∫ 1_______∘ 1+∘ _______ 1+√ __

Optimal. Leaf size=128 \[ \sqrt {x+1} \left (\frac {8}{7} \sqrt {\sqrt {x+1}+1} \sqrt {\sqrt {\sqrt {x+1}+1}+1}-\frac {48}{35} \sqrt {\sqrt {\sqrt {x+1}+1}+1}\right )+\frac {32}{105} \sqrt {\sqrt {x+1}+1} \sqrt {\sqrt {\sqrt {x+1}+1}+1}+\frac {32}{105} \sqrt {\sqrt {\sqrt {x+1}+1}+1} \]

________________________________________________________________________________________

Rubi [A] time = 0.07, antiderivative size = 70, normalized size of antiderivative = 0.55, number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {371, 1398, 772} \begin {gather*} \frac {8}{7} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{7/2}-\frac {24}{5} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{5/2}+\frac {16}{3} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{3/2} \end {gather*}

(16*(1 + Sqrt[1 + Sqrt[1 + x]])^(3/2))/3 - (24*(1 + Sqrt[1 + Sqrt[1 + x]])^(5/2))/5 + (8*(1 + Sqrt[1 + Sqrt[1

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coefficient[v, x, 0], d = Coefficient[v,

x, 1]}, Dist[1/d^(m + 1), Subst[Int[SimplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr

and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, D

ist[g, Subst[Int[x^(g - 1)*(d + e*x^(g*n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p

\begin {align*} \int \frac {1}{\sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+\sqrt {1+x}}} \, dx,x,\sqrt {1+x}\right )\\ &=2 \operatorname {Subst}\left (\int \frac {-1+x}{\sqrt {1+\sqrt {x}}} \, dx,x,1+\sqrt {1+x}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {x \left (-1+x^2\right )}{\sqrt {1+x}} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=4 \operatorname {Subst}\left (\int \left (2 \sqrt {1+x}-3 (1+x)^{3/2}+(1+x)^{5/2}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=\frac {16}{3} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{3/2}-\frac {24}{5} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{5/2}+\frac {8}{7} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{7/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A] time = 0.03, size = 49, normalized size = 0.38 \begin {gather*} \frac {8}{105} \left (15 \sqrt {x+1}-33 \sqrt {\sqrt {x+1}+1}+37\right ) \left (\sqrt {\sqrt {x+1}+1}+1\right )^{3/2} \end {gather*}

________________________________________________________________________________________

IntegrateAlgebraic [A] time = 0.06, size = 82, normalized size = 0.64 \begin {gather*} -\frac {16}{105} \left (-2+9 \sqrt {1+x}\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\frac {8}{105} \sqrt {1+\sqrt {1+x}} \left (4+15 \sqrt {1+x}\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}} \end {gather*}

(-16*(-2 + 9*Sqrt[1 + x])*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/105 + (8*Sqrt[1 + Sqrt[1 + x]]*(4 + 15*Sqrt[1 + x])

________________________________________________________________________________________

fricas [A] time = 0.46, size = 43, normalized size = 0.34 \begin {gather*} \frac {8}{105} \, {\left ({\left (15 \, \sqrt {x + 1} + 4\right )} \sqrt {\sqrt {x + 1} + 1} - 18 \, \sqrt {x + 1} + 4\right )} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} \end {gather*}

8/105*((15*sqrt(x + 1) + 4)*sqrt(sqrt(x + 1) + 1) - 18*sqrt(x + 1) + 4)*sqrt(sqrt(sqrt(x + 1) + 1) + 1)

________________________________________________________________________________________

giac [C] time = 1.32, size = 79, normalized size = 0.62 \begin {gather*} \frac {8 \, {\left (15 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {7}{2}} - 63 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {5}{2}} + 70 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {3}{2}}\right )}}{105 \, \mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right ) \mathrm {sgn}\left (4 \, x + 1\right )} \end {gather*}

8/105*(15*(sqrt(sqrt(x + 1) + 1) + 1)^(7/2) - 63*(sqrt(sqrt(x + 1) + 1) + 1)^(5/2) + 70*(sqrt(sqrt(x + 1) + 1)

________________________________________________________________________________________


method	result	size

derivativedivides	\(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {7}{2}}}{7}-\frac {24 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}+\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{3}\)	\(47\)
default	\(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {7}{2}}}{7}-\frac {24 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}+\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{3}\)	\(47\)

8/7*(1+(1+(1+x)^(1/2))^(1/2))^(7/2)-24/5*(1+(1+(1+x)^(1/2))^(1/2))^(5/2)+16/3*(1+(1+(1+x)^(1/2))^(1/2))^(3/2)

________________________________________________________________________________________

maxima [C] time = 0.32, size = 46, normalized size = 0.36 \begin {gather*} \frac {8}{7} \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {7}{2}} - \frac {24}{5} \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {5}{2}} + \frac {16}{3} \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {3}{2}} \end {gather*}

8/7*(sqrt(sqrt(x + 1) + 1) + 1)^(7/2) - 24/5*(sqrt(sqrt(x + 1) + 1) + 1)^(5/2) + 16/3*(sqrt(sqrt(x + 1) + 1) +

________________________________________________________________________________________

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

________________________________________________________________________________________

sympy [B] time = 1.72, size = 445, normalized size = 3.48 \begin {gather*} - \frac {336 \left (x + 1\right )^{\frac {13}{4}} \sqrt [4]{\sqrt {x + 1} + 1} \sin {\left (\frac {5 \operatorname {atan}{\left (\sqrt [4]{x + 1} \right )}}{2} \right )} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{105 \pi \left (x + 1\right )^{\frac {5}{2}} + 105 \pi \left (x + 1\right )^{2}} - \frac {112 \left (x + 1\right )^{\frac {11}{4}} \sqrt [4]{\sqrt {x + 1} + 1} \sin {\left (\frac {5 \operatorname {atan}{\left (\sqrt [4]{x + 1} \right )}}{2} \right )} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{105 \pi \left (x + 1\right )^{\frac {5}{2}} + 105 \pi \left (x + 1\right )^{2}} + \frac {224 \left (x + 1\right )^{\frac {9}{4}} \sqrt [4]{\sqrt {x + 1} + 1} \sin {\left (\frac {5 \operatorname {atan}{\left (\sqrt [4]{x + 1} \right )}}{2} \right )} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{105 \pi \left (x + 1\right )^{\frac {5}{2}} + 105 \pi \left (x + 1\right )^{2}} - \frac {152 \left (x + 1\right )^{\frac {5}{2}} \left (\sqrt {x + 1} + 1\right )^{\frac {3}{4}} \cos {\left (\frac {7 \operatorname {atan}{\left (\sqrt [4]{x + 1} \right )}}{2} \right )} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{105 \pi \left (x + 1\right )^{\frac {5}{2}} + 105 \pi \left (x + 1\right )^{2}} - \frac {64 \left (x + 1\right )^{\frac {5}{2}} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{105 \pi \left (x + 1\right )^{\frac {5}{2}} + 105 \pi \left (x + 1\right )^{2}} - \frac {216 \left (x + 1\right )^{3} \left (\sqrt {x + 1} + 1\right )^{\frac {3}{4}} \cos {\left (\frac {7 \operatorname {atan}{\left (\sqrt [4]{x + 1} \right )}}{2} \right )} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{105 \pi \left (x + 1\right )^{\frac {5}{2}} + 105 \pi \left (x + 1\right )^{2}} + \frac {64 \left (x + 1\right )^{2} \left (\sqrt {x + 1} + 1\right )^{\frac {3}{4}} \cos {\left (\frac {7 \operatorname {atan}{\left (\sqrt [4]{x + 1} \right )}}{2} \right )} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{105 \pi \left (x + 1\right )^{\frac {5}{2}} + 105 \pi \left (x + 1\right )^{2}} - \frac {64 \left (x + 1\right )^{2} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{105 \pi \left (x + 1\right )^{\frac {5}{2}} + 105 \pi \left (x + 1\right )^{2}} \end {gather*}

-336*(x + 1)**(13/4)*(sqrt(x + 1) + 1)**(1/4)*sin(5*atan((x + 1)**(1/4))/2)*gamma(1/4)*gamma(3/4)/(105*pi*(x +

1)**(5/2) + 105*pi*(x + 1)**2) - 112*(x + 1)**(11/4)*(sqrt(x + 1) + 1)**(1/4)*sin(5*atan((x + 1)**(1/4))/2)*g

amma(1/4)*gamma(3/4)/(105*pi*(x + 1)**(5/2) + 105*pi*(x + 1)**2) + 224*(x + 1)**(9/4)*(sqrt(x + 1) + 1)**(1/4)

*sin(5*atan((x + 1)**(1/4))/2)*gamma(1/4)*gamma(3/4)/(105*pi*(x + 1)**(5/2) + 105*pi*(x + 1)**2) - 152*(x + 1)

**(5/2)*(sqrt(x + 1) + 1)**(3/4)*cos(7*atan((x + 1)**(1/4))/2)*gamma(1/4)*gamma(3/4)/(105*pi*(x + 1)**(5/2) +

105*pi*(x + 1)**2) - 64*(x + 1)**(5/2)*gamma(1/4)*gamma(3/4)/(105*pi*(x + 1)**(5/2) + 105*pi*(x + 1)**2) - 216

*(x + 1)**3*(sqrt(x + 1) + 1)**(3/4)*cos(7*atan((x + 1)**(1/4))/2)*gamma(1/4)*gamma(3/4)/(105*pi*(x + 1)**(5/2

) + 105*pi*(x + 1)**2) + 64*(x + 1)**2*(sqrt(x + 1) + 1)**(3/4)*cos(7*atan((x + 1)**(1/4))/2)*gamma(1/4)*gamma

(3/4)/(105*pi*(x + 1)**(5/2) + 105*pi*(x + 1)**2) - 64*(x + 1)**2*gamma(1/4)*gamma(3/4)/(105*pi*(x + 1)**(5/2)

________________________________________________________________________________________

3.19.70 \(\int \frac {1}{\sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx\)