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

Optimal. Leaf size=112 \[ -\frac{9 a c^2 (1-a x)^{3/2}}{\sqrt{a x+1} (c-a c x)^{3/2}}-\frac{c^2 (1-a x)^{3/2}}{x \sqrt{a x+1} (c-a c x)^{3/2}}+\frac{7 a c^2 (1-a x)^{3/2} \tanh ^{-1}\left (\sqrt{a x+1}\right )}{(c-a c x)^{3/2}} \]

[Out]

(-9*a*c^2*(1 - a*x)^(3/2))/(Sqrt[1 + a*x]*(c - a*c*x)^(3/2)) - (c^2*(1 - a*x)^(3/2))/(x*Sqrt[1 + a*x]*(c - a*c
*x)^(3/2)) + (7*a*c^2*(1 - a*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]])/(c - a*c*x)^(3/2)

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Rubi [A]  time = 0.118861, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {6130, 23, 89, 78, 63, 208} \[ -\frac{9 a c^2 (1-a x)^{3/2}}{\sqrt{a x+1} (c-a c x)^{3/2}}-\frac{c^2 (1-a x)^{3/2}}{x \sqrt{a x+1} (c-a c x)^{3/2}}+\frac{7 a c^2 (1-a x)^{3/2} \tanh ^{-1}\left (\sqrt{a x+1}\right )}{(c-a c x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-9*a*c^2*(1 - a*x)^(3/2))/(Sqrt[1 + a*x]*(c - a*c*x)^(3/2)) - (c^2*(1 - a*x)^(3/2))/(x*Sqrt[1 + a*x]*(c - a*c
*x)^(3/2)) + (7*a*c^2*(1 - a*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]])/(c - a*c*x)^(3/2)

Rule 6130

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

Rule 23

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((c_) + (d_.)*(v_))^(n_), x_Symbol] :> Dist[(a + b*v)^m/(c + d*v)^m, Int[u*
(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] &&  !(IntegerQ[m] || IntegerQ[n
] || GtQ[b/d, 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 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{e^{-3 \tanh ^{-1}(a x)} \sqrt{c-a c x}}{x^2} \, dx &=\int \frac{(1-a x)^{3/2} \sqrt{c-a c x}}{x^2 (1+a x)^{3/2}} \, dx\\ &=\frac{(1-a x)^{3/2} \int \frac{(c-a c x)^2}{x^2 (1+a x)^{3/2}} \, dx}{(c-a c x)^{3/2}}\\ &=-\frac{c^2 (1-a x)^{3/2}}{x \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{(1-a x)^{3/2} \int \frac{-\frac{7 a c^2}{2}+a^2 c^2 x}{x (1+a x)^{3/2}} \, dx}{(c-a c x)^{3/2}}\\ &=-\frac{9 a c^2 (1-a x)^{3/2}}{\sqrt{1+a x} (c-a c x)^{3/2}}-\frac{c^2 (1-a x)^{3/2}}{x \sqrt{1+a x} (c-a c x)^{3/2}}-\frac{\left (7 a c^2 (1-a x)^{3/2}\right ) \int \frac{1}{x \sqrt{1+a x}} \, dx}{2 (c-a c x)^{3/2}}\\ &=-\frac{9 a c^2 (1-a x)^{3/2}}{\sqrt{1+a x} (c-a c x)^{3/2}}-\frac{c^2 (1-a x)^{3/2}}{x \sqrt{1+a x} (c-a c x)^{3/2}}-\frac{\left (7 c^2 (1-a x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{a}+\frac{x^2}{a}} \, dx,x,\sqrt{1+a x}\right )}{(c-a c x)^{3/2}}\\ &=-\frac{9 a c^2 (1-a x)^{3/2}}{\sqrt{1+a x} (c-a c x)^{3/2}}-\frac{c^2 (1-a x)^{3/2}}{x \sqrt{1+a x} (c-a c x)^{3/2}}+\frac{7 a c^2 (1-a x)^{3/2} \tanh ^{-1}\left (\sqrt{1+a x}\right )}{(c-a c x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0292104, size = 64, normalized size = 0.57 \[ \frac{c \sqrt{1-a x} \left (-9 a x+7 a x \sqrt{a x+1} \tanh ^{-1}\left (\sqrt{a x+1}\right )-1\right )}{x \sqrt{a x+1} \sqrt{c-a c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*Sqrt[1 - a*x]*(-1 - 9*a*x + 7*a*x*Sqrt[1 + a*x]*ArcTanh[Sqrt[1 + a*x]]))/(x*Sqrt[1 + a*x]*Sqrt[c - a*c*x])

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Maple [A]  time = 0.128, size = 82, normalized size = 0.7 \begin{align*}{\frac{1}{ \left ( ax+1 \right ) \left ( ax-1 \right ) x} \left ( -7\,{\it Artanh} \left ({\frac{\sqrt{c \left ( ax+1 \right ) }}{\sqrt{c}}} \right ) xa\sqrt{c \left ( ax+1 \right ) }+9\,xa\sqrt{c}+\sqrt{c} \right ) \sqrt{-c \left ( ax-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}{\frac{1}{\sqrt{c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-7*arctanh((c*(a*x+1))^(1/2)/c^(1/2))*x*a*(c*(a*x+1))^(1/2)+9*x*a*c^(1/2)+c^(1/2))*(-c*(a*x-1))^(1/2)*(-a^2*x
^2+1)^(1/2)/(a*x-1)/c^(1/2)/(a*x+1)/x

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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{-a c x + c}}{{\left (a x + 1\right )}^{3} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [A]  time = 1.68149, size = 483, normalized size = 4.31 \begin{align*} \left [\frac{7 \,{\left (a^{3} x^{3} - a x\right )} \sqrt{c} \log \left (-\frac{a^{2} c x^{2} + a c x - 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{a x^{2} - x}\right ) + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}{\left (9 \, a x + 1\right )}}{2 \,{\left (a^{2} x^{3} - x\right )}}, \frac{7 \,{\left (a^{3} x^{3} - a x\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) + \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}{\left (9 \, a x + 1\right )}}{a^{2} x^{3} - x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(7*(a^3*x^3 - a*x)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x - 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)
/(a*x^2 - x)) + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(9*a*x + 1))/(a^2*x^3 - x), (7*(a^3*x^3 - a*x)*sqrt(-c)*
arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(9*
a*x + 1))/(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 (a x - 1\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((-a*c*x+c)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**2,x)

[Out]

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

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Giac [A]  time = 1.23867, size = 144, normalized size = 1.29 \begin{align*} -a{\left (\frac{7 \, \arctan \left (\frac{\sqrt{a c x + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} + \frac{9 \, a c x + c}{{\left (a c x + c\right )}^{\frac{3}{2}} - \sqrt{a c x + c} c}\right )}{\left | c \right |} + \frac{\sqrt{2}{\left (7 \, \sqrt{2} a \sqrt{c}{\left | c \right |} \arctan \left (\frac{\sqrt{2} \sqrt{c}}{\sqrt{-c}}\right ) + 10 \, a \sqrt{-c}{\left | c \right |}\right )}}{2 \, \sqrt{-c} \sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*(7*arctan(sqrt(a*c*x + c)/sqrt(-c))/sqrt(-c) + (9*a*c*x + c)/((a*c*x + c)^(3/2) - sqrt(a*c*x + c)*c))*abs(c
) + 1/2*sqrt(2)*(7*sqrt(2)*a*sqrt(c)*abs(c)*arctan(sqrt(2)*sqrt(c)/sqrt(-c)) + 10*a*sqrt(-c)*abs(c))/(sqrt(-c)
*sqrt(c))