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

Optimal. Leaf size=77 \[ \frac {2 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{5 \sqrt {c-\frac {c}{a x}}}-\frac {2 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{15 \left (c-\frac {c}{a x}\right )^{3/2}} \]

[Out]

-2/15*a^2*c^2*(1-1/a^2/x^2)^(3/2)/(c-c/a/x)^(3/2)+2/5*a^2*c*(1-1/a^2/x^2)^(3/2)/(c-c/a/x)^(1/2)

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Rubi [A]  time = 0.19, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6178, 795, 649} \[ \frac {2 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{5 \sqrt {c-\frac {c}{a x}}}-\frac {2 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{15 \left (c-\frac {c}{a x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^2*c^2*(1 - 1/(a^2*x^2))^(3/2))/(15*(c - c/(a*x))^(3/2)) + (2*a^2*c*(1 - 1/(a^2*x^2))^(3/2))/(5*Sqrt[c -
c/(a*x)])

Rule 649

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p,
0]

Rule 795

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(g*(d + e*x)^m
*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[(m*(d*g + e*f) + 2*e*f*(p + 1))/(e*(m + 2*p + 2)), Int[(d +
e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && NeQ[m + 2*p +
2, 0] && NeQ[m, 2]

Rule 6178

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_.), x_Symbol] :> -Dist[c^n, Subst[Int[((c
+ d*x)^(p - n)*(1 - x^2/a^2)^(n/2))/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] &&
IntegerQ[(n - 1)/2] && IntegerQ[m] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p, n/2 + 1] || LtQ[-5, m, -1]) && In
tegerQ[2*p]

Rubi steps

\begin {align*} \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx &=-\left (c \operatorname {Subst}\left (\int \frac {x \sqrt {1-\frac {x^2}{a^2}}}{\sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {2 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{5 \sqrt {c-\frac {c}{a x}}}-\frac {1}{5} (a c) \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{\sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{15 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {2 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{5 \sqrt {c-\frac {c}{a x}}}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 58, normalized size = 0.75 \[ \frac {2 a \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a^2 x^2-a x-3\right ) \sqrt {c-\frac {c}{a x}}}{15 x (a x-1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(-3 - a*x + 2*a^2*x^2))/(15*x*(-1 + a*x))

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fricas [A]  time = 0.51, size = 68, normalized size = 0.88 \[ \frac {2 \, {\left (2 \, a^{3} x^{3} + a^{2} x^{2} - 4 \, a x - 3\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{15 \, {\left (a x^{3} - x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(2*a^3*x^3 + a^2*x^2 - 4*a*x - 3)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(a*x^3 - x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Warning, integrat
ion of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [abs(a*x+1)]Warn
ing, choosing root of [1,0,%%%{-2,[2,1,2]%%%}+%%%{-2,[2,0,2]%%%}+%%%{2,[1,1,1]%%%}+%%%{2,[0,0,0]%%%},0,%%%{1,[
4,2,4]%%%}+%%%{-2,[4,1,4]%%%}+%%%{1,[4,0,4]%%%}+%%%{-2,[3,2,3]%%%}+%%%{2,[3,1,3]%%%}+%%%{1,[2,2,2]%%%}+%%%{2,[
2,1,2]%%%}+%%%{-2,[2,0,2]%%%}+%%%{-2,[1,1,1]%%%}+%%%{1,[0,0,0]%%%}] at parameters values [-89,63,-49]Warning,
choosing root of [1,0,%%%{-2,[2,1,2]%%%}+%%%{-2,[2,0,2]%%%}+%%%{-2,[1,1,1]%%%}+%%%{2,[0,0,0]%%%},0,%%%{1,[4,2,
4]%%%}+%%%{-2,[4,1,4]%%%}+%%%{1,[4,0,4]%%%}+%%%{2,[3,2,3]%%%}+%%%{-2,[3,1,3]%%%}+%%%{1,[2,2,2]%%%}+%%%{2,[2,1,
2]%%%}+%%%{-2,[2,0,2]%%%}+%%%{2,[1,1,1]%%%}+%%%{1,[0,0,0]%%%}] at parameters values [-86,-64,-30]sym2poly/r2sy
m(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.04, size = 47, normalized size = 0.61 \[ \frac {2 \left (a x +1\right ) \left (2 a x -3\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{15 x^{2} \sqrt {\frac {a x -1}{a x +1}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(a*x+1)*(2*a*x-3)*(c*(a*x-1)/a/x)^(1/2)/x^2/((a*x-1)/(a*x+1))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.39, size = 53, normalized size = 0.69 \[ \frac {2\,\sqrt {c-\frac {c}{a\,x}}\,{\left (a\,x+1\right )}^2\,\left (2\,a\,x-3\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{15\,x^2\,\left (a\,x-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(c - c/(a*x))^(1/2)*(a*x + 1)^2*(2*a*x - 3)*((a*x - 1)/(a*x + 1))^(1/2))/(15*x^2*(a*x - 1))

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

[Out]

Timed out

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