### 3.238 $$\int e^{2 \coth ^{-1}(a x)} \sqrt{c-a c x} \, dx$$

Optimal. Leaf size=38 $\frac{4 \sqrt{c-a c x}}{a}-\frac{2 (c-a c x)^{3/2}}{3 a c}$

[Out]

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

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Rubi [A]  time = 0.0781572, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.2, Rules used = {6167, 6130, 21, 43} $\frac{4 \sqrt{c-a c x}}{a}-\frac{2 (c-a c x)^{3/2}}{3 a c}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 6167

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rule 6130

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int e^{2 \coth ^{-1}(a x)} \sqrt{c-a c x} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \sqrt{c-a c x} \, dx\\ &=-\int \frac{(1+a x) \sqrt{c-a c x}}{1-a x} \, dx\\ &=-\left (c \int \frac{1+a x}{\sqrt{c-a c x}} \, dx\right )\\ &=-\left (c \int \left (\frac{2}{\sqrt{c-a c x}}-\frac{\sqrt{c-a c x}}{c}\right ) \, dx\right )\\ &=\frac{4 \sqrt{c-a c x}}{a}-\frac{2 (c-a c x)^{3/2}}{3 a c}\\ \end{align*}

Mathematica [A]  time = 0.0248467, size = 23, normalized size = 0.61 $\frac{2 (a x+5) \sqrt{c-a c x}}{3 a}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(5 + a*x)*Sqrt[c - a*c*x])/(3*a)

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Maple [A]  time = 0.046, size = 20, normalized size = 0.5 \begin{align*}{\frac{2\,ax+10}{3\,a}\sqrt{-acx+c}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.994603, size = 41, normalized size = 1.08 \begin{align*} -\frac{2 \,{\left ({\left (-a c x + c\right )}^{\frac{3}{2}} - 6 \, \sqrt{-a c x + c} c\right )}}{3 \, a c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.58634, size = 46, normalized size = 1.21 \begin{align*} \frac{2 \, \sqrt{-a c x + c}{\left (a x + 5\right )}}{3 \, a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sqrt(-a*c*x + c)*(a*x + 5)/a

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Sympy [A]  time = 4.98112, size = 31, normalized size = 0.82 \begin{align*} - \frac{2 \left (- 2 c \sqrt{- a c x + c} + \frac{\left (- a c x + c\right )^{\frac{3}{2}}}{3}\right )}{a c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.1624, size = 41, normalized size = 1.08 \begin{align*} -\frac{2 \,{\left ({\left (-a c x + c\right )}^{\frac{3}{2}} - 6 \, \sqrt{-a c x + c} c\right )}}{3 \, a c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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