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

Optimal. Leaf size=147 \[ \frac{3 a \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{2 x \sqrt{1-a^2 x^2}}+\frac{4 a^2 x \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}}-\frac{4 a^2 x \sqrt{c-\frac{c}{a^2 x^2}} \log (a x+1)}{\sqrt{1-a^2 x^2}} \]

[Out]

(3*a*Sqrt[c - c/(a^2*x^2)])/Sqrt[1 - a^2*x^2] - Sqrt[c - c/(a^2*x^2)]/(2*x*Sqrt[1 - a^2*x^2]) + (4*a^2*Sqrt[c
- c/(a^2*x^2)]*x*Log[x])/Sqrt[1 - a^2*x^2] - (4*a^2*Sqrt[c - c/(a^2*x^2)]*x*Log[1 + a*x])/Sqrt[1 - a^2*x^2]

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Rubi [A]  time = 0.268698, antiderivative size = 147, 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 = {6160, 6150, 88} \[ \frac{3 a \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{2 x \sqrt{1-a^2 x^2}}+\frac{4 a^2 x \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}}-\frac{4 a^2 x \sqrt{c-\frac{c}{a^2 x^2}} \log (a x+1)}{\sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a*Sqrt[c - c/(a^2*x^2)])/Sqrt[1 - a^2*x^2] - Sqrt[c - c/(a^2*x^2)]/(2*x*Sqrt[1 - a^2*x^2]) + (4*a^2*Sqrt[c
- c/(a^2*x^2)]*x*Log[x])/Sqrt[1 - a^2*x^2] - (4*a^2*Sqrt[c - c/(a^2*x^2)]*x*Log[1 + a*x])/Sqrt[1 - a^2*x^2]

Rule 6160

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

Rule 6150

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

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{e^{-3 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^2} \, dx &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{e^{-3 \tanh ^{-1}(a x)} \sqrt{1-a^2 x^2}}{x^3} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{(1-a x)^2}{x^3 (1+a x)} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \left (\frac{1}{x^3}-\frac{3 a}{x^2}+\frac{4 a^2}{x}-\frac{4 a^3}{1+a x}\right ) \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{3 a \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{2 x \sqrt{1-a^2 x^2}}+\frac{4 a^2 \sqrt{c-\frac{c}{a^2 x^2}} x \log (x)}{\sqrt{1-a^2 x^2}}-\frac{4 a^2 \sqrt{c-\frac{c}{a^2 x^2}} x \log (1+a x)}{\sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.040549, size = 63, normalized size = 0.43 \[ \frac{x \sqrt{c-\frac{c}{a^2 x^2}} \left (4 a^2 \log (x)-4 a^2 \log (a x+1)+\frac{3 a}{x}-\frac{1}{2 x^2}\right )}{\sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c - c/(a^2*x^2)]*x*(-1/(2*x^2) + (3*a)/x + 4*a^2*Log[x] - 4*a^2*Log[1 + a*x]))/Sqrt[1 - a^2*x^2]

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Maple [A]  time = 0.151, size = 78, normalized size = 0.5 \begin{align*} -{\frac{8\,{a}^{2}\ln \left ( x \right ){x}^{2}-8\,\ln \left ( ax+1 \right ){a}^{2}{x}^{2}+6\,ax-1}{2\,x \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}}\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(c*(a^2*x^2-1)/a^2/x^2)^(1/2)/x*(-a^2*x^2+1)^(1/2)*(8*a^2*ln(x)*x^2-8*ln(a*x+1)*a^2*x^2+6*a*x-1)/(a^2*x^2
-1)

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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^{2} x^{2}}}}{{\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^2/x^2)^(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^2*x^2))/((a*x + 1)^3*x^2), x)

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Fricas [A]  time = 2.70146, size = 1025, normalized size = 6.97 \begin{align*} \left [\frac{4 \,{\left (a^{3} x^{3} - a x\right )} \sqrt{-c} \log \left (\frac{4 \, a^{5} c x^{5} +{\left (2 \, a^{6} + 4 \, a^{5} + 6 \, a^{4} + 4 \, a^{3} + a^{2}\right )} c x^{6} +{\left (4 \, a^{4} - 4 \, a^{3} - 6 \, a^{2} - 4 \, a - 1\right )} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, a c x -{\left (4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} -{\left (4 \, a^{4} + 6 \, a^{3} + 4 \, a^{2} + a\right )} x^{5} + 4 \, a^{2} x^{2} + a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{4} x^{6} + 2 \, a^{3} x^{5} - 2 \, a x^{3} - x^{2}}\right ) + \sqrt{-a^{2} x^{2} + 1}{\left ({\left (6 \, a - 1\right )} x^{2} - 6 \, a x + 1\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \,{\left (a^{2} x^{3} - x\right )}}, -\frac{8 \,{\left (a^{3} x^{3} - a x\right )} \sqrt{c} \arctan \left (-\frac{{\left (2 \, a^{2} x^{2} +{\left (2 \, a^{3} + 2 \, a^{2} + a\right )} x^{3} + a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, a^{3} c x^{3} -{\left (2 \, a^{3} + a^{2}\right )} c x^{4} +{\left (a^{2} + 2 \, a + 1\right )} c x^{2} - 2 \, a c x - c}\right ) - \sqrt{-a^{2} x^{2} + 1}{\left ({\left (6 \, a - 1\right )} x^{2} - 6 \, a x + 1\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \,{\left (a^{2} x^{3} - x\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(4*(a^3*x^3 - a*x)*sqrt(-c)*log((4*a^5*c*x^5 + (2*a^6 + 4*a^5 + 6*a^4 + 4*a^3 + a^2)*c*x^6 + (4*a^4 - 4*a
^3 - 6*a^2 - 4*a - 1)*c*x^4 - 5*a^2*c*x^2 - 4*a*c*x - (4*a^4*x^4 + 6*a^3*x^3 - (4*a^4 + 6*a^3 + 4*a^2 + a)*x^5
+ 4*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^4*x^6 + 2*a^3*x^5 - 2*
a*x^3 - x^2)) + sqrt(-a^2*x^2 + 1)*((6*a - 1)*x^2 - 6*a*x + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x^3 - x),
-1/2*(8*(a^3*x^3 - a*x)*sqrt(c)*arctan(-(2*a^2*x^2 + (2*a^3 + 2*a^2 + a)*x^3 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(c
)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(2*a^3*c*x^3 - (2*a^3 + a^2)*c*x^4 + (a^2 + 2*a + 1)*c*x^2 - 2*a*c*x - c)) -
sqrt(-a^2*x^2 + 1)*((6*a - 1)*x^2 - 6*a*x + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x^3 - 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**2/x**2)**(1/2)/(a*x+1)**3*(-a**2*x**2+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{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{{\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^2/x^2)^(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^2*x^2))/((a*x + 1)^3*x^2), x)