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

Optimal. Leaf size=123 \[ \frac{46 a^2 x \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-a x} \sqrt{a x+1}}+\frac{20 a \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-a x} \sqrt{a x+1}}-\frac{2 \sqrt{c-\frac{c}{a x}}}{3 x \sqrt{1-a x} \sqrt{a x+1}} \]

[Out]

(20*a*Sqrt[c - c/(a*x)])/(3*Sqrt[1 - a*x]*Sqrt[1 + a*x]) - (2*Sqrt[c - c/(a*x)])/(3*x*Sqrt[1 - a*x]*Sqrt[1 + a
*x]) + (46*a^2*Sqrt[c - c/(a*x)]*x)/(3*Sqrt[1 - a*x]*Sqrt[1 + a*x])

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Rubi [A]  time = 0.223131, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {6134, 6129, 89, 78, 37} \[ \frac{46 a^2 x \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-a x} \sqrt{a x+1}}+\frac{20 a \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-a x} \sqrt{a x+1}}-\frac{2 \sqrt{c-\frac{c}{a x}}}{3 x \sqrt{1-a x} \sqrt{a x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(20*a*Sqrt[c - c/(a*x)])/(3*Sqrt[1 - a*x]*Sqrt[1 + a*x]) - (2*Sqrt[c - c/(a*x)])/(3*x*Sqrt[1 - a*x]*Sqrt[1 + a
*x]) + (46*a^2*Sqrt[c - c/(a*x)]*x)/(3*Sqrt[1 - a*x]*Sqrt[1 + a*x])

Rule 6134

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

Rule 6129

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[c^p, Int[(u*(1 + (d*x)/c)
^p*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ
[p] || GtQ[c, 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 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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{e^{-3 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^2} \, dx &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{e^{-3 \tanh ^{-1}(a x)} \sqrt{1-a x}}{x^{5/2}} \, dx}{\sqrt{1-a x}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{(1-a x)^2}{x^{5/2} (1+a x)^{3/2}} \, dx}{\sqrt{1-a x}}\\ &=-\frac{2 \sqrt{c-\frac{c}{a x}}}{3 x \sqrt{1-a x} \sqrt{1+a x}}+\frac{\left (2 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{-5 a+\frac{3 a^2 x}{2}}{x^{3/2} (1+a x)^{3/2}} \, dx}{3 \sqrt{1-a x}}\\ &=\frac{20 a \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-a x} \sqrt{1+a x}}-\frac{2 \sqrt{c-\frac{c}{a x}}}{3 x \sqrt{1-a x} \sqrt{1+a x}}+\frac{\left (23 a^2 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{1}{\sqrt{x} (1+a x)^{3/2}} \, dx}{3 \sqrt{1-a x}}\\ &=\frac{20 a \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-a x} \sqrt{1+a x}}-\frac{2 \sqrt{c-\frac{c}{a x}}}{3 x \sqrt{1-a x} \sqrt{1+a x}}+\frac{46 a^2 \sqrt{c-\frac{c}{a x}} x}{3 \sqrt{1-a x} \sqrt{1+a x}}\\ \end{align*}

Mathematica [A]  time = 0.0296321, size = 50, normalized size = 0.41 \[ \frac{2 \left (23 a^2 x^2+10 a x-1\right ) \sqrt{c-\frac{c}{a x}}}{3 x \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.087, size = 61, normalized size = 0.5 \begin{align*}{\frac{46\,{a}^{2}{x}^{2}+20\,ax-2}{3\, \left ( ax+1 \right ) ^{2}x \left ( ax-1 \right ) ^{2}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( -{a}^{2}{x}^{2}+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)^3*(-a^2*x^2+1)^(3/2)/x^2,x)

[Out]

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

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

[Out]

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

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Fricas [A]  time = 2.12165, size = 120, normalized size = 0.98 \begin{align*} -\frac{2 \,{\left (23 \, a^{2} x^{2} + 10 \, a x - 1\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{3 \,{\left (a^{2} x^{3} - 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)^3*(-a^2*x^2+1)^(3/2)/x^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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