3.111 \(\int \frac {\log (-1+4 x+4 \sqrt {(-1+x) x})}{x^{3/2}} \, dx\)

Optimal. Leaf size=114 \[ -\frac {2 \log \left (4 \sqrt {x^2-x}+4 x-1\right )}{\sqrt {x}}-\frac {4 \sqrt {2} \sqrt {x^2-x} \tan ^{-1}\left (\frac {2}{3} \sqrt {2} \sqrt {x-1}\right )}{\sqrt {x-1} \sqrt {x}}-8 \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {x^2-x}}\right )+4 \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right ) \]

[Out]

-8*arctan(x^(1/2)/(x^2-x)^(1/2))+4*arctan(2*2^(1/2)*x^(1/2))*2^(1/2)-2*ln(-1+4*x+4*(x^2-x)^(1/2))/x^(1/2)-4*ar
ctan(2/3*2^(1/2)*(-1+x)^(1/2))*(x^2-x)^(1/2)*2^(1/2)/(-1+x)^(1/2)/x^(1/2)

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Rubi [A]  time = 0.31, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 13, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {2537, 2535, 6733, 6742, 203, 1588, 2021, 2008, 1146, 444, 50, 63, 204} \[ -\frac {2 \log \left (4 \sqrt {x^2-x}+4 x-1\right )}{\sqrt {x}}-\frac {4 \sqrt {2} \sqrt {x^2-x} \tan ^{-1}\left (\frac {2}{3} \sqrt {2} \sqrt {x-1}\right )}{\sqrt {x-1} \sqrt {x}}-8 \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {x^2-x}}\right )+4 \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Log[-1 + 4*x + 4*Sqrt[(-1 + x)*x]]/x^(3/2),x]

[Out]

(-4*Sqrt[2]*Sqrt[-x + x^2]*ArcTan[(2*Sqrt[2]*Sqrt[-1 + x])/3])/(Sqrt[-1 + x]*Sqrt[x]) + 4*Sqrt[2]*ArcTan[2*Sqr
t[2]*Sqrt[x]] - 8*ArcTan[Sqrt[x]/Sqrt[-x + x^2]] - (2*Log[-1 + 4*x + 4*Sqrt[-x + x^2]])/Sqrt[x]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

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 203

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])

Rule 204

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

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 1146

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(b*x^2 + c*x^4)^FracPart
[p]/(x^(2*FracPart[p])*(b + c*x^2)^FracPart[p]), Int[x^(2*p)*(d + e*x^2)^q*(b + c*x^2)^p, x], x] /; FreeQ[{b,
c, d, e, p, q}, x] &&  !IntegerQ[p]

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rule 2008

Int[1/Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[2/(2 - n), Subst[Int[1/(1 - a*x^2), x], x, x/Sq
rt[a*x^2 + b*x^n]], x] /; FreeQ[{a, b, n}, x] && NeQ[n, 2]

Rule 2021

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a*x^j + b
*x^n)^p)/(c*(m + n*p + 1)), x] + Dist[(a*(n - j)*p)/(c^j*(m + n*p + 1)), Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p
- 1), x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && G
tQ[p, 0] && NeQ[m + n*p + 1, 0]

Rule 2535

Int[Log[(d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]]*((g_.)*(x_))^(m_.), x_Symbol] :> S
imp[((g*x)^(m + 1)*Log[d + e*x + f*Sqrt[a + b*x + c*x^2]])/(g*(m + 1)), x] + Dist[(f^2*(b^2 - 4*a*c))/(2*g*(m
+ 1)), Int[(g*x)^(m + 1)/((2*d*e - b*f^2)*(a + b*x + c*x^2) - f*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x +
c*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[e^2 - c*f^2, 0] && NeQ[m, -1] && IntegerQ[2*m]

Rule 2537

Int[Log[(d_.) + (f_.)*Sqrt[u_] + (e_.)*(x_)]*(v_.), x_Symbol] :> Int[v*Log[d + e*x + f*Sqrt[ExpandToSum[u, x]]
], x] /; FreeQ[{d, e, f}, x] && QuadraticQ[u, x] &&  !QuadraticMatchQ[u, x] && (EqQ[v, 1] || MatchQ[v, ((g_.)*
x)^(m_.) /; FreeQ[{g, m}, x]])

Rule 6733

Int[(u_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(u /. x -> x^k
), x], x, x^(1/k)], x]] /; FractionQ[m]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x^{3/2}} \, dx &=\int \frac {\log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{x^{3/2}} \, dx\\ &=-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{\sqrt {x}}-16 \int \frac {1}{\sqrt {x} \left (-4 (1+2 x) \sqrt {-x+x^2}+8 \left (-x+x^2\right )\right )} \, dx\\ &=-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{\sqrt {x}}-32 \operatorname {Subst}\left (\int \frac {1}{-4 \left (1+2 x^2\right ) \sqrt {-x^2+x^4}+8 \left (-x^2+x^4\right )} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{\sqrt {x}}-32 \operatorname {Subst}\left (\int \left (-\frac {1}{2 \left (1+8 x^2\right )}-\frac {x^2}{12 \sqrt {-x^2+x^4}}+\frac {\sqrt {-x^2+x^4}}{4 x^2}+\frac {4 \sqrt {-x^2+x^4}}{3 \left (-1-8 x^2\right )}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{\sqrt {x}}+\frac {8}{3} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-x^2+x^4}} \, dx,x,\sqrt {x}\right )-8 \operatorname {Subst}\left (\int \frac {\sqrt {-x^2+x^4}}{x^2} \, dx,x,\sqrt {x}\right )+16 \operatorname {Subst}\left (\int \frac {1}{1+8 x^2} \, dx,x,\sqrt {x}\right )-\frac {128}{3} \operatorname {Subst}\left (\int \frac {\sqrt {-x^2+x^4}}{-1-8 x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {16 \sqrt {-x+x^2}}{3 \sqrt {x}}+4 \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{\sqrt {x}}+8 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-x^2+x^4}} \, dx,x,\sqrt {x}\right )-\frac {\left (128 \sqrt {-x+x^2}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt {-1+x^2}}{-1-8 x^2} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-1+x} \sqrt {x}}\\ &=-\frac {16 \sqrt {-x+x^2}}{3 \sqrt {x}}+4 \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{\sqrt {x}}-8 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {-x+x^2}}\right )-\frac {\left (64 \sqrt {-x+x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{-1-8 x} \, dx,x,x\right )}{3 \sqrt {-1+x} \sqrt {x}}\\ &=4 \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )-8 \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-x+x^2}}\right )-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{\sqrt {x}}+\frac {\left (24 \sqrt {-x+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{(-1-8 x) \sqrt {-1+x}} \, dx,x,x\right )}{\sqrt {-1+x} \sqrt {x}}\\ &=4 \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )-8 \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-x+x^2}}\right )-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{\sqrt {x}}+\frac {\left (48 \sqrt {-x+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-9-8 x^2} \, dx,x,\sqrt {-1+x}\right )}{\sqrt {-1+x} \sqrt {x}}\\ &=-\frac {4 \sqrt {2} \sqrt {-x+x^2} \tan ^{-1}\left (\frac {2}{3} \sqrt {2} \sqrt {-1+x}\right )}{\sqrt {-1+x} \sqrt {x}}+4 \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )-8 \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-x+x^2}}\right )-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{\sqrt {x}}\\ \end {align*}

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Mathematica [C]  time = 0.46, size = 177, normalized size = 1.55 \[ -2 i \sqrt {2} \log \left (4 (8 x+1)^2\right )+i \sqrt {2} \log \left ((8 x+1) \left (-10 x-6 \sqrt {(x-1) x}+1\right )\right )-\frac {2 \log \left (4 x+4 \sqrt {(x-1) x}-1\right )}{\sqrt {x}}+i \sqrt {2} \log \left ((8 x+1) \left (-10 x+6 \sqrt {(x-1) x}+1\right )\right )+4 \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )+8 \tan ^{-1}\left (\frac {\sqrt {(x-1) x}}{\sqrt {x}}\right )-4 \sqrt {2} \tan ^{-1}\left (\frac {2 \sqrt {2} \sqrt {(x-1) x}}{3 \sqrt {x}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[-1 + 4*x + 4*Sqrt[(-1 + x)*x]]/x^(3/2),x]

[Out]

4*Sqrt[2]*ArcTan[2*Sqrt[2]*Sqrt[x]] + 8*ArcTan[Sqrt[(-1 + x)*x]/Sqrt[x]] - 4*Sqrt[2]*ArcTan[(2*Sqrt[2]*Sqrt[(-
1 + x)*x])/(3*Sqrt[x])] - (2*I)*Sqrt[2]*Log[4*(1 + 8*x)^2] + I*Sqrt[2]*Log[(1 + 8*x)*(1 - 10*x - 6*Sqrt[(-1 +
x)*x])] - (2*Log[-1 + 4*x + 4*Sqrt[(-1 + x)*x]])/Sqrt[x] + I*Sqrt[2]*Log[(1 + 8*x)*(1 - 10*x + 6*Sqrt[(-1 + x)
*x])]

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fricas [A]  time = 0.81, size = 84, normalized size = 0.74 \[ \frac {2 \, {\left (2 \, \sqrt {2} x \arctan \left (2 \, \sqrt {2} \sqrt {x}\right ) + 2 \, \sqrt {2} x \arctan \left (\frac {3 \, \sqrt {2} \sqrt {x}}{4 \, \sqrt {x^{2} - x}}\right ) - 4 \, x \arctan \left (\frac {\sqrt {x}}{\sqrt {x^{2} - x}}\right ) - \sqrt {x} \log \left (4 \, x + 4 \, \sqrt {x^{2} - x} - 1\right )\right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(-1+4*x+4*((-1+x)*x)^(1/2))/x^(3/2),x, algorithm="fricas")

[Out]

2*(2*sqrt(2)*x*arctan(2*sqrt(2)*sqrt(x)) + 2*sqrt(2)*x*arctan(3/4*sqrt(2)*sqrt(x)/sqrt(x^2 - x)) - 4*x*arctan(
sqrt(x)/sqrt(x^2 - x)) - sqrt(x)*log(4*x + 4*sqrt(x^2 - x) - 1))/x

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giac [A]  time = 0.48, size = 166, normalized size = 1.46 \[ 2 \, \sqrt {2} \pi i - 4 \, \pi i + 4 \, \pi \mathrm {sgn}\left (\sqrt {x - 1} - \sqrt {x}\right ) - 2 \, \sqrt {2} {\left (\pi \mathrm {sgn}\left (\sqrt {x - 1} - \sqrt {x}\right ) + 2 \, \arctan \left (\frac {\sqrt {2} {\left ({\left (\sqrt {x - 1} - \sqrt {x}\right )}^{2} - 1\right )}}{3 \, {\left (\sqrt {x - 1} - \sqrt {x}\right )}}\right )\right )} + 4 \, \sqrt {2} \arctan \left (\frac {2}{3} \, \sqrt {2} i\right ) + 4 \, \sqrt {2} \arctan \left (2 \, \sqrt {2} \sqrt {x}\right ) - \frac {2 \, \log \left (4 \, x + 4 \, \sqrt {{\left (x - 1\right )} x} - 1\right )}{\sqrt {x}} - 8 \, \arctan \relax (i) + 8 \, \arctan \left (\frac {{\left (\sqrt {x - 1} - \sqrt {x}\right )}^{2} - 1}{2 \, {\left (\sqrt {x - 1} - \sqrt {x}\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(-1+4*x+4*((-1+x)*x)^(1/2))/x^(3/2),x, algorithm="giac")

[Out]

2*sqrt(2)*pi*i - 4*pi*i + 4*pi*sgn(sqrt(x - 1) - sqrt(x)) - 2*sqrt(2)*(pi*sgn(sqrt(x - 1) - sqrt(x)) + 2*arcta
n(1/3*sqrt(2)*((sqrt(x - 1) - sqrt(x))^2 - 1)/(sqrt(x - 1) - sqrt(x)))) + 4*sqrt(2)*arctan(2/3*sqrt(2)*i) + 4*
sqrt(2)*arctan(2*sqrt(2)*sqrt(x)) - 2*log(4*x + 4*sqrt((x - 1)*x) - 1)/sqrt(x) - 8*arctan(i) + 8*arctan(1/2*((
sqrt(x - 1) - sqrt(x))^2 - 1)/(sqrt(x - 1) - sqrt(x)))

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maple [F]  time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (4 x -1+4 \sqrt {\left (x -1\right ) x}\right )}{x^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(4*x-1+4*((x-1)*x)^(1/2))/x^(3/2),x)

[Out]

int(ln(4*x-1+4*((x-1)*x)^(1/2))/x^(3/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, \log \left (4 \, \sqrt {x - 1} \sqrt {x} + 4 \, x - 1\right )}{\sqrt {x}} - \frac {2}{\sqrt {x}} - \int \frac {2 \, x^{2} + x}{4 \, x^{\frac {9}{2}} - 5 \, x^{\frac {7}{2}} + x^{\frac {5}{2}} + 4 \, {\left (x^{4} - x^{3}\right )} \sqrt {x - 1}}\,{d x} - \log \left (\sqrt {x} + 1\right ) + \log \left (\sqrt {x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(-1+4*x+4*((-1+x)*x)^(1/2))/x^(3/2),x, algorithm="maxima")

[Out]

-2*log(4*sqrt(x - 1)*sqrt(x) + 4*x - 1)/sqrt(x) - 2/sqrt(x) - integrate((2*x^2 + x)/(4*x^(9/2) - 5*x^(7/2) + x
^(5/2) + 4*(x^4 - x^3)*sqrt(x - 1)), x) - log(sqrt(x) + 1) + log(sqrt(x) - 1)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\ln \left (4\,x+4\,\sqrt {x\,\left (x-1\right )}-1\right )}{x^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(4*x + 4*(x*(x - 1))^(1/2) - 1)/x^(3/2),x)

[Out]

int(log(4*x + 4*(x*(x - 1))^(1/2) - 1)/x^(3/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(-1+4*x+4*((-1+x)*x)**(1/2))/x**(3/2),x)

[Out]

Timed out

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