3.357 \(\int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx\)

Optimal. Leaf size=238 \[ \frac {119 a^{5/2} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \sinh ^{-1}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{8 \sqrt {1-\frac {1}{a x}}}-\frac {119 a^2 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{8 x \sqrt {1-\frac {1}{a x}}}-\frac {\sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{3 x^3 \sqrt {1-\frac {1}{a x}}}-\frac {8 \sqrt {c-a c x}}{x^3 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}+\frac {119 a \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{12 x^2 \sqrt {1-\frac {1}{a x}}} \]

[Out]

-8*(-a*c*x+c)^(1/2)/x^3/(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2)-1/3*(1+1/a/x)^(1/2)*(-a*c*x+c)^(1/2)/x^3/(1-1/a/x)^(1/
2)+119/12*a*(1+1/a/x)^(1/2)*(-a*c*x+c)^(1/2)/x^2/(1-1/a/x)^(1/2)-119/8*a^2*(1+1/a/x)^(1/2)*(-a*c*x+c)^(1/2)/x/
(1-1/a/x)^(1/2)+119/8*a^(5/2)*arcsinh((1/x)^(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 = 0.25, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6176, 6181, 89, 80, 50, 54, 215} \[ -\frac {119 a^2 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{8 x \sqrt {1-\frac {1}{a x}}}+\frac {119 a^{5/2} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \sinh ^{-1}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{8 \sqrt {1-\frac {1}{a x}}}+\frac {119 a \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{12 x^2 \sqrt {1-\frac {1}{a x}}}-\frac {\sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{3 x^3 \sqrt {1-\frac {1}{a x}}}-\frac {8 \sqrt {c-a c x}}{x^3 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c - a*c*x]/(E^(3*ArcCoth[a*x])*x^4),x]

[Out]

(-8*Sqrt[c - a*c*x])/(Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x^3) - (Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(3*Sqrt[1
- 1/(a*x)]*x^3) + (119*a*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(12*Sqrt[1 - 1/(a*x)]*x^2) - (119*a^2*Sqrt[1 + 1/
(a*x)]*Sqrt[c - a*c*x])/(8*Sqrt[1 - 1/(a*x)]*x) + (119*a^(5/2)*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcSinh[Sqrt[x^(-1
)]/Sqrt[a]])/(8*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 80

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

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^4} \, dx &=\frac {\sqrt {c-a c x} \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}}}{x^{7/2}} \, 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 {x^{3/2} \left (1-\frac {x}{a}\right )^2}{\left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=-\frac {8 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^3}+\frac {\left (2 a^2 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {x^{3/2} \left (\frac {19}{2 a^2}-\frac {x}{2 a^3}\right )}{\sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=-\frac {8 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^3}-\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} x^3}+\frac {\left (119 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {x^{3/2}}{\sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{6 \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {8 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^3}-\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} x^3}+\frac {119 a \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{12 \sqrt {1-\frac {1}{a x}} x^2}-\frac {\left (119 a \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {x}}{\sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{8 \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {8 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^3}-\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} x^3}+\frac {119 a \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{12 \sqrt {1-\frac {1}{a x}} x^2}-\frac {119 a^2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{8 \sqrt {1-\frac {1}{a x}} x}+\frac {\left (119 a^2 \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 )}{16 \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {8 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^3}-\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} x^3}+\frac {119 a \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{12 \sqrt {1-\frac {1}{a x}} x^2}-\frac {119 a^2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{8 \sqrt {1-\frac {1}{a x}} x}+\frac {\left (119 a^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 )}{8 \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {8 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^3}-\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} x^3}+\frac {119 a \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{12 \sqrt {1-\frac {1}{a x}} x^2}-\frac {119 a^2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{8 \sqrt {1-\frac {1}{a x}} x}+\frac {119 a^{5/2} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \sinh ^{-1}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{8 \sqrt {1-\frac {1}{a x}}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 98, normalized size = 0.41 \[ \frac {\sqrt {c-a c x} \left (\frac {357 a^{7/2} \sqrt {\frac {1}{a x}+1} \sinh ^{-1}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\left (\frac {1}{x}\right )^{7/2}}-357 a^3 x^3-119 a^2 x^2+38 a x-8\right )}{24 a x^4 \sqrt {1-\frac {1}{a^2 x^2}}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[c - a*c*x]/(E^(3*ArcCoth[a*x])*x^4),x]

[Out]

(Sqrt[c - a*c*x]*(-8 + 38*a*x - 119*a^2*x^2 - 357*a^3*x^3 + (357*a^(7/2)*Sqrt[1 + 1/(a*x)]*ArcSinh[Sqrt[x^(-1)
]/Sqrt[a]])/(x^(-1))^(7/2)))/(24*a*Sqrt[1 - 1/(a^2*x^2)]*x^4)

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fricas [A]  time = 0.54, size = 278, normalized size = 1.17 \[ \left [\frac {357 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + a c x - 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) - 2 \, {\left (357 \, a^{3} x^{3} + 119 \, a^{2} x^{2} - 38 \, a x + 8\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{48 \, {\left (a x^{4} - x^{3}\right )}}, \frac {357 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - {\left (357 \, a^{3} x^{3} + 119 \, a^{2} x^{2} - 38 \, a x + 8\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{24 \, {\left (a x^{4} - x^{3}\right )}}\right ] \]

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^4,x, algorithm="fricas")

[Out]

[1/48*(357*(a^4*x^4 - a^3*x^3)*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*(357*a^3*x^3 + 119*a^2*x^2 - 38*a*x + 8)*sqrt(-a*c*x + c)*sqrt((a*
x - 1)/(a*x + 1)))/(a*x^4 - x^3), 1/24*(357*(a^4*x^4 - a^3*x^3)*sqrt(c)*arctan(sqrt(-a*c*x + c)*sqrt(c)*sqrt((
a*x - 1)/(a*x + 1))/(a*c*x - c)) - (357*a^3*x^3 + 119*a^2*x^2 - 38*a*x + 8)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a
*x + 1)))/(a*x^4 - x^3)]

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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^4,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 = 0.07, size = 114, normalized size = 0.48 \[ -\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (357 \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}}{\sqrt {c}}\right ) x^{3} a^{3} \sqrt {-c \left (a x +1\right )}+357 x^{3} a^{3} \sqrt {c}+119 x^{2} a^{2} \sqrt {c}-38 x a \sqrt {c}+8 \sqrt {c}\right )}{24 \left (a x -1\right )^{2} \sqrt {c}\, x^{3}} \]

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^4,x)

[Out]

-1/24*((a*x-1)/(a*x+1))^(3/2)*(a*x+1)/(a*x-1)^2*(-c*(a*x-1))^(1/2)*(357*arctan((-c*(a*x+1))^(1/2)/c^(1/2))*x^3
*a^3*(-c*(a*x+1))^(1/2)+357*x^3*a^3*c^(1/2)+119*x^2*a^2*c^(1/2)-38*x*a*c^(1/2)+8*c^(1/2))/c^(1/2)/x^3

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

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^4,x, algorithm="maxima")

[Out]

integrate(sqrt(-a*c*x + c)*((a*x - 1)/(a*x + 1))^(3/2)/x^4, x)

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

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^4,x)

[Out]

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

[Out]

Timed out

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