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

Optimal. Leaf size=68 \[ \frac{2 c \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}-2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.151622, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {6128, 865, 875, 208} \[ \frac{2 c \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}-2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6128

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

Rule 865

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*
x)^m*(f + g*x)^(n + 1)*(a + c*x^2)^p)/(g*(m - n - 1)), x] - Dist[(c*m*(e*f + d*g))/(e^2*g*(m - n - 1)), Int[(d
+ e*x)^(m + 1)*(f + g*x)^n*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0
] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0] && GtQ[p, 0] && NeQ[m - n - 1, 0] &&  !IGtQ[n, 0]
&&  !(IntegerQ[n + p] && LtQ[n + p + 2, 0]) && RationalQ[n]

Rule 875

Int[Sqrt[(d_) + (e_.)*(x_)]/(((f_.) + (g_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e^2, Subst[I
nt[1/(c*(e*f + d*g) + e^2*g*x^2), x], x, Sqrt[a + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x] &&
NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0]

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

Mathematica [A]  time = 0.0195518, size = 46, normalized size = 0.68 \[ \frac{\sqrt{c-a c x} \left (2 \sqrt{a x+1}-2 \tanh ^{-1}\left (\sqrt{a x+1}\right )\right )}{\sqrt{1-a x}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.097, size = 71, normalized size = 1. \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 ( \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)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^(1/2)/x,x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.89636, size = 417, normalized size = 6.13 \begin{align*} \left [\frac{{\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 ({\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)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^(1/2)/x,x, algorithm="fricas")

[Out]

[((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*((a*x - 1)*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))/(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 )}{x \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.26251, size = 108, normalized size = 1.59 \begin{align*} \frac{2 \, c^{2}{\left (\frac{\arctan \left (\frac{\sqrt{a c x + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} - \frac{\sqrt{c} \arctan \left (\frac{\sqrt{2} \sqrt{c}}{\sqrt{-c}}\right ) + \sqrt{2} \sqrt{-c}}{\sqrt{-c} \sqrt{c}} + \frac{\sqrt{a c x + c}}{c}\right )}}{{\left | c \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*c^2*(arctan(sqrt(a*c*x + c)/sqrt(-c))/sqrt(-c) - (sqrt(c)*arctan(sqrt(2)*sqrt(c)/sqrt(-c)) + sqrt(2)*sqrt(-c
))/(sqrt(-c)*sqrt(c)) + sqrt(a*c*x + c)/c)/abs(c)