3.181 \(\int e^{3 \tanh ^{-1}(a x)} (c-a c x) \, dx\)

Optimal. Leaf size=65 \[ -\frac{c \left (1-a^2 x^2\right )^{3/2}}{2 a (1-a x)}-\frac{3 c \sqrt{1-a^2 x^2}}{2 a}+\frac{3 c \sin ^{-1}(a x)}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.047755, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {6127, 665, 216} \[ -\frac{c \left (1-a^2 x^2\right )^{3/2}}{2 a (1-a x)}-\frac{3 c \sqrt{1-a^2 x^2}}{2 a}+\frac{3 c \sin ^{-1}(a x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])*(c - a*c*x),x]

[Out]

(-3*c*Sqrt[1 - a^2*x^2])/(2*a) - (c*(1 - a^2*x^2)^(3/2))/(2*a*(1 - a*x)) + (3*c*ArcSin[a*x])/(2*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 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 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

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

Mathematica [A]  time = 0.0333457, size = 48, normalized size = 0.74 \[ -\frac{c \left (\sqrt{1-a^2 x^2} (a x+4)+6 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{2 a} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(3*ArcTanh[a*x])*(c - a*c*x),x]

[Out]

-(c*((4 + a*x)*Sqrt[1 - a^2*x^2] + 6*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(2*a)

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Maple [A]  time = 0.039, size = 104, normalized size = 1.6 \begin{align*}{\frac{{a}^{2}c{x}^{3}}{2}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{cx}{2}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{3\,c}{2}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+2\,{\frac{ac{x}^{2}}{\sqrt{-{a}^{2}{x}^{2}+1}}}-2\,{\frac{c}{a\sqrt{-{a}^{2}{x}^{2}+1}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.44198, size = 127, normalized size = 1.95 \begin{align*} \frac{a^{2} c x^{3}}{2 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{2 \, a c x^{2}}{\sqrt{-a^{2} x^{2} + 1}} - \frac{c x}{2 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{3 \, c \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{2 \, \sqrt{a^{2}}} - \frac{2 \, c}{\sqrt{-a^{2} x^{2} + 1} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.96266, size = 119, normalized size = 1.83 \begin{align*} -\frac{6 \, c \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt{-a^{2} x^{2} + 1}{\left (a c x + 4 \, c\right )}}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c),x, algorithm="fricas")

[Out]

-1/2*(6*c*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + sqrt(-a^2*x^2 + 1)*(a*c*x + 4*c))/a

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Sympy [A]  time = 8.49674, size = 165, normalized size = 2.54 \begin{align*} a^{2} c \left (\begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left (a x \right )}}{2 a^{3}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left (a x \right )}}{2 a^{3}} & \text{otherwise} \end{cases}\right ) + 2 a c \left (\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right ) + c \left (\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left (x \sqrt{a^{2}} \right )} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left (x \sqrt{- a^{2}} \right )} & \text{for}\: a^{2} < 0 \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*c*Piecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) > 1), (x**3/(2*sqr
t(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True)) + 2*a*c*Piecewise((x**2/2, E
q(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) + c*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (
sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0))

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Giac [A]  time = 1.20616, size = 51, normalized size = 0.78 \begin{align*} \frac{3 \, c \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{2 \,{\left | a \right |}} - \frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1}{\left (c x + \frac{4 \, c}{a}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c),x, algorithm="giac")

[Out]

3/2*c*arcsin(a*x)*sgn(a)/abs(a) - 1/2*sqrt(-a^2*x^2 + 1)*(c*x + 4*c/a)