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

Optimal. Leaf size=83 \[ \frac{2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{a \sqrt{c}}-\frac{2 \sqrt{1-a^2 x^2}}{a \sqrt{c-a c x}} \]

[Out]

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

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Rubi [A]  time = 0.0816572, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6127, 665, 661, 208} \[ \frac{2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{a \sqrt{c}}-\frac{2 \sqrt{1-a^2 x^2}}{a \sqrt{c-a c x}} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]/Sqrt[c - a*c*x],x]

[Out]

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

Rule 6127

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

Rule 665

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

Rule 661

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e, Subst[Int[1/(2*c*d + e^2*x^2
), x], x, Sqrt[a + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e}, x] && 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}} \, dx &=c \int \frac{\sqrt{1-a^2 x^2}}{(c-a c x)^{3/2}} \, dx\\ &=-\frac{2 \sqrt{1-a^2 x^2}}{a \sqrt{c-a c x}}+2 \int \frac{1}{\sqrt{c-a c x} \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{2 \sqrt{1-a^2 x^2}}{a \sqrt{c-a c x}}-(4 a c) \operatorname{Subst}\left (\int \frac{1}{-2 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 \sqrt{1-a^2 x^2}}{a \sqrt{c-a c x}}+\frac{2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{a \sqrt{c}}\\ \end{align*}

Mathematica [A]  time = 0.0248053, size = 62, normalized size = 0.75 \[ -\frac{2 \sqrt{c-a c x} \left (\sqrt{a x+1}-\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a x+1}}{\sqrt{2}}\right )\right )}{a c \sqrt{1-a x}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.11, size = 84, 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 ) }ca} \left ( \sqrt{c}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( ax+1 \right ) }\sqrt{2}}{\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)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}\,{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, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.94142, size = 491, normalized size = 5.92 \begin{align*} \left [\frac{\frac{\sqrt{2}{\left (a c x - c\right )} \log \left (-\frac{a^{2} x^{2} + 2 \, a x - \frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{\sqrt{c}} - 3}{a^{2} x^{2} - 2 \, a x + 1}\right )}{\sqrt{c}} + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{a^{2} c x - a c}, \frac{2 \,{\left (\sqrt{2}{\left (a c x - c\right )} \sqrt{-\frac{1}{c}} \arctan \left (\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-\frac{1}{c}}}{a^{2} x^{2} - 1}\right ) + \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}\right )}}{a^{2} c x - a 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, algorithm="fricas")

[Out]

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

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

[Out]

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

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Giac [A]  time = 1.29129, size = 123, normalized size = 1.48 \begin{align*} -\frac{2 \, c{\left (\frac{\frac{\sqrt{2} c \arctan \left (\frac{\sqrt{2} \sqrt{a c x + c}}{2 \, \sqrt{-c}}\right )}{\sqrt{-c}} + \sqrt{a c x + c}}{c} - \frac{\sqrt{2} c \arctan \left (\frac{\sqrt{c}}{\sqrt{-c}}\right ) + \sqrt{2} \sqrt{-c} \sqrt{c}}{\sqrt{-c} c}\right )}}{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, algorithm="giac")

[Out]

-2*c*((sqrt(2)*c*arctan(1/2*sqrt(2)*sqrt(a*c*x + c)/sqrt(-c))/sqrt(-c) + sqrt(a*c*x + c))/c - (sqrt(2)*c*arcta
n(sqrt(c)/sqrt(-c)) + sqrt(2)*sqrt(-c)*sqrt(c))/(sqrt(-c)*c))/(a*abs(c))