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

Optimal. Leaf size=66 \[ \frac{8 c \sqrt{1-a^2 x^2}}{3 a \sqrt{c-a c x}}+\frac{2 \sqrt{1-a^2 x^2} \sqrt{c-a c x}}{3 a} \]

[Out]

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

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Rubi [A]  time = 0.0652532, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6127, 657, 649} \[ \frac{8 c \sqrt{1-a^2 x^2}}{3 a \sqrt{c-a c x}}+\frac{2 \sqrt{1-a^2 x^2} \sqrt{c-a c x}}{3 a} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c - a*c*x]/E^ArcTanh[a*x],x]

[Out]

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

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 657

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*(m + 2*p + 1)), x] + Dist[(2*c*d*Simplify[m + p])/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^
2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p]
, 0]

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

Mathematica [A]  time = 0.0189387, size = 38, normalized size = 0.58 \[ -\frac{2 c (a x-5) \sqrt{1-a^2 x^2}}{3 a \sqrt{c-a c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*c*(-5 + a*x)*Sqrt[1 - a^2*x^2])/(3*a*Sqrt[c - a*c*x])

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Maple [A]  time = 0.03, size = 39, normalized size = 0.6 \begin{align*}{\frac{2\,ax-10}{ \left ( 3\,ax-3 \right ) a}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-acx+c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^(1/2)*(a*x-5)/(a*x-1)/a

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Maxima [A]  time = 0.999403, size = 50, normalized size = 0.76 \begin{align*} -\frac{2 \,{\left (a \sqrt{c} x - 5 \, \sqrt{c}\right )} \sqrt{a x + 1}{\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*c*x+c)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

-2/3*(a*sqrt(c)*x - 5*sqrt(c))*sqrt(a*x + 1)*(a*x - 1)/(a^2*x - a)

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Fricas [A]  time = 1.79553, size = 85, normalized size = 1.29 \begin{align*} \frac{2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}{\left (a x - 5\right )}}{3 \,{\left (a^{2} x - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(a*x - 5)/(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 )} \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}{a x + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19726, size = 58, normalized size = 0.88 \begin{align*} -\frac{2 \,{\left (\frac{4 \, \sqrt{2} c^{\frac{3}{2}}}{a} + \frac{{\left (a c x + c\right )}^{\frac{3}{2}} - 6 \, \sqrt{a c x + c} c}{a}\right )}{\left | c \right |}}{3 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(4*sqrt(2)*c^(3/2)/a + ((a*c*x + c)^(3/2) - 6*sqrt(a*c*x + c)*c)/a)*abs(c)/c^2