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

Optimal. Leaf size=126 \[ \frac{a x^2 \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{3/2}}{(1-a x)^{3/2}}-\frac{5 \sqrt{a} x^{3/2} \left (c-\frac{c}{a x}\right )^{3/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{3/2}}-\frac{2 x \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{3/2}}{(1-a x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.151152, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6134, 6129, 89, 80, 54, 215} \[ \frac{a x^2 \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{3/2}}{(1-a x)^{3/2}}-\frac{5 \sqrt{a} x^{3/2} \left (c-\frac{c}{a x}\right )^{3/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{3/2}}-\frac{2 x \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{3/2}}{(1-a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 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]

Rubi steps

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

Mathematica [A]  time = 0.0462589, size = 71, normalized size = 0.56 \[ -\frac{c \sqrt{c-\frac{c}{a x}} \left ((a x-2) \sqrt{a x+1}-5 \sqrt{a} \sqrt{x} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )\right )}{a \sqrt{1-a x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((c*Sqrt[c - c/(a*x)]*((-2 + a*x)*Sqrt[1 + a*x] - 5*Sqrt[a]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]]))/(a*Sqrt[1 - a*
x]))

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Maple [A]  time = 0.14, size = 109, normalized size = 0.9 \begin{align*}{\frac{c}{2\,ax-2}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\sqrt{-{a}^{2}{x}^{2}+1} \left ( 2\,{a}^{3/2}x\sqrt{- \left ( ax+1 \right ) x}+5\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) xa-4\,\sqrt{a}\sqrt{- \left ( ax+1 \right ) x} \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{- \left ( ax+1 \right ) x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(c*(a*x-1)/a/x)^(1/2)*c/a^(3/2)*(-a^2*x^2+1)^(1/2)*(2*a^(3/2)*x*(-(a*x+1)*x)^(1/2)+5*arctan(1/2/a^(1/2)*(2
*a*x+1)/(-(a*x+1)*x)^(1/2))*x*a-4*a^(1/2)*(-(a*x+1)*x)^(1/2))/(a*x-1)/(-(a*x+1)*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.14547, size = 566, normalized size = 4.49 \begin{align*} \left [\frac{5 \,{\left (a c x - c\right )} \sqrt{-c} \log \left (-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \,{\left (2 \, a^{2} x^{2} + a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, \sqrt{-a^{2} x^{2} + 1}{\left (a c x - 2 \, c\right )} \sqrt{\frac{a c x - c}{a x}}}{4 \,{\left (a^{2} x - a\right )}}, -\frac{5 \,{\left (a c x - c\right )} \sqrt{c} \arctan \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, \sqrt{-a^{2} x^{2} + 1}{\left (a c x - 2 \, c\right )} \sqrt{\frac{a c x - c}{a x}}}{2 \,{\left (a^{2} x - a\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(5*(a*c*x - c)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqr
t((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*sqrt(-a^2*x^2 + 1)*(a*c*x - 2*c)*sqrt((a*c*x - c)/(a*x)))/(a^2*x - a)
, -1/2*(5*(a*c*x - c)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a
*c*x - c)) - 2*sqrt(-a^2*x^2 + 1)*(a*c*x - 2*c)*sqrt((a*c*x - c)/(a*x)))/(a^2*x - a)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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