3.354 \(\int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx\)

Optimal. Leaf size=140 \[ \frac {2 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}+\frac {10 \sqrt {c-a c x}}{a x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}-\frac {2 \sqrt {\frac {1}{x}} \sqrt {c-a c x} \sinh ^{-1}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {1-\frac {1}{a x}}} \]

[Out]

2*(-a*c*x+c)^(1/2)/(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2)+10*(-a*c*x+c)^(1/2)/a/x/(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2)-2*a
rcsinh((1/x)^(1/2)/a^(1/2))*(1/x)^(1/2)*(-a*c*x+c)^(1/2)/a^(1/2)/(1-1/a/x)^(1/2)

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Rubi [A]  time = 0.22, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6176, 6181, 89, 78, 54, 215} \[ \frac {2 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}+\frac {10 \sqrt {c-a c x}}{a x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}-\frac {2 \sqrt {\frac {1}{x}} \sqrt {c-a c x} \sinh ^{-1}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {1-\frac {1}{a x}}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c - a*c*x]/(E^(3*ArcCoth[a*x])*x),x]

[Out]

(2*Sqrt[c - a*c*x])/(Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (10*Sqrt[c - a*c*x])/(a*Sqrt[1 - 1/(a*x)]*Sqrt[1 +
1/(a*x)]*x) - (2*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(Sqrt[a]*Sqrt[1 - 1/(a*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 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

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^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx &=\frac {\sqrt {c-a c x} \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}}}{\sqrt {x}} \, 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 {\left (1-\frac {x}{a}\right )^2}{x^{3/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=\frac {2 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\left (2 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {-\frac {2}{a}+\frac {x}{2 a^2}}{\sqrt {x} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=\frac {2 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {10 \sqrt {c-a c x}}{a \sqrt {1-\frac {1}{a 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 )}{a \sqrt {1-\frac {1}{a x}}}\\ &=\frac {2 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {10 \sqrt {c-a c x}}{a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x}-\frac {\left (2 \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 )}{a \sqrt {1-\frac {1}{a x}}}\\ &=\frac {2 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {10 \sqrt {c-a c x}}{a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x}-\frac {2 \sqrt {\frac {1}{x}} \sqrt {c-a c x} \sinh ^{-1}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {1-\frac {1}{a x}}}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 78, normalized size = 0.56 \[ \frac {2 \sqrt {c-a c x} \left (-\sqrt {a} \sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1} \sinh ^{-1}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )+a+\frac {5}{x}\right )}{a \sqrt {1-\frac {1}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c - a*c*x]/(E^(3*ArcCoth[a*x])*x),x]

[Out]

(2*Sqrt[c - a*c*x]*(a + 5/x - Sqrt[a]*Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]]))/(a*Sqrt[1
- 1/(a^2*x^2)])

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fricas [A]  time = 0.64, size = 207, normalized size = 1.48 \[ \left [\frac {{\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) + 2 \, \sqrt {-a c x + c} {\left (a x + 5\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x - 1}, -\frac {2 \, {\left ({\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - \sqrt {-a c x + c} {\left (a x + 5\right )} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a x - 1}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[((a*x - 1)*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 + 5)*sqrt((a*x - 1)/(a*x + 1)))/(a*x - 1), -2*((a*x - 1)*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 + 5)*sqrt((a*
x - 1)/(a*x + 1)))/(a*x - 1)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(a*x+1)]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.06, size = 80, normalized size = 0.57 \[ \frac {2 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (\sqrt {c}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}}{\sqrt {c}}\right ) \sqrt {-c \left (a x +1\right )}+a c x +5 c \right )}{\left (a x -1\right )^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x,x)

[Out]

2*((a*x-1)/(a*x+1))^(3/2)*(a*x+1)/(a*x-1)^2*(-c*(a*x-1))^(1/2)*(c^(1/2)*arctan((-c*(a*x+1))^(1/2)/c^(1/2))*(-c
*(a*x+1))^(1/2)+a*c*x+5*c)/c

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-a*c*x + c)*((a*x - 1)/(a*x + 1))^(3/2)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c - a*c*x)^(1/2)*((a*x - 1)/(a*x + 1))^(3/2))/x,x)

[Out]

int(((c - a*c*x)^(1/2)*((a*x - 1)/(a*x + 1))^(3/2))/x, 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((-a*c*x+c)**(1/2)*((a*x-1)/(a*x+1))**(3/2)/x,x)

[Out]

Timed out

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