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

Optimal. Leaf size=35 \[ \frac{2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0410775, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6127, 649} \[ \frac{2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]*Sqrt[c - a*c*x],x]

[Out]

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

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 649

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

Rubi steps

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

Mathematica [A]  time = 0.0131721, size = 37, normalized size = 1.06 \[ \frac{2 (a x+1)^{3/2} \sqrt{c-a c x}}{3 a \sqrt{1-a x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.027, size = 34, normalized size = 1. \begin{align*}{\frac{2\, \left ( ax+1 \right ) ^{2}}{3\,a}\sqrt{-acx+c}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \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]

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

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Maxima [A]  time = 1.03323, size = 78, normalized size = 2.23 \begin{align*} \frac{2 \,{\left (a^{2} \sqrt{c} x^{2} - a \sqrt{c} x - 2 \, \sqrt{c}\right )}}{3 \, \sqrt{a x + 1} a} + \frac{2 \,{\left (a \sqrt{c} x + \sqrt{c}\right )}}{\sqrt{a x + 1} a} \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]

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

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Fricas [A]  time = 1.86509, size = 86, normalized size = 2.46 \begin{align*} -\frac{2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}{\left (a x + 1\right )}}{3 \,{\left (a^{2} x - a\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]

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

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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 )}{\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(sqrt(-c*(a*x - 1))*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)

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Giac [A]  time = 1.18932, size = 43, normalized size = 1.23 \begin{align*} -\frac{2 \,{\left (2 \, \sqrt{2} \sqrt{c} - \frac{{\left (a c x + c\right )}^{\frac{3}{2}}}{c}\right )} c}{3 \, 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]

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