Integrand size = 28, antiderivative size = 73 \[ \int \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x} \, dx=-\frac {1}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}+\frac {1}{2} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {\text {arccosh}\left (\sqrt {x}\right )}{4} \]
-1/4*arccosh(x^(1/2))+1/2*x^(3/2)*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)-1/4 *x^(1/2)*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(404\) vs. \(2(73)=146\).
Time = 1.93 (sec) , antiderivative size = 404, normalized size of antiderivative = 5.53 \[ \int \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x} \, dx=\frac {-4 \sqrt {1+\sqrt {x}} \left (-18816+28224 \sqrt {x}+55360 x+17296 x^{3/2}+7240 x^2-1096 x^{5/2}-4752 x^3-1136 x^{7/2}\right )-4 \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \left (32592+74488 \sqrt {x}+38632 x+6992 x^{3/2}-104 x^2-6079 x^{5/2}-3120 x^3-194 x^{7/2}\right )+\sqrt {3} \left (-4 \sqrt {-1+\sqrt {x}} \left (-18816-52416 \sqrt {x}-41472 x-10928 x^{3/2}-1192 x^2+3832 x^{5/2}+3408 x^3+656 x^{7/2}\right )-4 \left (10864-10872 \sqrt {x}-41440 x-23268 x^{3/2}-6678 x^2-1148 x^{5/2}+3416 x^3+1800 x^{7/2}+112 x^4\right )\right )}{-12416+13312 \sqrt {x}+49408 x+24960 x^{3/2}+1552 x^2+\sqrt {3} \sqrt {1+\sqrt {x}} \left (7168-11264 \sqrt {x}-22016 x-5248 x^{3/2}\right )+\sqrt {-1+\sqrt {x}} \left (21504+60416 \sqrt {x}+47104 x+9088 x^{3/2}+\sqrt {3} \sqrt {1+\sqrt {x}} \left (-12416-28672 \sqrt {x}-14400 x-896 x^{3/2}\right )\right )}+\text {arctanh}\left (\frac {-1+\sqrt {-1+\sqrt {x}}}{\sqrt {3}-\sqrt {1+\sqrt {x}}}\right ) \]
(-4*Sqrt[1 + Sqrt[x]]*(-18816 + 28224*Sqrt[x] + 55360*x + 17296*x^(3/2) + 7240*x^2 - 1096*x^(5/2) - 4752*x^3 - 1136*x^(7/2)) - 4*Sqrt[-1 + Sqrt[x]]* Sqrt[1 + Sqrt[x]]*(32592 + 74488*Sqrt[x] + 38632*x + 6992*x^(3/2) - 104*x^ 2 - 6079*x^(5/2) - 3120*x^3 - 194*x^(7/2)) + Sqrt[3]*(-4*Sqrt[-1 + Sqrt[x] ]*(-18816 - 52416*Sqrt[x] - 41472*x - 10928*x^(3/2) - 1192*x^2 + 3832*x^(5 /2) + 3408*x^3 + 656*x^(7/2)) - 4*(10864 - 10872*Sqrt[x] - 41440*x - 23268 *x^(3/2) - 6678*x^2 - 1148*x^(5/2) + 3416*x^3 + 1800*x^(7/2) + 112*x^4)))/ (-12416 + 13312*Sqrt[x] + 49408*x + 24960*x^(3/2) + 1552*x^2 + Sqrt[3]*Sqr t[1 + Sqrt[x]]*(7168 - 11264*Sqrt[x] - 22016*x - 5248*x^(3/2)) + Sqrt[-1 + Sqrt[x]]*(21504 + 60416*Sqrt[x] + 47104*x + 9088*x^(3/2) + Sqrt[3]*Sqrt[1 + Sqrt[x]]*(-12416 - 28672*Sqrt[x] - 14400*x - 896*x^(3/2)))) + ArcTanh[( -1 + Sqrt[-1 + Sqrt[x]])/(Sqrt[3] - Sqrt[1 + Sqrt[x]])]
Time = 0.19 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {812, 845, 852, 43}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x} \, dx\) |
\(\Big \downarrow \) 812 |
\(\displaystyle \frac {1}{2} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}-\frac {1}{4} \int \frac {\sqrt {x}}{\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}dx\) |
\(\Big \downarrow \) 845 |
\(\displaystyle \frac {1}{4} \left (-\frac {1}{2} \int \frac {1}{\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}}dx-\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}\right )+\frac {1}{2} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}\) |
\(\Big \downarrow \) 852 |
\(\displaystyle \frac {1}{4} \left (-\int \frac {1}{\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}d\sqrt {x}-\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}\right )+\frac {1}{2} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}\) |
\(\Big \downarrow \) 43 |
\(\displaystyle \frac {1}{4} \left (-\text {arccosh}\left (\sqrt {x}\right )-\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}\right )+\frac {1}{2} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}\) |
(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(3/2))/2 + (-(Sqrt[-1 + Sqrt[x]]*S qrt[1 + Sqrt[x]]*Sqrt[x]) - ArcCosh[Sqrt[x]])/4
3.11.4.3.1 Defintions of rubi rules used
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a *d, 0] && GtQ[a, 0] && GtQ[d/b, 0]
Int[((c_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_) ^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*(a1 + b1*x^n)^p*((a2 + b2*x^n) ^p/(c*(m + 2*n*p + 1))), x] + Simp[2*a1*a2*n*(p/(m + 2*n*p + 1)) Int[(c*x )^m*(a1 + b1*x^n)^(p - 1)*(a2 + b2*x^n)^(p - 1), x], x] /; FreeQ[{a1, b1, a 2, b2, c, m}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && GtQ[p, 0] && N eQ[m + 2*n*p + 1, 0] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]
Int[((c_.)*(x_))^(m_)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^ (n_))^(p_), x_Symbol] :> Simp[c^(2*n - 1)*(c*x)^(m - 2*n + 1)*(a1 + b1*x^n) ^(p + 1)*((a2 + b2*x^n)^(p + 1)/(b1*b2*(m + 2*n*p + 1))), x] - Simp[a1*a2*c ^(2*n)*((m - 2*n + 1)/(b1*b2*(m + 2*n*p + 1))) Int[(c*x)^(m - 2*n)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, p}, x] && Eq Q[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && GtQ[m, 2*n - 1] && NeQ[m + 2*n*p + 1 , 0] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]
Int[((c_.)*(x_))^(m_)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^ (n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/c Subst[Int[x^ (k*(m + 1) - 1)*(a1 + b1*(x^(k*n)/c^n))^p*(a2 + b2*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a1, b1, a2, b2, c, p}, x] && EqQ[a2*b1 + a1*b2 , 0] && IGtQ[2*n, 0] && FractionQ[m] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]
Time = 4.59 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(-\frac {\sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (-2 x^{\frac {3}{2}} \sqrt {-1+x}+\sqrt {x}\, \sqrt {-1+x}+\ln \left (\sqrt {x}+\sqrt {-1+x}\right )\right )}{4 \sqrt {-1+x}}\) | \(52\) |
default | \(-\frac {\sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (-2 x^{\frac {3}{2}} \sqrt {-1+x}+\sqrt {x}\, \sqrt {-1+x}+\ln \left (\sqrt {x}+\sqrt {-1+x}\right )\right )}{4 \sqrt {-1+x}}\) | \(52\) |
-1/4*(x^(1/2)-1)^(1/2)*(x^(1/2)+1)^(1/2)*(-2*x^(3/2)*(-1+x)^(1/2)+x^(1/2)* (-1+x)^(1/2)+ln(x^(1/2)+(-1+x)^(1/2)))/(-1+x)^(1/2)
Time = 0.28 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.71 \[ \int \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x} \, dx=\frac {1}{4} \, {\left (2 \, x - 1\right )} \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \frac {1}{8} \, \log \left (2 \, \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - 2 \, x + 1\right ) \]
1/4*(2*x - 1)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) + 1/8*log(2*sqrt (x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 2*x + 1)
\[ \int \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x} \, dx=\int \sqrt {x} \sqrt {\sqrt {x} - 1} \sqrt {\sqrt {x} + 1}\, dx \]
Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.51 \[ \int \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x} \, dx=\frac {1}{2} \, {\left (x - 1\right )}^{\frac {3}{2}} \sqrt {x} + \frac {1}{4} \, \sqrt {x - 1} \sqrt {x} - \frac {1}{4} \, \log \left (2 \, \sqrt {x - 1} + 2 \, \sqrt {x}\right ) \]
Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (45) = 90\).
Time = 0.29 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.26 \[ \int \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x} \, dx=\frac {1}{12} \, {\left ({\left (2 \, {\left (3 \, \sqrt {x} - 10\right )} {\left (\sqrt {x} + 1\right )} + 43\right )} {\left (\sqrt {x} + 1\right )} - 39\right )} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \frac {1}{3} \, {\left ({\left (2 \, \sqrt {x} - 5\right )} {\left (\sqrt {x} + 1\right )} + 9\right )} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \frac {1}{2} \, \log \left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right ) \]
1/12*((2*(3*sqrt(x) - 10)*(sqrt(x) + 1) + 43)*(sqrt(x) + 1) - 39)*sqrt(sqr t(x) + 1)*sqrt(sqrt(x) - 1) + 1/3*((2*sqrt(x) - 5)*(sqrt(x) + 1) + 9)*sqrt (sqrt(x) + 1)*sqrt(sqrt(x) - 1) + 1/2*log(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))
Timed out. \[ \int \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x} \, dx=\int \sqrt {x}\,\sqrt {\sqrt {x}-1}\,\sqrt {\sqrt {x}+1} \,d x \]