3.11.2 \(\int \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2} \, dx\) [1002]

Optimal. Leaf size=135 \[ -\frac {5}{64} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {5}{96} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{24} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {1}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{7/2}-\frac {5}{64} \cosh ^{-1}\left (\sqrt {x}\right ) \]

[Out]

-5/64*arccosh(x^(1/2))-5/96*x^(3/2)*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)-1/24*x^(5/2)*(-1+x^(1/2))^(1/2)*(1+x^
(1/2))^(1/2)+1/4*x^(7/2)*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)-5/64*x^(1/2)*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2
)

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Rubi [A]
time = 0.04, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {286, 329, 336, 54} \begin {gather*} \frac {1}{4} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{7/2}-\frac {1}{24} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{5/2}-\frac {5}{96} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}-\frac {5}{64} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}-\frac {5}{64} \cosh ^{-1}\left (\sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(5/2),x]

[Out]

(-5*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x])/64 - (5*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(3/2))/96 - (
Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(5/2))/24 + (Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(7/2))/4 - (5*ArcCo
sh[Sqrt[x]])/64

Rule 54

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[b*(x/a)]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rule 286

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] + Dist[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, a2, b2, c, m}, x] && EqQ[a2*b1 +
a1*b2, 0] && IGtQ[2*n, 0] && GtQ[p, 0] && NeQ[m + 2*n*p + 1, 0] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x
]

Rule 329

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] - Dist[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] && EqQ[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]

Rule 336

Int[((c_.)*(x_))^(m_)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k =
Denominator[m]}, Dist[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] && Fractio
nQ[m] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]

Rubi steps

\begin {align*} \int \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2} \, dx &=\frac {1}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{7/2}-\frac {1}{8} \int \frac {x^{5/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=-\frac {1}{24} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {1}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{7/2}-\frac {5}{48} \int \frac {x^{3/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=-\frac {5}{96} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{24} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {1}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{7/2}-\frac {5}{64} \int \frac {\sqrt {x}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=-\frac {5}{64} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {5}{96} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{24} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {1}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{7/2}-\frac {5}{128} \int \frac {1}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}} \, dx\\ &=-\frac {5}{64} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {5}{96} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{24} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {1}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{7/2}-\frac {5}{64} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {5}{64} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {5}{96} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{24} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {1}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{7/2}-\frac {5}{64} \cosh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}

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Mathematica [A]
time = 1.55, size = 99, normalized size = 0.73 \begin {gather*} \frac {1}{192} \left (\sqrt {\frac {-1+\sqrt {x}}{1+\sqrt {x}}} \sqrt {x} \left (-15-15 \sqrt {x}-10 x-10 x^{3/2}-8 x^2-8 x^{5/2}+48 x^3+48 x^{7/2}\right )-30 \tanh ^{-1}\left (\sqrt {\frac {-1+\sqrt {x}}{1+\sqrt {x}}}\right )\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(5/2),x]

[Out]

(Sqrt[(-1 + Sqrt[x])/(1 + Sqrt[x])]*Sqrt[x]*(-15 - 15*Sqrt[x] - 10*x - 10*x^(3/2) - 8*x^2 - 8*x^(5/2) + 48*x^3
+ 48*x^(7/2)) - 30*ArcTanh[Sqrt[(-1 + Sqrt[x])/(1 + Sqrt[x])]])/192

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Maple [A]
time = 0.35, size = 75, normalized size = 0.56

method	result	size
derivativedivides	\(-\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {\sqrt {x}+1}\, \left (-48 \sqrt {x -1}\, x^{\frac {7}{2}}+8 x^{\frac {5}{2}} \sqrt {x -1}+10 x^{\frac {3}{2}} \sqrt {x -1}+15 \sqrt {x}\, \sqrt {x -1}+15 \ln \left (\sqrt {x}+\sqrt {x -1}\right )\right )}{192 \sqrt {x -1}}\)	\(75\)
default	\(-\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {\sqrt {x}+1}\, \left (-48 \sqrt {x -1}\, x^{\frac {7}{2}}+8 x^{\frac {5}{2}} \sqrt {x -1}+10 x^{\frac {3}{2}} \sqrt {x -1}+15 \sqrt {x}\, \sqrt {x -1}+15 \ln \left (\sqrt {x}+\sqrt {x -1}\right )\right )}{192 \sqrt {x -1}}\)	\(75\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(-1+x^(1/2))^(1/2)*(x^(1/2)+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/192*(-1+x^(1/2))^(1/2)*(x^(1/2)+1)^(1/2)*(-48*(x-1)^(1/2)*x^(7/2)+8*x^(5/2)*(x-1)^(1/2)+10*x^(3/2)*(x-1)^(1
/2)+15*x^(1/2)*(x-1)^(1/2)+15*ln(x^(1/2)+(x-1)^(1/2)))/(x-1)^(1/2)

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Maxima [A]
time = 0.29, size = 57, normalized size = 0.42 \begin {gather*} \frac {1}{4} \, {\left (x - 1\right )}^{\frac {3}{2}} x^{\frac {5}{2}} + \frac {5}{24} \, {\left (x - 1\right )}^{\frac {3}{2}} x^{\frac {3}{2}} + \frac {5}{32} \, {\left (x - 1\right )}^{\frac {3}{2}} \sqrt {x} + \frac {5}{64} \, \sqrt {x - 1} \sqrt {x} - \frac {5}{64} \, \log \left (2 \, \sqrt {x - 1} + 2 \, \sqrt {x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2),x, algorithm="maxima")

[Out]

1/4*(x - 1)^(3/2)*x^(5/2) + 5/24*(x - 1)^(3/2)*x^(3/2) + 5/32*(x - 1)^(3/2)*sqrt(x) + 5/64*sqrt(x - 1)*sqrt(x)
- 5/64*log(2*sqrt(x - 1) + 2*sqrt(x))

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Fricas [A]
time = 1.30, size = 62, normalized size = 0.46 \begin {gather*} \frac {1}{192} \, {\left (48 \, x^{3} - 8 \, x^{2} - 10 \, x - 15\right )} \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \frac {5}{128} \, \log \left (2 \, \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - 2 \, x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2),x, algorithm="fricas")

[Out]

1/192*(48*x^3 - 8*x^2 - 10*x - 15)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) + 5/128*log(2*sqrt(x)*sqrt(sqrt
(x) + 1)*sqrt(sqrt(x) - 1) - 2*x + 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{\frac {5}{2}} \sqrt {\sqrt {x} - 1} \sqrt {\sqrt {x} + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(-1+x**(1/2))**(1/2)*(1+x**(1/2))**(1/2),x)

[Out]

Integral(x**(5/2)*sqrt(sqrt(x) - 1)*sqrt(sqrt(x) + 1), x)

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Giac [A]
time = 1.38, size = 162, normalized size = 1.20 \begin {gather*} \frac {1}{6720} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, {\left (7 \, \sqrt {x} - 50\right )} {\left (\sqrt {x} + 1\right )} + 1219\right )} {\left (\sqrt {x} + 1\right )} - 12463\right )} {\left (\sqrt {x} + 1\right )} + 64233\right )} {\left (\sqrt {x} + 1\right )} - 53963\right )} {\left (\sqrt {x} + 1\right )} + 59465\right )} {\left (\sqrt {x} + 1\right )} - 23205\right )} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \frac {1}{840} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, \sqrt {x} - 37\right )} {\left (\sqrt {x} + 1\right )} + 661\right )} {\left (\sqrt {x} + 1\right )} - 4551\right )} {\left (\sqrt {x} + 1\right )} + 4781\right )} {\left (\sqrt {x} + 1\right )} - 6335\right )} {\left (\sqrt {x} + 1\right )} + 2835\right )} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \frac {5}{32} \, \log \left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2),x, algorithm="giac")

[Out]

1/6720*((2*((4*(5*(6*(7*sqrt(x) - 50)*(sqrt(x) + 1) + 1219)*(sqrt(x) + 1) - 12463)*(sqrt(x) + 1) + 64233)*(sqr
t(x) + 1) - 53963)*(sqrt(x) + 1) + 59465)*(sqrt(x) + 1) - 23205)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) + 1/840*(
(2*((4*(5*(6*sqrt(x) - 37)*(sqrt(x) + 1) + 661)*(sqrt(x) + 1) - 4551)*(sqrt(x) + 1) + 4781)*(sqrt(x) + 1) - 63
35)*(sqrt(x) + 1) + 2835)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) + 5/32*log(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1)
)

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Mupad [B]
time = 52.03, size = 831, normalized size = 6.16 \begin {gather*} -\frac {5\,\mathrm {atanh}\left (\frac {\sqrt {\sqrt {x}-1}-\mathrm {i}}{\sqrt {\sqrt {x}+1}-1}\right )}{16}+\frac {-\frac {235\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^3}{48\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^3}+\frac {1723\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^5}{48\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^5}+\frac {72283\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^7}{16\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^7}+\frac {848801\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^9}{16\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^9}+\frac {4181067\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{11}}{16\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{11}}+\frac {10994181\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{13}}{16\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{13}}+\frac {17457599\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{15}}{16\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{15}}+\frac {17457599\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{17}}{16\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{17}}+\frac {10994181\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{19}}{16\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{19}}+\frac {4181067\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{21}}{16\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{21}}+\frac {848801\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{23}}{16\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{23}}+\frac {72283\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{25}}{16\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{25}}+\frac {1723\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{27}}{48\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{27}}-\frac {235\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{29}}{48\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{29}}+\frac {5\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{31}}{16\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{31}}+\frac {5\,\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}{16\,\left (\sqrt {\sqrt {x}+1}-1\right )}}{1+\frac {120\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^4}-\frac {560\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^6}+\frac {1820\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^8}-\frac {4368\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{10}}+\frac {8008\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{12}}-\frac {11440\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{14}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{14}}+\frac {12870\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{16}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{16}}-\frac {11440\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{18}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{18}}+\frac {8008\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{20}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{20}}-\frac {4368\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{22}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{22}}+\frac {1820\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{24}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{24}}-\frac {560\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{26}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{26}}+\frac {120\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{28}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{28}}-\frac {16\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{30}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{30}}+\frac {{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{32}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{32}}-\frac {16\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(x^(1/2) - 1)^(1/2)*(x^(1/2) + 1)^(1/2),x)

[Out]

((1723*((x^(1/2) - 1)^(1/2) - 1i)^5)/(48*((x^(1/2) + 1)^(1/2) - 1)^5) - (235*((x^(1/2) - 1)^(1/2) - 1i)^3)/(48
*((x^(1/2) + 1)^(1/2) - 1)^3) + (72283*((x^(1/2) - 1)^(1/2) - 1i)^7)/(16*((x^(1/2) + 1)^(1/2) - 1)^7) + (84880
1*((x^(1/2) - 1)^(1/2) - 1i)^9)/(16*((x^(1/2) + 1)^(1/2) - 1)^9) + (4181067*((x^(1/2) - 1)^(1/2) - 1i)^11)/(16
*((x^(1/2) + 1)^(1/2) - 1)^11) + (10994181*((x^(1/2) - 1)^(1/2) - 1i)^13)/(16*((x^(1/2) + 1)^(1/2) - 1)^13) +
(17457599*((x^(1/2) - 1)^(1/2) - 1i)^15)/(16*((x^(1/2) + 1)^(1/2) - 1)^15) + (17457599*((x^(1/2) - 1)^(1/2) -
1i)^17)/(16*((x^(1/2) + 1)^(1/2) - 1)^17) + (10994181*((x^(1/2) - 1)^(1/2) - 1i)^19)/(16*((x^(1/2) + 1)^(1/2)
- 1)^19) + (4181067*((x^(1/2) - 1)^(1/2) - 1i)^21)/(16*((x^(1/2) + 1)^(1/2) - 1)^21) + (848801*((x^(1/2) - 1)^
(1/2) - 1i)^23)/(16*((x^(1/2) + 1)^(1/2) - 1)^23) + (72283*((x^(1/2) - 1)^(1/2) - 1i)^25)/(16*((x^(1/2) + 1)^(
1/2) - 1)^25) + (1723*((x^(1/2) - 1)^(1/2) - 1i)^27)/(48*((x^(1/2) + 1)^(1/2) - 1)^27) - (235*((x^(1/2) - 1)^(
1/2) - 1i)^29)/(48*((x^(1/2) + 1)^(1/2) - 1)^29) + (5*((x^(1/2) - 1)^(1/2) - 1i)^31)/(16*((x^(1/2) + 1)^(1/2)
- 1)^31) + (5*((x^(1/2) - 1)^(1/2) - 1i))/(16*((x^(1/2) + 1)^(1/2) - 1)))/((120*((x^(1/2) - 1)^(1/2) - 1i)^4)/
((x^(1/2) + 1)^(1/2) - 1)^4 - (16*((x^(1/2) - 1)^(1/2) - 1i)^2)/((x^(1/2) + 1)^(1/2) - 1)^2 - (560*((x^(1/2) -
1)^(1/2) - 1i)^6)/((x^(1/2) + 1)^(1/2) - 1)^6 + (1820*((x^(1/2) - 1)^(1/2) - 1i)^8)/((x^(1/2) + 1)^(1/2) - 1)
^8 - (4368*((x^(1/2) - 1)^(1/2) - 1i)^10)/((x^(1/2) + 1)^(1/2) - 1)^10 + (8008*((x^(1/2) - 1)^(1/2) - 1i)^12)/
((x^(1/2) + 1)^(1/2) - 1)^12 - (11440*((x^(1/2) - 1)^(1/2) - 1i)^14)/((x^(1/2) + 1)^(1/2) - 1)^14 + (12870*((x
^(1/2) - 1)^(1/2) - 1i)^16)/((x^(1/2) + 1)^(1/2) - 1)^16 - (11440*((x^(1/2) - 1)^(1/2) - 1i)^18)/((x^(1/2) + 1
)^(1/2) - 1)^18 + (8008*((x^(1/2) - 1)^(1/2) - 1i)^20)/((x^(1/2) + 1)^(1/2) - 1)^20 - (4368*((x^(1/2) - 1)^(1/
2) - 1i)^22)/((x^(1/2) + 1)^(1/2) - 1)^22 + (1820*((x^(1/2) - 1)^(1/2) - 1i)^24)/((x^(1/2) + 1)^(1/2) - 1)^24
- (560*((x^(1/2) - 1)^(1/2) - 1i)^26)/((x^(1/2) + 1)^(1/2) - 1)^26 + (120*((x^(1/2) - 1)^(1/2) - 1i)^28)/((x^(
1/2) + 1)^(1/2) - 1)^28 - (16*((x^(1/2) - 1)^(1/2) - 1i)^30)/((x^(1/2) + 1)^(1/2) - 1)^30 + ((x^(1/2) - 1)^(1/
2) - 1i)^32/((x^(1/2) + 1)^(1/2) - 1)^32 + 1) - (5*atanh(((x^(1/2) - 1)^(1/2) - 1i)/((x^(1/2) + 1)^(1/2) - 1))
)/16

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