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

Optimal. Leaf size=72 \[ -\frac{c \sqrt{1-a^2 x^2}}{x \sqrt{c-a c x}}-a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.156802, antiderivative size = 72, 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, 863, 875, 208} \[ -\frac{c \sqrt{1-a^2 x^2}}{x \sqrt{c-a c x}}-a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((c*Sqrt[1 - a^2*x^2])/(x*Sqrt[c - a*c*x])) - a*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 863

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*(n + 1)), x] + Dist[(c*m)/(e*g*(n + 1)), Int[(d + e*x)^(m + 1)*(f + g*
x)^(n + 1)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e
^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0] && GtQ[p, 0] && LtQ[n, -1] &&  !(IntegerQ[n + p] && LeQ[n + p + 2, 0]
)

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^2} \, dx &=c \int \frac{\sqrt{1-a^2 x^2}}{x^2 \sqrt{c-a c x}} \, dx\\ &=-\frac{c \sqrt{1-a^2 x^2}}{x \sqrt{c-a c x}}+\frac{1}{2} a \int \frac{\sqrt{c-a c x}}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{c \sqrt{1-a^2 x^2}}{x \sqrt{c-a c x}}+\left (a^3 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{c \sqrt{1-a^2 x^2}}{x \sqrt{c-a c x}}-a \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 = 0.0242757, size = 57, normalized size = 0.79 \[ -\frac{\sqrt{c-a c x} \left (a x+a x \sqrt{a x+1} \tanh ^{-1}\left (\sqrt{a x+1}\right )+1\right )}{x \sqrt{1-a^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.1, size = 78, normalized size = 1.1 \begin{align*}{\frac{1}{ \left ( ax-1 \right ) x}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ( ax-1 \right ) } \left ({\it Artanh} \left ({\sqrt{c \left ( ax+1 \right ) }{\frac{1}{\sqrt{c}}}} \right ) xac+\sqrt{c \left ( ax+1 \right ) }\sqrt{c} \right ){\frac{1}{\sqrt{c \left ( ax+1 \right ) }}}{\frac{1}{\sqrt{c}}}} \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^2,x)

[Out]

(-a^2*x^2+1)^(1/2)*(-c*(a*x-1))^(1/2)*(arctanh((c*(a*x+1))^(1/2)/c^(1/2))*x*a*c+(c*(a*x+1))^(1/2)*c^(1/2))/(a*
x-1)/(c*(a*x+1))^(1/2)/x/c^(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 )}}{\sqrt{-a^{2} x^{2} + 1} x^{2}}\,{d x} \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^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.93985, size = 441, normalized size = 6.12 \begin{align*} \left [\frac{{\left (a^{2} x^{2} - 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}}{2 \,{\left (a x^{2} - x\right )}}, -\frac{{\left (a^{2} x^{2} - 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}}{a x^{2} - x}\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^2,x, algorithm="fricas")

[Out]

[1/2*((a^2*x^2 - 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))/(a*x^2 - x), -((a^2*x^2 - 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))/(a*x^2 - 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 (a x + 1\right )}{x^{2} \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**2,x)

[Out]

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

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Giac [A]  time = 1.28845, size = 136, normalized size = 1.89 \begin{align*} \frac{{\left (a^{2} c{\left (\frac{\arctan \left (\frac{\sqrt{a c x + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} - \frac{\sqrt{a c x + c}}{a c x}\right )} - \frac{a^{2} c^{\frac{3}{2}} \arctan \left (\frac{\sqrt{2} \sqrt{c}}{\sqrt{-c}}\right ) - \sqrt{2} a^{2} \sqrt{-c} c}{\sqrt{-c} \sqrt{c}}\right )} c}{a{\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^2,x, algorithm="giac")

[Out]

(a^2*c*(arctan(sqrt(a*c*x + c)/sqrt(-c))/sqrt(-c) - sqrt(a*c*x + c)/(a*c*x)) - (a^2*c^(3/2)*arctan(sqrt(2)*sqr
t(c)/sqrt(-c)) - sqrt(2)*a^2*sqrt(-c)*c)/(sqrt(-c)*sqrt(c)))*c/(a*abs(c))