∫ 1______ x2∘ _______ x+√ __

Optimal. Leaf size=53 \[ -\frac {1}{x \sqrt {\sqrt {x^2+1}+x}}+\tan ^{-1}\left (\sqrt {\sqrt {x^2+1}+x}\right )+\tanh ^{-1}\left (\sqrt {\sqrt {x^2+1}+x}\right ) \]

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Rubi [A] time = 0.06, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2119, 457, 329, 212, 206, 203} \begin {gather*} -\frac {1}{x \sqrt {\sqrt {x^2+1}+x}}+\tan ^{-1}\left (\sqrt {\sqrt {x^2+1}+x}\right )+\tanh ^{-1}\left (\sqrt {\sqrt {x^2+1}+x}\right ) \end {gather*}

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

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])

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])

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]

]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&

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]

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d

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

*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &

& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] &&

Int[((g_.) + (h_.)*(x_))^(m_.)*((e_.)*(x_) + (f_.)*Sqrt[(a_.) + (c_.)*(x_)^2])^(n_.), x_Symbol] :> Dist[1/(2^(

m + 1)*e^(m + 1)), Subst[Int[x^(n - m - 2)*(a*f^2 + x^2)*(-(a*f^2*h) + 2*e*g*x + h*x^2)^m, x], x, e*x + f*Sqrt

[a + c*x^2]], x] /; FreeQ[{a, c, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[m]

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

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Mathematica [C] time = 14.56, size = 983, normalized size = 18.55 \begin {gather*} \frac {159120 \left (x+\sqrt {x^2+1}\right )^{19/2} \left (x^2+\sqrt {x^2+1} x+1\right ) \left (\frac {16}{585} \, _4F_3\left (\frac {5}{4},2,2,2;1,1,\frac {17}{4};\left (x+\sqrt {x^2+1}\right )^2\right ) \left (\left (x+\sqrt {x^2+1}\right )^3+x+\sqrt {x^2+1}\right )^2+\frac {681 \left (x+\sqrt {x^2+1}\right )^6-1483 \left (x+\sqrt {x^2+1}\right )^4-6769 \left (x+\sqrt {x^2+1}\right )^2+5 \left (\left (x+\sqrt {x^2+1}\right )^8-248 \left (x+\sqrt {x^2+1}\right )^6+102 \left (x+\sqrt {x^2+1}\right )^4+1208 \left (x+\sqrt {x^2+1}\right )^2+729\right ) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\left (x+\sqrt {x^2+1}\right )^2\right )-3645}{640 \left (x+\sqrt {x^2+1}\right )^4}\right )}{x \left (\frac {x}{\sqrt {x^2+1}}+1\right ) \left (\left (x+\sqrt {x^2+1}\right )^2+1\right ) \left (1989 \left (54400 x^{10}+54400 \sqrt {x^2+1} x^9+107808 x^8+80608 \sqrt {x^2+1} x^7+90944 x^6+57440 \sqrt {x^2+1} x^5+42462 x^4+20418 \sqrt {x^2+1} x^3+6909 x^2+10 \left (256 x^{11}+256 \sqrt {x^2+1} x^{10}-7296 x^9-7424 \sqrt {x^2+1} x^8-15296 x^7-11552 \sqrt {x^2+1} x^6-12608 x^5-7776 \sqrt {x^2+1} x^4-5590 x^3-2672 \sqrt {x^2+1} x^2-971 x-182 \sqrt {x^2+1}\right ) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\left (x+\sqrt {x^2+1}\right )^2\right ) x+967 \sqrt {x^2+1} x-140\right )+4352 x \left (26624 x^{13}+26624 \sqrt {x^2+1} x^{12}+89088 x^{11}+75776 \sqrt {x^2+1} x^{10}+115200 x^9+80640 \sqrt {x^2+1} x^8+71936 x^7+39424 \sqrt {x^2+1} x^6+22008 x^5+8680 \sqrt {x^2+1} x^4+2916 x^3+696 \sqrt {x^2+1} x^2+112 x+9 \sqrt {x^2+1}\right ) \, _4F_3\left (\frac {5}{4},2,2,2;1,1,\frac {17}{4};\left (x+\sqrt {x^2+1}\right )^2\right )+40960 x \left (8192 x^{15}+8192 \sqrt {x^2+1} x^{14}+32768 x^{13}+28672 \sqrt {x^2+1} x^{12}+52736 x^{11}+39424 \sqrt {x^2+1} x^{10}+43520 x^9+26880 \sqrt {x^2+1} x^8+19392 x^7+9408 \sqrt {x^2+1} x^6+4480 x^5+1568 \sqrt {x^2+1} x^4+462 x^3+98 \sqrt {x^2+1} x^2+14 x+\sqrt {x^2+1}\right ) \, _4F_3\left (\frac {9}{4},3,3,3;2,2,\frac {21}{4};\left (x+\sqrt {x^2+1}\right )^2\right )\right )}-\frac {\left (\left (x+\sqrt {x^2+1}\right )^2-1\right ) \left (-2 \tan ^{-1}\left (\sqrt {x+\sqrt {x^2+1}}\right )-2 \tanh ^{-1}\left (\sqrt {x+\sqrt {x^2+1}}\right )+2 \sqrt {x+\sqrt {x^2+1}}\right )}{2 x \left (\frac {x}{\sqrt {x^2+1}}+1\right ) \left (x+\sqrt {x^2+1}\right ) \left (1-\frac {\left (x+\sqrt {x^2+1}\right )^2-1}{2 \left (x+\sqrt {x^2+1}\right )^2}\right )} \end {gather*}

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

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

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

x + Sqrt[1 + x^2])^2))) + (159120*(x + Sqrt[1 + x^2])^(19/2)*(1 + x^2 + x*Sqrt[1 + x^2])*((-3645 - 6769*(x + S

qrt[1 + x^2])^2 - 1483*(x + Sqrt[1 + x^2])^4 + 681*(x + Sqrt[1 + x^2])^6 + 5*(729 + 1208*(x + Sqrt[1 + x^2])^2

+ 102*(x + Sqrt[1 + x^2])^4 - 248*(x + Sqrt[1 + x^2])^6 + (x + Sqrt[1 + x^2])^8)*Hypergeometric2F1[1/4, 1, 5/

4, (x + Sqrt[1 + x^2])^2])/(640*(x + Sqrt[1 + x^2])^4) + (16*(x + Sqrt[1 + x^2] + (x + Sqrt[1 + x^2])^3)^2*Hyp

ergeometricPFQ[{5/4, 2, 2, 2}, {1, 1, 17/4}, (x + Sqrt[1 + x^2])^2])/585))/(x*(1 + x/Sqrt[1 + x^2])*(1 + (x +

Sqrt[1 + x^2])^2)*(1989*(-140 + 6909*x^2 + 42462*x^4 + 90944*x^6 + 107808*x^8 + 54400*x^10 + 967*x*Sqrt[1 + x^

2] + 20418*x^3*Sqrt[1 + x^2] + 57440*x^5*Sqrt[1 + x^2] + 80608*x^7*Sqrt[1 + x^2] + 54400*x^9*Sqrt[1 + x^2] + 1

0*x*(-971*x - 5590*x^3 - 12608*x^5 - 15296*x^7 - 7296*x^9 + 256*x^11 - 182*Sqrt[1 + x^2] - 2672*x^2*Sqrt[1 + x

^2] - 7776*x^4*Sqrt[1 + x^2] - 11552*x^6*Sqrt[1 + x^2] - 7424*x^8*Sqrt[1 + x^2] + 256*x^10*Sqrt[1 + x^2])*Hype

rgeometric2F1[1/4, 1, 5/4, (x + Sqrt[1 + x^2])^2]) + 4352*x*(112*x + 2916*x^3 + 22008*x^5 + 71936*x^7 + 115200

*x^9 + 89088*x^11 + 26624*x^13 + 9*Sqrt[1 + x^2] + 696*x^2*Sqrt[1 + x^2] + 8680*x^4*Sqrt[1 + x^2] + 39424*x^6*

Sqrt[1 + x^2] + 80640*x^8*Sqrt[1 + x^2] + 75776*x^10*Sqrt[1 + x^2] + 26624*x^12*Sqrt[1 + x^2])*HypergeometricP

FQ[{5/4, 2, 2, 2}, {1, 1, 17/4}, (x + Sqrt[1 + x^2])^2] + 40960*x*(14*x + 462*x^3 + 4480*x^5 + 19392*x^7 + 435

20*x^9 + 52736*x^11 + 32768*x^13 + 8192*x^15 + Sqrt[1 + x^2] + 98*x^2*Sqrt[1 + x^2] + 1568*x^4*Sqrt[1 + x^2] +

9408*x^6*Sqrt[1 + x^2] + 26880*x^8*Sqrt[1 + x^2] + 39424*x^10*Sqrt[1 + x^2] + 28672*x^12*Sqrt[1 + x^2] + 8192

*x^14*Sqrt[1 + x^2])*HypergeometricPFQ[{9/4, 3, 3, 3}, {2, 2, 21/4}, (x + Sqrt[1 + x^2])^2]))

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IntegrateAlgebraic [A] time = 0.12, size = 53, normalized size = 1.00 \begin {gather*} -\frac {1}{x \sqrt {x+\sqrt {1+x^2}}}+\tan ^{-1}\left (\sqrt {x+\sqrt {1+x^2}}\right )+\tanh ^{-1}\left (\sqrt {x+\sqrt {1+x^2}}\right ) \end {gather*}

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

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

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fricas [A] time = 0.55, size = 78, normalized size = 1.47 \begin {gather*} \frac {2 \, x \arctan \left (\sqrt {x + \sqrt {x^{2} + 1}}\right ) + x \log \left (\sqrt {x + \sqrt {x^{2} + 1}} + 1\right ) - x \log \left (\sqrt {x + \sqrt {x^{2} + 1}} - 1\right ) + 2 \, \sqrt {x + \sqrt {x^{2} + 1}} {\left (x - \sqrt {x^{2} + 1}\right )}}{2 \, x} \end {gather*}

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

1/2*(2*x*arctan(sqrt(x + sqrt(x^2 + 1))) + x*log(sqrt(x + sqrt(x^2 + 1)) + 1) - x*log(sqrt(x + sqrt(x^2 + 1))

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

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

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

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

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method	result	size

meijerg	\(-\frac {\sqrt {2}\, \hypergeom \left (\left [\frac {1}{4}, \frac {3}{4}, \frac {3}{4}\right ], \left [\frac {3}{2}, \frac {7}{4}\right ], -\frac {1}{x^{2}}\right )}{3 x^{\frac {3}{2}}}\)	\(22\)

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

-1/3*2^(1/2)/x^(3/2)*hypergeom([1/4,3/4,3/4],[3/2,7/4],-1/x^2)

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

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

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

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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{x^2\,\sqrt {x+\sqrt {x^2+1}}} \,d x \end {gather*}

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sympy [C] time = 1.76, size = 44, normalized size = 0.83 \begin {gather*} - \frac {\Gamma \left (\frac {1}{4}\right ) \Gamma ^{2}\left (\frac {3}{4}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4}, \frac {3}{4} \\ \frac {3}{2}, \frac {7}{4} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{2}}} \right )}}{4 \pi x^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} \end {gather*}

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

-gamma(1/4)*gamma(3/4)**2*hyper((1/4, 3/4, 3/4), (3/2, 7/4), exp_polar(I*pi)/x**2)/(4*pi*x**(3/2)*gamma(7/4))

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