3.538 \(\int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^2} \, dx\)

Optimal. Leaf size=109 \[ -\frac{2 a \left (c-\frac{c}{a x}\right )^{5/2}}{3 c^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{16 a \left (c-\frac{c}{a x}\right )^{3/2}}{3 c \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{64 a \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-\frac{1}{a^2 x^2}}} \]

[Out]

(64*a*Sqrt[c - c/(a*x)])/(3*Sqrt[1 - 1/(a^2*x^2)]) - (16*a*(c - c/(a*x))^(3/2))/(3*c*Sqrt[1 - 1/(a^2*x^2)]) -
(2*a*(c - c/(a*x))^(5/2))/(3*c^2*Sqrt[1 - 1/(a^2*x^2)])

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Rubi [A]  time = 0.216214, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6178, 657, 649} \[ -\frac{2 a \left (c-\frac{c}{a x}\right )^{5/2}}{3 c^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{16 a \left (c-\frac{c}{a x}\right )^{3/2}}{3 c \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{64 a \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-\frac{1}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c - c/(a*x)]/(E^(3*ArcCoth[a*x])*x^2),x]

[Out]

(64*a*Sqrt[c - c/(a*x)])/(3*Sqrt[1 - 1/(a^2*x^2)]) - (16*a*(c - c/(a*x))^(3/2))/(3*c*Sqrt[1 - 1/(a^2*x^2)]) -
(2*a*(c - c/(a*x))^(5/2))/(3*c^2*Sqrt[1 - 1/(a^2*x^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]

Rule 657

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*(m + 2*p + 1)), x] + Dist[(2*c*d*Simplify[m + p])/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^
2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p]
, 0]

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]

Rubi steps

\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^{7/2}}{\left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=-\frac{2 a \left (c-\frac{c}{a x}\right )^{5/2}}{3 c^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{8 \operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^{5/2}}{\left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{3 c^2}\\ &=-\frac{16 a \left (c-\frac{c}{a x}\right )^{3/2}}{3 c \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{2 a \left (c-\frac{c}{a x}\right )^{5/2}}{3 c^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{32 \operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^{3/2}}{\left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{3 c}\\ &=\frac{64 a \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{16 a \left (c-\frac{c}{a x}\right )^{3/2}}{3 c \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{2 a \left (c-\frac{c}{a x}\right )^{5/2}}{3 c^2 \sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}

Mathematica [A]  time = 0.0955544, size = 58, normalized size = 0.53 \[ \frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} \left (23 a^2 x^2+10 a x-1\right ) \sqrt{c-\frac{c}{a x}}}{3 a^2 x^2-3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c - c/(a*x)]/(E^(3*ArcCoth[a*x])*x^2),x]

[Out]

(2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(-1 + 10*a*x + 23*a^2*x^2))/(-3 + 3*a^2*x^2)

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Maple [A]  time = 0.122, size = 62, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ( 23\,{a}^{2}{x}^{2}+10\,ax-1 \right ) }{3\,x \left ( ax-1 \right ) ^{2}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(a*x+1)*(23*a^2*x^2+10*a*x-1)*(c*(a*x-1)/a/x)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x/(a*x-1)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c - \frac{c}{a x}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.53207, size = 126, normalized size = 1.16 \begin{align*} \frac{2 \,{\left (23 \, a^{2} x^{2} + 10 \, a x - 1\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{3 \,{\left (a x^{2} - x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(23*a^2*x^2 + 10*a*x - 1)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(a*x^2 - x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c - \frac{c}{a x}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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