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

Optimal. Leaf size=185 \[ \frac {\left (\sqrt {x^2+1}+x\right )^{5/2}}{8 \left (x^2+\sqrt {x^2+1} x+1\right )^2}-\frac {3 \sqrt {\sqrt {x^2+1}+x}}{8 \left (x^2+\sqrt {x^2+1} x+1\right )^2}+\frac {3 \tan ^{-1}\left (\frac {\frac {\sqrt {x^2+1}}{\sqrt {2}}+\frac {x}{\sqrt {2}}-\frac {1}{\sqrt {2}}}{\sqrt {\sqrt {x^2+1}+x}}\right )}{4 \sqrt {2}}+\frac {3 \tanh ^{-1}\left (\frac {\frac {\sqrt {x^2+1}}{\sqrt {2}}+\frac {x}{\sqrt {2}}+\frac {1}{\sqrt {2}}}{\sqrt {\sqrt {x^2+1}+x}}\right )}{4 \sqrt {2}} \]

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Rubi [A]  time = 0.16, antiderivative size = 225, normalized size of antiderivative = 1.22, number of steps used = 13, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {2122, 288, 290, 329, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {\sqrt {\sqrt {x^2+1}+x}}{2 \left (\left (\sqrt {x^2+1}+x\right )^2+1\right )}-\frac {2 \sqrt {\sqrt {x^2+1}+x}}{\left (\left (\sqrt {x^2+1}+x\right )^2+1\right )^2}-\frac {3 \log \left (\sqrt {x^2+1}-\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+x+1\right )}{8 \sqrt {2}}+\frac {3 \log \left (\sqrt {x^2+1}+\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+x+1\right )}{8 \sqrt {2}}-\frac {3 \tan ^{-1}\left (1-\sqrt {2} \sqrt {\sqrt {x^2+1}+x}\right )}{4 \sqrt {2}}+\frac {3 \tan ^{-1}\left (\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+1\right )}{4 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[x + Sqrt[1 + x^2]])/(1 + (x + Sqrt[1 + x^2])^2)^2 + Sqrt[x + Sqrt[1 + x^2]]/(2*(1 + (x + Sqrt[1 + x^2
])^2)) - (3*ArcTan[1 - Sqrt[2]*Sqrt[x + Sqrt[1 + x^2]]])/(4*Sqrt[2]) + (3*ArcTan[1 + Sqrt[2]*Sqrt[x + Sqrt[1 +
x^2]]])/(4*Sqrt[2]) - (3*Log[1 + x + Sqrt[1 + x^2] - Sqrt[2]*Sqrt[x + Sqrt[1 + x^2]]])/(8*Sqrt[2]) + (3*Log[1
+ x + Sqrt[1 + x^2] + Sqrt[2]*Sqrt[x + Sqrt[1 + x^2]]])/(8*Sqrt[2])

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 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

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 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

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

Rule 2122

Int[((g_) + (i_.)*(x_)^2)^(m_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_.), x_Symbol] :> Dis
t[(1*(i/c)^m)/(2^(2*m + 1)*e*f^(2*m)), Subst[Int[(x^n*(d^2 + a*f^2 - 2*d*x + x^2)^(2*m + 1))/(-d + x)^(2*(m +
1)), x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /; FreeQ[{a, c, d, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0] && E
qQ[c*g - a*i, 0] && IntegerQ[2*m] && (IntegerQ[m] || GtQ[i/c, 0])

Rubi steps

\begin {align*} \int \frac {1}{\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx &=8 \operatorname {Subst}\left (\int \frac {x^{3/2}}{\left (1+x^2\right )^3} \, dx,x,x+\sqrt {1+x^2}\right )\\ &=-\frac {2 \sqrt {x+\sqrt {1+x^2}}}{\left (1+\left (x+\sqrt {1+x^2}\right )^2\right )^2}+\operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+x^2\right )^2} \, dx,x,x+\sqrt {1+x^2}\right )\\ &=-\frac {2 \sqrt {x+\sqrt {1+x^2}}}{\left (1+\left (x+\sqrt {1+x^2}\right )^2\right )^2}+\frac {\sqrt {x+\sqrt {1+x^2}}}{2 \left (1+\left (x+\sqrt {1+x^2}\right )^2\right )}+\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )\\ &=-\frac {2 \sqrt {x+\sqrt {1+x^2}}}{\left (1+\left (x+\sqrt {1+x^2}\right )^2\right )^2}+\frac {\sqrt {x+\sqrt {1+x^2}}}{2 \left (1+\left (x+\sqrt {1+x^2}\right )^2\right )}+\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )\\ &=-\frac {2 \sqrt {x+\sqrt {1+x^2}}}{\left (1+\left (x+\sqrt {1+x^2}\right )^2\right )^2}+\frac {\sqrt {x+\sqrt {1+x^2}}}{2 \left (1+\left (x+\sqrt {1+x^2}\right )^2\right )}+\frac {3}{4} \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )+\frac {3}{4} \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )\\ &=-\frac {2 \sqrt {x+\sqrt {1+x^2}}}{\left (1+\left (x+\sqrt {1+x^2}\right )^2\right )^2}+\frac {\sqrt {x+\sqrt {1+x^2}}}{2 \left (1+\left (x+\sqrt {1+x^2}\right )^2\right )}+\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )+\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-\frac {3 \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )}{8 \sqrt {2}}-\frac {3 \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )}{8 \sqrt {2}}\\ &=-\frac {2 \sqrt {x+\sqrt {1+x^2}}}{\left (1+\left (x+\sqrt {1+x^2}\right )^2\right )^2}+\frac {\sqrt {x+\sqrt {1+x^2}}}{2 \left (1+\left (x+\sqrt {1+x^2}\right )^2\right )}-\frac {3 \log \left (1+x+\sqrt {1+x^2}-\sqrt {2} \sqrt {x+\sqrt {1+x^2}}\right )}{8 \sqrt {2}}+\frac {3 \log \left (1+x+\sqrt {1+x^2}+\sqrt {2} \sqrt {x+\sqrt {1+x^2}}\right )}{8 \sqrt {2}}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {x+\sqrt {1+x^2}}\right )}{4 \sqrt {2}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {x+\sqrt {1+x^2}}\right )}{4 \sqrt {2}}\\ &=-\frac {2 \sqrt {x+\sqrt {1+x^2}}}{\left (1+\left (x+\sqrt {1+x^2}\right )^2\right )^2}+\frac {\sqrt {x+\sqrt {1+x^2}}}{2 \left (1+\left (x+\sqrt {1+x^2}\right )^2\right )}-\frac {3 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x+\sqrt {1+x^2}}\right )}{4 \sqrt {2}}+\frac {3 \tan ^{-1}\left (1+\sqrt {2} \sqrt {x+\sqrt {1+x^2}}\right )}{4 \sqrt {2}}-\frac {3 \log \left (1+x+\sqrt {1+x^2}-\sqrt {2} \sqrt {x+\sqrt {1+x^2}}\right )}{8 \sqrt {2}}+\frac {3 \log \left (1+x+\sqrt {1+x^2}+\sqrt {2} \sqrt {x+\sqrt {1+x^2}}\right )}{8 \sqrt {2}}\\ \end {align*}

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Mathematica [A]  time = 0.23, size = 220, normalized size = 1.19 \begin {gather*} \frac {1}{16} \left (\frac {8 \sqrt {\sqrt {x^2+1}+x}}{\left (\sqrt {x^2+1}+x\right )^2+1}-\frac {8 \sqrt {\sqrt {x^2+1}+x}}{\left (x^2+\sqrt {x^2+1} x+1\right )^2}-3 \sqrt {2} \log \left (\sqrt {x^2+1}-\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+x+1\right )+3 \sqrt {2} \log \left (\sqrt {x^2+1}+\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+x+1\right )-6 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\sqrt {x^2+1}+x}\right )+6 \sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+1\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-8*Sqrt[x + Sqrt[1 + x^2]])/(1 + x^2 + x*Sqrt[1 + x^2])^2 + (8*Sqrt[x + Sqrt[1 + x^2]])/(1 + (x + Sqrt[1 + x
^2])^2) - 6*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[x + Sqrt[1 + x^2]]] + 6*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[x + Sqrt[1
+ x^2]]] - 3*Sqrt[2]*Log[1 + x + Sqrt[1 + x^2] - Sqrt[2]*Sqrt[x + Sqrt[1 + x^2]]] + 3*Sqrt[2]*Log[1 + x + Sqr
t[1 + x^2] + Sqrt[2]*Sqrt[x + Sqrt[1 + x^2]]])/16

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IntegrateAlgebraic [A]  time = 0.31, size = 185, normalized size = 1.00 \begin {gather*} -\frac {3 \sqrt {x+\sqrt {1+x^2}}}{8 \left (1+x^2+x \sqrt {1+x^2}\right )^2}+\frac {\left (x+\sqrt {1+x^2}\right )^{5/2}}{8 \left (1+x^2+x \sqrt {1+x^2}\right )^2}+\frac {3 \tan ^{-1}\left (\frac {-\frac {1}{\sqrt {2}}+\frac {x}{\sqrt {2}}+\frac {\sqrt {1+x^2}}{\sqrt {2}}}{\sqrt {x+\sqrt {1+x^2}}}\right )}{4 \sqrt {2}}+\frac {3 \tanh ^{-1}\left (\frac {\frac {1}{\sqrt {2}}+\frac {x}{\sqrt {2}}+\frac {\sqrt {1+x^2}}{\sqrt {2}}}{\sqrt {x+\sqrt {1+x^2}}}\right )}{4 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Sqrt[x + Sqrt[1 + x^2]])/(8*(1 + x^2 + x*Sqrt[1 + x^2])^2) + (x + Sqrt[1 + x^2])^(5/2)/(8*(1 + x^2 + x*Sqr
t[1 + x^2])^2) + (3*ArcTan[(-(1/Sqrt[2]) + x/Sqrt[2] + Sqrt[1 + x^2]/Sqrt[2])/Sqrt[x + Sqrt[1 + x^2]]])/(4*Sqr
t[2]) + (3*ArcTanh[(1/Sqrt[2] + x/Sqrt[2] + Sqrt[1 + x^2]/Sqrt[2])/Sqrt[x + Sqrt[1 + x^2]]])/(4*Sqrt[2])

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fricas [A]  time = 0.56, size = 248, normalized size = 1.34 \begin {gather*} -\frac {12 \, \sqrt {2} {\left (x^{2} + 1\right )} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} \sqrt {x + \sqrt {x^{2} + 1}} + x + \sqrt {x^{2} + 1} + 1} - \sqrt {2} \sqrt {x + \sqrt {x^{2} + 1}} - 1\right ) + 12 \, \sqrt {2} {\left (x^{2} + 1\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-4 \, \sqrt {2} \sqrt {x + \sqrt {x^{2} + 1}} + 4 \, x + 4 \, \sqrt {x^{2} + 1} + 4} - \sqrt {2} \sqrt {x + \sqrt {x^{2} + 1}} + 1\right ) - 3 \, \sqrt {2} {\left (x^{2} + 1\right )} \log \left (4 \, \sqrt {2} \sqrt {x + \sqrt {x^{2} + 1}} + 4 \, x + 4 \, \sqrt {x^{2} + 1} + 4\right ) + 3 \, \sqrt {2} {\left (x^{2} + 1\right )} \log \left (-4 \, \sqrt {2} \sqrt {x + \sqrt {x^{2} + 1}} + 4 \, x + 4 \, \sqrt {x^{2} + 1} + 4\right ) + 4 \, {\left (3 \, x^{2} - 3 \, \sqrt {x^{2} + 1} x + 1\right )} \sqrt {x + \sqrt {x^{2} + 1}}}{16 \, {\left (x^{2} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(12*sqrt(2)*(x^2 + 1)*arctan(sqrt(2)*sqrt(sqrt(2)*sqrt(x + sqrt(x^2 + 1)) + x + sqrt(x^2 + 1) + 1) - sqr
t(2)*sqrt(x + sqrt(x^2 + 1)) - 1) + 12*sqrt(2)*(x^2 + 1)*arctan(1/2*sqrt(2)*sqrt(-4*sqrt(2)*sqrt(x + sqrt(x^2
+ 1)) + 4*x + 4*sqrt(x^2 + 1) + 4) - sqrt(2)*sqrt(x + sqrt(x^2 + 1)) + 1) - 3*sqrt(2)*(x^2 + 1)*log(4*sqrt(2)*
sqrt(x + sqrt(x^2 + 1)) + 4*x + 4*sqrt(x^2 + 1) + 4) + 3*sqrt(2)*(x^2 + 1)*log(-4*sqrt(2)*sqrt(x + sqrt(x^2 +
1)) + 4*x + 4*sqrt(x^2 + 1) + 4) + 4*(3*x^2 - 3*sqrt(x^2 + 1)*x + 1)*sqrt(x + sqrt(x^2 + 1)))/(x^2 + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (x^{2}+1\right )^{2} \sqrt {x +\sqrt {x^{2}+1}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (x^2+1\right )}^2\,\sqrt {x+\sqrt {x^2+1}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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