\(\int \frac {-1+x}{(1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx\) [1526]

Optimal result

Integrand size = 28, antiderivative size = 105 \[ \int \frac {-1+x}{(1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\frac {-8 (-128-15 x)+64 \sqrt {1+\sqrt {1+x}}+\sqrt {1+x} \left (8-24 \sqrt {1+\sqrt {1+x}}\right )}{105 \sqrt {1+\sqrt {1+\sqrt {1+x}}}}+4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}}{\sqrt {2}}\right ) \]

[Out]

1/105*(1024+120*x+64*(1+(1+x)^(1/2))^(1/2)+(1+x)^(1/2)*(8-24*(1+(1+x)^(1/2))^(1/2)))/(1+(1+(1+x)^(1/2))^(1/2))
^(1/2)+4*2^(1/2)*arctanh(1/2*(1+(1+(1+x)^(1/2))^(1/2))^(1/2)*2^(1/2))

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.18, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {1643, 65, 212} \[ \int \frac {-1+x}{(1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {\sqrt {\sqrt {x+1}+1}+1}}{\sqrt {2}}\right )+\frac {8}{7} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{7/2}-\frac {24}{5} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{5/2}+\frac {16}{3} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{3/2}+\frac {8}{\sqrt {\sqrt {\sqrt {x+1}+1}+1}} \]

[In]

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

[Out]

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 212

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

Rule 1643

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*
Pq*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {-2+x^2}{x \sqrt {1+\sqrt {1+x}}} \, dx,x,\sqrt {1+x}\right ) \\ & = 4 \text {Subst}\left (\int \frac {x \left (-2+\left (-1+x^2\right )^2\right )}{\sqrt {1+x} \left (-1+x^2\right )} \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = 4 \text {Subst}\left (\int \left (-\frac {1}{(1+x)^{3/2}}+\frac {1}{(1-x) \sqrt {1+x}}+2 \sqrt {1+x}-3 (1+x)^{3/2}+(1+x)^{5/2}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = \frac {8}{\sqrt {1+\sqrt {1+\sqrt {1+x}}}}+\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}+4 \text {Subst}\left (\int \frac {1}{(1-x) \sqrt {1+x}} \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = \frac {8}{\sqrt {1+\sqrt {1+\sqrt {1+x}}}}+\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}+8 \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right ) \\ & = \frac {8}{\sqrt {1+\sqrt {1+\sqrt {1+x}}}}+\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}+4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}}{\sqrt {2}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.97 \[ \int \frac {-1+x}{(1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\frac {-8 \sqrt {1+\sqrt {1+x}} \left (-8+3 \sqrt {1+x}\right )-8 \left (-113-\sqrt {1+x}-15 (1+x)\right )}{105 \sqrt {1+\sqrt {1+\sqrt {1+x}}}}+4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}}{\sqrt {2}}\right ) \]

[In]

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

[Out]

(-8*Sqrt[1 + Sqrt[1 + x]]*(-8 + 3*Sqrt[1 + x]) - 8*(-113 - Sqrt[1 + x] - 15*(1 + x)))/(105*Sqrt[1 + Sqrt[1 + S
qrt[1 + x]]]) + 4*Sqrt[2]*ArcTanh[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]/Sqrt[2]]

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.82

method	result	size
derivativedivides	\(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {7}{2}}}{7}-\frac {24 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}+\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{3}+\frac {8}{\sqrt {1+\sqrt {1+\sqrt {1+x}}}}+4 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}\, \sqrt {2}}{2}\right )\)	\(86\)
default	\(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {7}{2}}}{7}-\frac {24 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}+\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{3}+\frac {8}{\sqrt {1+\sqrt {1+\sqrt {1+x}}}}+4 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}\, \sqrt {2}}{2}\right )\)	\(86\)

[In]

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

[Out]

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)+
8/(1+(1+(1+x)^(1/2))^(1/2))^(1/2)+4*2^(1/2)*arctanh(1/2*(1+(1+(1+x)^(1/2))^(1/2))^(1/2)*2^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.43 \[ \int \frac {-1+x}{(1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\frac {2 \, {\left (105 \, \sqrt {2} {\left (x + 1\right )} \log \left (\frac {2 \, {\left (\sqrt {2} \sqrt {x + 1} \sqrt {\sqrt {x + 1} + 1} + \sqrt {2} \sqrt {x + 1}\right )} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} + x + 4 \, \sqrt {x + 1} \sqrt {\sqrt {x + 1} + 1} + 4 \, \sqrt {x + 1} + 1}{x + 1}\right ) - 4 \, {\left (3 \, {\left (6 \, x + 41\right )} \sqrt {x + 1} - {\left (15 \, {\left (x + 8\right )} \sqrt {x + 1} + 4 \, x + 4\right )} \sqrt {\sqrt {x + 1} + 1} - 4 \, x - 4\right )} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}\right )}}{105 \, {\left (x + 1\right )}} \]

[In]

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

[Out]

2/105*(105*sqrt(2)*(x + 1)*log((2*(sqrt(2)*sqrt(x + 1)*sqrt(sqrt(x + 1) + 1) + sqrt(2)*sqrt(x + 1))*sqrt(sqrt(
sqrt(x + 1) + 1) + 1) + x + 4*sqrt(x + 1)*sqrt(sqrt(x + 1) + 1) + 4*sqrt(x + 1) + 1)/(x + 1)) - 4*(3*(6*x + 41
)*sqrt(x + 1) - (15*(x + 8)*sqrt(x + 1) + 4*x + 4)*sqrt(sqrt(x + 1) + 1) - 4*x - 4)*sqrt(sqrt(sqrt(x + 1) + 1)
+ 1))/(x + 1)

Sympy [A] (verification not implemented)

Time = 2.89 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.27 \[ \int \frac {-1+x}{(1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\frac {8 \left (\sqrt {\sqrt {x + 1} + 1} + 1\right )^{\frac {7}{2}}}{7} - \frac {24 \left (\sqrt {\sqrt {x + 1} + 1} + 1\right )^{\frac {5}{2}}}{5} + \frac {16 \left (\sqrt {\sqrt {x + 1} + 1} + 1\right )^{\frac {3}{2}}}{3} - 2 \sqrt {2} \left (\log {\left (\sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} - \sqrt {2} \right )} - \log {\left (\sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} + \sqrt {2} \right )}\right ) + \frac {8}{\sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.02 \[ \int \frac {-1+x}{(1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\frac {8}{7} \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {7}{2}} - \frac {24}{5} \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {5}{2}} + \frac {16}{3} \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {3}{2}} - 2 \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}}{\sqrt {2} + \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}}\right ) + \frac {8}{\sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}} \]

[In]

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

[Out]

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) +
1)^(3/2) - 2*sqrt(2)*log(-(sqrt(2) - sqrt(sqrt(sqrt(x + 1) + 1) + 1))/(sqrt(2) + sqrt(sqrt(sqrt(x + 1) + 1) +
1))) + 8/sqrt(sqrt(sqrt(x + 1) + 1) + 1)

Giac [A] (verification not implemented)

none

Time = 7.03 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.05 \[ \int \frac {-1+x}{(1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\frac {8}{7} \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {7}{2}} - \frac {24}{5} \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {5}{2}} + \frac {16}{3} \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {3}{2}} - 2 \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} \right |}}{2 \, {\left (\sqrt {2} + \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}\right )}}\right ) + \frac {8}{\sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}} \]

[In]

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

[Out]

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) +
1)^(3/2) - 2*sqrt(2)*log(1/2*abs(-2*sqrt(2) + 2*sqrt(sqrt(sqrt(x + 1) + 1) + 1))/(sqrt(2) + sqrt(sqrt(sqrt(x
+ 1) + 1) + 1))) + 8/sqrt(sqrt(sqrt(x + 1) + 1) + 1)

Mupad [F(-1)]

Timed out. \[ \int \frac {-1+x}{(1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\int \frac {x-1}{\sqrt {\sqrt {\sqrt {x+1}+1}+1}\,\left (x+1\right )} \,d x \]

[In]

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

[Out]

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