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

Optimal. Leaf size=119 \[ -\frac{2 \sqrt{a x+1} (c-a c x)^{3/2}}{c (1-a x)^{3/2}}-\frac{2 (c-a c x)^{3/2} \tanh ^{-1}\left (\sqrt{a x+1}\right )}{c (1-a x)^{3/2}}+\frac{4 \sqrt{2} (c-a c x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a x+1}}{\sqrt{2}}\right )}{c (1-a x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.136005, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {6130, 23, 84, 156, 63, 208} \[ -\frac{2 \sqrt{a x+1} (c-a c x)^{3/2}}{c (1-a x)^{3/2}}-\frac{2 (c-a c x)^{3/2} \tanh ^{-1}\left (\sqrt{a x+1}\right )}{c (1-a x)^{3/2}}+\frac{4 \sqrt{2} (c-a c x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a x+1}}{\sqrt{2}}\right )}{c (1-a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 + a*x]*(c - a*c*x)^(3/2))/(c*(1 - a*x)^(3/2)) - (2*(c - a*c*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]])/(c*(1
- a*x)^(3/2)) + (4*Sqrt[2]*(c - a*c*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]/Sqrt[2]])/(c*(1 - a*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 84

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[(f*(e + f*x)^(p -
1))/(b*d*(p - 1)), x] + Dist[1/(b*d), Int[((b*d*e^2 - a*c*f^2 + f*(2*b*d*e - b*c*f - a*d*f)*x)*(e + f*x)^(p -
2))/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 1]

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

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} \, dx &=\int \frac{(1+a x)^{3/2} \sqrt{c-a c x}}{x (1-a x)^{3/2}} \, dx\\ &=\frac{(c-a c x)^{3/2} \int \frac{(1+a x)^{3/2}}{x (c-a c x)} \, dx}{(1-a x)^{3/2}}\\ &=-\frac{2 \sqrt{1+a x} (c-a c x)^{3/2}}{c (1-a x)^{3/2}}-\frac{(c-a c x)^{3/2} \int \frac{-a c-3 a^2 c x}{x \sqrt{1+a x} (c-a c x)} \, dx}{a c (1-a x)^{3/2}}\\ &=-\frac{2 \sqrt{1+a x} (c-a c x)^{3/2}}{c (1-a x)^{3/2}}+\frac{\left (4 a (c-a c x)^{3/2}\right ) \int \frac{1}{\sqrt{1+a x} (c-a c x)} \, dx}{(1-a x)^{3/2}}+\frac{(c-a c x)^{3/2} \int \frac{1}{x \sqrt{1+a x}} \, dx}{c (1-a x)^{3/2}}\\ &=-\frac{2 \sqrt{1+a x} (c-a c x)^{3/2}}{c (1-a x)^{3/2}}+\frac{\left (8 (c-a c x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{2 c-c x^2} \, dx,x,\sqrt{1+a x}\right )}{(1-a x)^{3/2}}+\frac{\left (2 (c-a c x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{a}+\frac{x^2}{a}} \, dx,x,\sqrt{1+a x}\right )}{a c (1-a x)^{3/2}}\\ &=-\frac{2 \sqrt{1+a x} (c-a c x)^{3/2}}{c (1-a x)^{3/2}}-\frac{2 (c-a c x)^{3/2} \tanh ^{-1}\left (\sqrt{1+a x}\right )}{c (1-a x)^{3/2}}+\frac{4 \sqrt{2} (c-a c x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{1+a x}}{\sqrt{2}}\right )}{c (1-a x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0322904, size = 66, normalized size = 0.55 \[ -\frac{2 \sqrt{c-a c x} \left (\sqrt{a x+1}+\tanh ^{-1}\left (\sqrt{a x+1}\right )-2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a x+1}}{\sqrt{2}}\right )\right )}{\sqrt{1-a x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.104, size = 98, normalized size = 0.8 \begin{align*} -2\,{\frac{\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ( ax-1 \right ) }}{ \left ( ax-1 \right ) \sqrt{c \left ( ax+1 \right ) }} \left ( 2\,\sqrt{c}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) -\sqrt{c}{\it Artanh} \left ({\frac{\sqrt{c \left ( ax+1 \right ) }}{\sqrt{c}}} \right ) -\sqrt{c \left ( ax+1 \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.94456, size = 753, normalized size = 6.33 \begin{align*} \left [\frac{2 \, \sqrt{2}{\left (a x - 1\right )} \sqrt{c} \log \left (-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) +{\left (a x - 1\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}}{a x - 1}, \frac{2 \,{\left (2 \, \sqrt{2}{\left (a x - 1\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) -{\left (a x - 1\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}\right )}}{a x - 1}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[(2*sqrt(2)*(a*x - 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c
) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + (a*x - 1)*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))/(a*x - 1), 2*(2*sqrt(2)*(a*x - 1)*s
qrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) - (a*x - 1)*sqrt(-c)*arct
an(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))/(a*x -
1)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.23092, size = 188, normalized size = 1.58 \begin{align*} -2 \, c{\left (\frac{2 \, \sqrt{2} c \arctan \left (\frac{\sqrt{2} \sqrt{a c x + c}}{2 \, \sqrt{-c}}\right )}{\sqrt{-c}{\left | c \right |}} - \frac{c \arctan \left (\frac{\sqrt{a c x + c}}{\sqrt{-c}}\right )}{\sqrt{-c}{\left | c \right |}} + \frac{\sqrt{a c x + c}}{{\left | c \right |}}\right )} - \frac{\sqrt{2}{\left (\sqrt{2} c^{2} \arctan \left (\frac{\sqrt{2} \sqrt{c}}{\sqrt{-c}}\right ) - 4 \, c^{2} \arctan \left (\frac{\sqrt{c}}{\sqrt{-c}}\right ) - 2 \, \sqrt{-c} c^{\frac{3}{2}}\right )}}{\sqrt{-c}{\left | c \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*c*(2*sqrt(2)*c*arctan(1/2*sqrt(2)*sqrt(a*c*x + c)/sqrt(-c))/(sqrt(-c)*abs(c)) - c*arctan(sqrt(a*c*x + c)/sq
rt(-c))/(sqrt(-c)*abs(c)) + sqrt(a*c*x + c)/abs(c)) - sqrt(2)*(sqrt(2)*c^2*arctan(sqrt(2)*sqrt(c)/sqrt(-c)) -
4*c^2*arctan(sqrt(c)/sqrt(-c)) - 2*sqrt(-c)*c^(3/2))/(sqrt(-c)*abs(c))