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

Optimal. Leaf size=36 \[ \frac {2 \sqrt {c-a c x}}{a c}+\frac {4}{a \sqrt {c-a c x}} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6130, 21, 43} \[ \frac {2 \sqrt {c-a c x}}{a c}+\frac {4}{a \sqrt {c-a c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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])

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]
)

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 21, normalized size = 0.58 \[ \frac {6-2 a x}{a \sqrt {c-a c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6 - 2*a*x)/(a*Sqrt[c - a*c*x])

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fricas [A]  time = 0.51, size = 29, normalized size = 0.81 \[ \frac {2 \, \sqrt {-a c x + c} {\left (a x - 3\right )}}{a^{2} c x - a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.25, size = 32, normalized size = 0.89 \[ \frac {4}{\sqrt {-a c x + c} a} + \frac {2 \, \sqrt {-a c x + c}}{a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 20, normalized size = 0.56 \[ -\frac {2 \left (a x -3\right )}{\sqrt {-a c x +c}\, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.31, size = 30, normalized size = 0.83 \[ \frac {2 \, {\left (\sqrt {-a c x + c} + \frac {2 \, c}{\sqrt {-a c x + c}}\right )}}{a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.04, size = 20, normalized size = 0.56 \[ -\frac {2\,a\,x-6}{a\,\sqrt {c-a\,c\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*a*x - 6)/(a*(c - a*c*x)^(1/2))

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sympy [A]  time = 36.40, size = 48, normalized size = 1.33 \[ \begin {cases} \frac {\frac {2}{\sqrt {- a c x + c}} - \frac {2 \left (- \frac {c}{\sqrt {- a c x + c}} - \sqrt {- a c x + c}\right )}{c}}{a} & \text {for}\: a \neq 0 \\\frac {x}{\sqrt {c}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((2/sqrt(-a*c*x + c) - 2*(-c/sqrt(-a*c*x + c) - sqrt(-a*c*x + c))/c)/a, Ne(a, 0)), (x/sqrt(c), True)
)

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