56/25*I + 8/25) + 6*I + 2) - 240*(3*x - 1)*sqrt(x + 1) - 560*x - 80)*sqrt(x + sqrt(x + 1)) + ((40*sqrt(

5)*(11*x + 8)*sqrt(x + 1) - (2*sqrt(5)*(33*x + 19)*sqrt(x + 1) + sqrt(5)*(31*x^2 + 88*x + 35))*(5*sqrt(-56/25*

I + 8/25) + 6*I + 2) + 20*sqrt(5)*(7*x^2 + 36*x + 15))*(5*sqrt(56/25*I + 8/25) - 6*I + 2)^2 + 20*(2*sqrt(5)*(1

1*x + 8)*sqrt(x + 1) + sqrt(5)*(7*x^2 + 36*x + 15))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 3200*sqrt(5)*(6*x

+ 13)*sqrt(x + 1) - ((2*sqrt(5)*(33*x + 19)*sqrt(x + 1) + sqrt(5)*(31*x^2 + 88*x + 35))*(5*sqrt(-56/25*I + 8/2

5) + 6*I + 2)^2 - 80*sqrt(5)*(x + 3)*sqrt(x + 1) - 8*(2*sqrt(5)*(33*x + 19)*sqrt(x + 1) + sqrt(5)*(31*x^2 + 88

*x + 35))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) + 40*sqrt(5)*(13*x^2 + 24*x - 15))*(5*sqrt(56/25*I + 8/25) - 6*I

+ 2) + 40*(2*sqrt(5)*(x + 3)*sqrt(x + 1) - sqrt(5)*(13*x^2 + 24*x - 15))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)

- 3200*sqrt(5)*(x^2 + 8*x + 5) + 2*(400*sqrt(5)*(9*x + 7)*sqrt(x + 1) - (40*sqrt(5)*(11*x + 8)*sqrt(x + 1) - (

2*sqrt(5)*(33*x + 19)*sqrt(x + 1) + sqrt(5)*(31*x^2 + 88*x + 35))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) + 20*sqr

t(5)*(7*x^2 + 36*x + 15))*(5*sqrt(56/25*I + 8/25) - 6*I + 2) - 20*(2*sqrt(5)*(11*x + 8)*sqrt(x + 1) + sqrt(5)*

(7*x^2 + 36*x + 15))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) + 600*sqrt(5)*(x^2 + 8*x + 5))*sqrt(-3/4*(5*sqrt(56/2

5*I + 8/25) - 6*I + 2)^2 - 3/4*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 1/2*(5*sqrt(56/25*I + 8/25) - 6*I + 2)*

(5*sqrt(-56/25*I + 8/25) + 6*I - 6) + 20*sqrt(-56/25*I + 8/25) + 24*I - 56))*sqrt(5*sqrt(56/25*I + 8/25) + 5*s

qrt(-56/25*I + 8/25) + 2*sqrt(-3/4*(5*sqrt(56/25*I + 8/25) - 6*I + 2)^2 - 3/4*(5*sqrt(-56/25*I + 8/25) + 6*I +

2)^2 - 1/2*(5*sqrt(56/25*I + 8/25) - 6*I + 2)*(5*sqrt(-56/25*I + 8/25) + 6*I - 6) + 20*sqrt(-56/25*I + 8/25)

+ 24*I - 56) - 4))/(x^2 + 1)) - 1/20*sqrt(5)*sqrt(5*sqrt(56/25*I + 8/25) + 5*sqrt(-56/25*I + 8/25) + 2*sqrt(-3

/4*(5*sqrt(56/25*I + 8/25) - 6*I + 2)^2 - 3/4*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 1/2*(5*sqrt(56/25*I + 8/

25) - 6*I + 2)*(5*sqrt(-56/25*I + 8/25) + 6*I - 6) + 20*sqrt(-56/25*I + 8/25) + 24*I - 56) - 4)*log(-1/200*(40

*(((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 30*(3*x - 1)*sqrt(x + 1) - 70*x +

90)*sqrt(x + sqrt(x + 1))*(5*sqrt(56/25*I + 8/25) - 6*I + 2)^2 + 40*(((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(5*sq

rt(-56/25*I + 8/25) + 6*I + 2)^2 - 8*((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) +

40*(3*x - 1)*sqrt(x + 1) - 240*x - 120)*sqrt(x + sqrt(x + 1))*(5*sqrt(56/25*I + 8/25) - 6*I + 2) - 80*((((11*

x - 7)*sqrt(x + 1) + 8*x - 11)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 30*(3*x - 1)*sqrt(x + 1) - 70*x + 90)*sqr

t(x + sqrt(x + 1))*(5*sqrt(56/25*I + 8/25) - 6*I + 2) - 10*((3*(3*x - 1)*sqrt(x + 1) + 7*x - 9)*(5*sqrt(-56/25

*I + 8/25) + 6*I + 2) - 20*(3*x - 1)*sqrt(x + 1) - 80*x + 60)*sqrt(x + sqrt(x + 1)))*sqrt(-3/4*(5*sqrt(56/25*I

+ 8/25) - 6*I + 2)^2 - 3/4*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 1/2*(5*sqrt(56/25*I + 8/25) - 6*I + 2)*(5*

sqrt(-56/25*I + 8/25) + 6*I - 6) + 20*sqrt(-56/25*I + 8/25) + 24*I - 56) - 400*((3*(3*x - 1)*sqrt(x + 1) + 7*x

- 9)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 4*((3*x - 1)*sqrt(x + 1) - 6*x - 3)*(5*sqrt(-56/25*I + 8/25) + 6

*I + 2) - 240*(3*x - 1)*sqrt(x + 1) - 560*x - 80)*sqrt(x + sqrt(x + 1)) - ((40*sqrt(5)*(11*x + 8)*sqrt(x + 1)

- (2*sqrt(5)*(33*x + 19)*sqrt(x + 1) + sqrt(5)*(31*x^2 + 88*x + 35))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) + 20*

sqrt(5)*(7*x^2 + 36*x + 15))*(5*sqrt(56/25*I + 8/25) - 6*I + 2)^2 + 20*(2*sqrt(5)*(11*x + 8)*sqrt(x + 1) + sqr

t(5)*(7*x^2 + 36*x + 15))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 3200*sqrt(5)*(6*x + 13)*sqrt(x + 1) - ((2*sq

rt(5)*(33*x + 19)*sqrt(x + 1) + sqrt(5)*(31*x^2 + 88*x + 35))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 80*sqrt(

5)*(x + 3)*sqrt(x + 1) - 8*(2*sqrt(5)*(33*x + 19)*sqrt(x + 1) + sqrt(5)*(31*x^2 + 88*x + 35))*(5*sqrt(-56/25*I

+ 8/25) + 6*I + 2) + 40*sqrt(5)*(13*x^2 + 24*x - 15))*(5*sqrt(56/25*I + 8/25) - 6*I + 2) + 40*(2*sqrt(5)*(x +

3)*sqrt(x + 1) - sqrt(5)*(13*x^2 + 24*x - 15))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 3200*sqrt(5)*(x^2 + 8*x

+ 5) + 2*(400*sqrt(5)*(9*x + 7)*sqrt(x + 1) - (40*sqrt(5)*(11*x + 8)*sqrt(x + 1) - (2*sqrt(5)*(33*x + 19)*sqrt

(x + 1) + sqrt(5)*(31*x^2 + 88*x + 35))*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) + 20*sqrt(5)*(7*x^2 + 36*x + 15))*

(5*sqrt(56/25*I + 8/25) - 6*I + 2) - 20*(2*sqrt(5)*(11*x + 8)*sqrt(x + 1) + sqrt(5)*(7*x^2 + 36*x + 15))*(5*sq

rt(-56/25*I + 8/25) + 6*I + 2) + 600*sqrt(5)*(x^2 + 8*x + 5))*sqrt(-3/4*(5*sqrt(56/25*I + 8/25) - 6*I + 2)^2 -

3/4*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 1/2*(5*sqrt(56/25*I + 8/25) - 6*I + 2)*(5*sqrt(-56/25*I + 8/25) +

6*I - 6) + 20*sqrt(-56/25*I + 8/25) + 24*I - 56))*sqrt(5*sqrt(56/25*I + 8/25) + 5*sqrt(-56/25*I + 8/25) + 2*s

qrt(-3/4*(5*sqrt(56/25*I + 8/25) - 6*I + 2)^2 - 3/4*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 1/2*(5*sqrt(56/25*

I + 8/25) - 6*I + 2)*(5*sqrt(-56/25*I + 8/25) + 6*I - 6) + 20*sqrt(-56/25*I + 8/25) + 24*I - 56) - 4))/(x^2 +

1)) + 1/2*sqrt(x + sqrt(x + 1))*(2*sqrt(x + 1) - 3) - 1/2*sqrt(-1/2*sqrt(56/25*I + 8/25) + 3/5*I - 1/5)*log(1/

100*(4*(((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 30*(3*x - 1)*sqrt(x + 1) - 7

0*x + 90)*sqrt(x + sqrt(x + 1))*(5*sqrt(56/25*I + 8/25) - 6*I + 2)^2 + 4*(((11*x - 7)*sqrt(x + 1) + 8*x - 11)*

(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 8*((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(5*sqrt(-56/25*I + 8/25) + 6*I +

2) + 40*(3*x - 1)*sqrt(x + 1) - 240*x - 120)*sqrt(x + sqrt(x + 1))*(5*sqrt(56/25*I + 8/25) - 6*I + 2) + 4*(((

11*x - 7)*sqrt(x + 1) + 8*x - 11)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^3 - 8*((11*x - 7)*sqrt(x + 1) + 8*x - 11

)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 + 80*((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(5*sqrt(-56/25*I + 8/25) + 6*

I + 2) - 2800*(3*x - 1)*sqrt(x + 1) - 5200*x + 4400)*sqrt(x + sqrt(x + 1)) + ((31*x^2 + 2*(33*x + 19)*sqrt(x +

1) + 88*x + 35)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^3 - (140*x^2 - (31*x^2 + 2*(33*x + 19)*sqrt(x + 1) + 88*x

+ 35)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) + 40*(11*x + 8)*sqrt(x + 1) + 720*x + 300)*(5*sqrt(56/25*I + 8/25)

- 6*I + 2)^2 - 8*(31*x^2 + 2*(33*x + 19)*sqrt(x + 1) + 88*x + 35)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 1040

0*x^2 + ((31*x^2 + 2*(33*x + 19)*sqrt(x + 1) + 88*x + 35)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 + 520*x^2 - 8*

(31*x^2 + 2*(33*x + 19)*sqrt(x + 1) + 88*x + 35)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 80*(x + 3)*sqrt(x + 1)

+ 960*x - 600)*(5*sqrt(56/25*I + 8/25) - 6*I + 2) + 80*(31*x^2 + 2*(33*x + 19)*sqrt(x + 1) + 88*x + 35)*(5*sqr

t(-56/25*I + 8/25) + 6*I + 2) - 1600*(19*x + 22)*sqrt(x + 1) - 51200*x - 20000)*sqrt(-1/2*sqrt(56/25*I + 8/25)

+ 3/5*I - 1/5))/(x^2 + 1)) + 1/2*sqrt(-1/2*sqrt(56/25*I + 8/25) + 3/5*I - 1/5)*log(1/100*(4*(((11*x - 7)*sqrt

(x + 1) + 8*x - 11)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 30*(3*x - 1)*sqrt(x + 1) - 70*x + 90)*sqrt(x + sqrt(

x + 1))*(5*sqrt(56/25*I + 8/25) - 6*I + 2)^2 + 4*(((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(5*sqrt(-56/25*I + 8/25)

+ 6*I + 2)^2 - 8*((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) + 40*(3*x - 1)*sqrt(

x + 1) - 240*x - 120)*sqrt(x + sqrt(x + 1))*(5*sqrt(56/25*I + 8/25) - 6*I + 2) + 4*(((11*x - 7)*sqrt(x + 1) +

8*x - 11)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^3 - 8*((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(5*sqrt(-56/25*I + 8/2

5) + 6*I + 2)^2 + 80*((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 2800*(3*x - 1)*

sqrt(x + 1) - 5200*x + 4400)*sqrt(x + sqrt(x + 1)) - ((31*x^2 + 2*(33*x + 19)*sqrt(x + 1) + 88*x + 35)*(5*sqrt

(-56/25*I + 8/25) + 6*I + 2)^3 - (140*x^2 - (31*x^2 + 2*(33*x + 19)*sqrt(x + 1) + 88*x + 35)*(5*sqrt(-56/25*I

+ 8/25) + 6*I + 2) + 40*(11*x + 8)*sqrt(x + 1) + 720*x + 300)*(5*sqrt(56/25*I + 8/25) - 6*I + 2)^2 - 8*(31*x^2

+ 2*(33*x + 19)*sqrt(x + 1) + 88*x + 35)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 10400*x^2 + ((31*x^2 + 2*(33

*x + 19)*sqrt(x + 1) + 88*x + 35)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 + 520*x^2 - 8*(31*x^2 + 2*(33*x + 19)*

sqrt(x + 1) + 88*x + 35)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 80*(x + 3)*sqrt(x + 1) + 960*x - 600)*(5*sqrt(5

6/25*I + 8/25) - 6*I + 2) + 80*(31*x^2 + 2*(33*x + 19)*sqrt(x + 1) + 88*x + 35)*(5*sqrt(-56/25*I + 8/25) + 6*I

+ 2) - 1600*(19*x + 22)*sqrt(x + 1) - 51200*x - 20000)*sqrt(-1/2*sqrt(56/25*I + 8/25) + 3/5*I - 1/5))/(x^2 +

1)) - 1/2*sqrt(-1/2*sqrt(-56/25*I + 8/25) - 3/5*I - 1/5)*log(-1/100*(4*(((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(5

*sqrt(-56/25*I + 8/25) + 6*I + 2)^3 + 2*((x + 13)*sqrt(x + 1) + 3*x - 1)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2

+ 40*((19*x - 13)*sqrt(x + 1) + 22*x - 19)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 400*(23*x - 21)*sqrt(x + 1)

- 7600*x + 13200)*sqrt(x + sqrt(x + 1)) + ((31*x^2 + 2*(33*x + 19)*sqrt(x + 1) + 88*x + 35)*(5*sqrt(-56/25*I +

8/25) + 6*I + 2)^3 - 4*(27*x^2 + 2*(11*x - 2)*sqrt(x + 1) - 4*x - 5)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 -

39200*x^2 + 40*(49*x^2 + 2*(67*x + 41)*sqrt(x + 1) + 152*x + 85)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 1600*(4

7*x + 16)*sqrt(x + 1) - 89600*x - 36000)*sqrt(-1/2*sqrt(-56/25*I + 8/25) - 3/5*I - 1/5))/(x^2 + 1)) + 1/2*sqrt

(-1/2*sqrt(-56/25*I + 8/25) - 3/5*I - 1/5)*log(-1/100*(4*(((11*x - 7)*sqrt(x + 1) + 8*x - 11)*(5*sqrt(-56/25*I

+ 8/25) + 6*I + 2)^3 + 2*((x + 13)*sqrt(x + 1) + 3*x - 1)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 + 40*((19*x -

13)*sqrt(x + 1) + 22*x - 19)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 400*(23*x - 21)*sqrt(x + 1) - 7600*x + 132

00)*sqrt(x + sqrt(x + 1)) - ((31*x^2 + 2*(33*x + 19)*sqrt(x + 1) + 88*x + 35)*(5*sqrt(-56/25*I + 8/25) + 6*I +

2)^3 - 4*(27*x^2 + 2*(11*x - 2)*sqrt(x + 1) - 4*x - 5)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2)^2 - 39200*x^2 + 40

*(49*x^2 + 2*(67*x + 41)*sqrt(x + 1) + 152*x + 85)*(5*sqrt(-56/25*I + 8/25) + 6*I + 2) - 1600*(47*x + 16)*sqrt

(x + 1) - 89600*x - 36000)*sqrt(-1/2*sqrt(-56/25*I + 8/25) - 3/5*I - 1/5))/(x^2 + 1)) + 7/8*log(4*sqrt(x + sqr

t(x + 1))*(2*sqrt(x + 1) + 1) + 8*x + 8*sqrt(x + 1) + 5)

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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [B] time = 0.17, size = 161, normalized size = 0.48


method	result	size

derivativedivides	$\sqrt {1+x}\, \sqrt {x +\sqrt {1+x}}-\frac {3 \sqrt {x +\sqrt {1+x}}}{2}+\frac {7 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{4}+\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+8 \textit {\_Z}^{5}+20 \textit {\_Z}^{4}-48 \textit {\_Z}^{3}+40 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )}{\sum }\frac {\left (2 \textit {\_R}^{5}-\textit {\_R}^{4}+4 \textit {\_R}^{3}-2 \textit {\_R}^{2}+2 \textit {\_R} -1\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+5 \textit {\_R}^{4}+10 \textit {\_R}^{3}-18 \textit {\_R}^{2}+10 \textit {\_R} -1}\right )$	$161$
default	$\sqrt {1+x}\, \sqrt {x +\sqrt {1+x}}-\frac {3 \sqrt {x +\sqrt {1+x}}}{2}+\frac {7 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{4}+\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+8 \textit {\_Z}^{5}+20 \textit {\_Z}^{4}-48 \textit {\_Z}^{3}+40 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )}{\sum }\frac {\left (2 \textit {\_R}^{5}-\textit {\_R}^{4}+4 \textit {\_R}^{3}-2 \textit {\_R}^{2}+2 \textit {\_R} -1\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+5 \textit {\_R}^{4}+10 \textit {\_R}^{3}-18 \textit {\_R}^{2}+10 \textit {\_R} -1}\right )$	$161$

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1+x)^(1/2)*(x+(1+x)^(1/2))^(1/2)-3/2*(x+(1+x)^(1/2))^(1/2)+7/4*ln(1/2+(1+x)^(1/2)+(x+(1+x)^(1/2))^(1/2))+sum(

(2*_R^5-_R^4+4*_R^3-2*_R^2+2*_R-1)/(_R^7-3*_R^5+5*_R^4+10*_R^3-18*_R^2+10*_R-1)*ln((x+(1+x)^(1/2))^(1/2)-(1+x)

^(1/2)-_R),_R=RootOf(_Z^8-4*_Z^6+8*_Z^5+20*_Z^4-48*_Z^3+40*_Z^2-8*_Z+1))

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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{2} - 1\right )} \sqrt {x + 1}}{{\left (x^{2} + 1\right )} \sqrt {x + \sqrt {x + 1}}}\,{d x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^2 - 1)*sqrt(x + 1)/((x^2 + 1)*sqrt(x + sqrt(x + 1))), x)

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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (x^2-1\right )\,\sqrt {x+1}}{\sqrt {x+\sqrt {x+1}}\,\left (x^2+1\right )} \,d x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x + 1\right )^{\frac {3}{2}}}{\sqrt {x + \sqrt {x + 1}} \left (x^{2} + 1\right )}\, dx \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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