3.300 \(\int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx\)

Optimal. Leaf size=97 \[ -\frac {\sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{x \sqrt {1-\frac {1}{a x}}}-\frac {\sqrt {a} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \sinh ^{-1}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\sqrt {1-\frac {1}{a x}}} \]

[Out]

-(1+1/a/x)^(1/2)*(-a*c*x+c)^(1/2)/x/(1-1/a/x)^(1/2)-arcsinh((1/x)^(1/2)/a^(1/2))*a^(1/2)*(1/x)^(1/2)*(-a*c*x+c
)^(1/2)/(1-1/a/x)^(1/2)

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Rubi [A]  time = 0.19, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6176, 6181, 50, 54, 215} \[ -\frac {\sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{x \sqrt {1-\frac {1}{a x}}}-\frac {\sqrt {a} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \sinh ^{-1}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\sqrt {1-\frac {1}{a x}}} \]

Antiderivative was successfully verified.

[In]

Int[(E^ArcCoth[a*x]*Sqrt[c - a*c*x])/x^2,x]

[Out]

-((Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(Sqrt[1 - 1/(a*x)]*x)) - (Sqrt[a]*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcSinh[S
qrt[x^(-1)]/Sqrt[a]])/Sqrt[1 - 1/(a*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 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 215

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

Rule 6176

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_), x_Symbol] :> Dist[(c + d*x)^p/(x^p*(1 + c/(d
*x))^p), Int[u*x^p*(1 + c/(d*x))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^
2, 0] &&  !IntegerQ[n/2] &&  !IntegerQ[p]

Rule 6181

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_), x_Symbol] :> -Dist[c^p*x^m*(1/x)^m, Sub
st[Int[((1 + (d*x)/c)^p*(1 + x/a)^(n/2))/(x^(m + 2)*(1 - x/a)^(n/2)), x], x, 1/x], x] /; FreeQ[{a, c, d, m, n,
p}, x] && EqQ[c^2 - a^2*d^2, 0] &&  !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) &&  !IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx &=\frac {\sqrt {c-a c x} \int \frac {e^{\coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}}}{x^{3/2}} \, dx}{\sqrt {1-\frac {1}{a x}} \sqrt {x}}\\ &=-\frac {\left (\sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{\sqrt {x}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=-\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} x}-\frac {\left (\sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} x}-\frac {\left (\sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{a}}} \, dx,x,\sqrt {\frac {1}{x}}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=-\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} x}-\frac {\sqrt {a} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \sinh ^{-1}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\sqrt {1-\frac {1}{a x}}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 76, normalized size = 0.78 \[ -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}+\sqrt {a} \sinh ^{-1}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )\right )}{\sqrt {1-\frac {1}{a x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^ArcCoth[a*x]*Sqrt[c - a*c*x])/x^2,x]

[Out]

-((Sqrt[x^(-1)]*Sqrt[c - a*c*x]*(Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)] + Sqrt[a]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]]))/Sqrt
[1 - 1/(a*x)])

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fricas [A]  time = 0.44, size = 229, normalized size = 2.36 \[ \left [\frac {{\left (a^{2} x^{2} - a x\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) - 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, {\left (a x^{2} - x\right )}}, -\frac {{\left (a^{2} x^{2} - a x\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) + \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x^{2} - x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*((a^2*x^2 - a*x)*sqrt(-c)*log(-(a^2*c*x^2 + a*c*x + 2*sqrt(-a*c*x + c)*(a*x + 1)*sqrt(-c)*sqrt((a*x - 1)/
(a*x + 1)) - 2*c)/(a*x^2 - x)) - 2*sqrt(-a*c*x + c)*(a*x + 1)*sqrt((a*x - 1)/(a*x + 1)))/(a*x^2 - x), -((a^2*x
^2 - a*x)*sqrt(c)*arctan(sqrt(-a*c*x + c)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))/(a*c*x - c)) + sqrt(-a*c*x + c)*(a
*x + 1)*sqrt((a*x - 1)/(a*x + 1)))/(a*x^2 - x)]

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giac [C]  time = 0.20, size = 100, normalized size = 1.03 \[ -\frac {\frac {a^{2} \sqrt {c} \arctan \left (\frac {\sqrt {-a c x - c}}{\sqrt {c}}\right )}{\mathrm {sgn}\left (-a c x - c\right )} + \frac {a^{2} \sqrt {c} \arctan \left (-i \, \sqrt {2}\right ) + \sqrt {2} a^{2} \sqrt {-c}}{\mathrm {sgn}\relax (c)} + \frac {\sqrt {-a c x - c} a}{x \mathrm {sgn}\left (-a c x - c\right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a^2*sqrt(c)*arctan(sqrt(-a*c*x - c)/sqrt(c))/sgn(-a*c*x - c) + (a^2*sqrt(c)*arctan(-I*sqrt(2)) + sqrt(2)*a^2
*sqrt(-c))/sgn(c) + sqrt(-a*c*x - c)*a/(x*sgn(-a*c*x - c)))/a

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maple [A]  time = 0.06, size = 78, normalized size = 0.80 \[ -\frac {\left (\arctan \left (\frac {\sqrt {-c \left (a x +1\right )}}{\sqrt {c}}\right ) x a c +\sqrt {-c \left (a x +1\right )}\, \sqrt {c}\right ) \sqrt {-c \left (a x -1\right )}}{\sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x +1\right )}\, x \sqrt {c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(arctan((-c*(a*x+1))^(1/2)/c^(1/2))*x*a*c+(-c*(a*x+1))^(1/2)*c^(1/2))*(-c*(a*x-1))^(1/2)/((a*x-1)/(a*x+1))^(1
/2)/(-c*(a*x+1))^(1/2)/x/c^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-a c x + c}}{x^{2} \sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-a*c*x + c)/(x^2*sqrt((a*x - 1)/(a*x + 1))), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {c-a\,c\,x}}{x^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- c \left (a x - 1\right )}}{x^{2} \sqrt {\frac {a x - 1}{a x + 1}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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