3.180 \(\int e^{3 \tanh ^{-1}(a x)} (c-a c x)^2 \, dx\)

Optimal. Leaf size=61 \[ -\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac{1}{2} c^2 x \sqrt{1-a^2 x^2}+\frac{c^2 \sin ^{-1}(a x)}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.0458223, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6127, 665, 195, 216} \[ -\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac{1}{2} c^2 x \sqrt{1-a^2 x^2}+\frac{c^2 \sin ^{-1}(a x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])*(c - a*c*x)^2,x]

[Out]

(c^2*x*Sqrt[1 - a^2*x^2])/2 - (c^2*(1 - a^2*x^2)^(3/2))/(3*a) + (c^2*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 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[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)^2 \, dx &=c^3 \int \frac{\left (1-a^2 x^2\right )^{3/2}}{c-a c x} \, dx\\ &=-\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a}+c^2 \int \sqrt{1-a^2 x^2} \, dx\\ &=\frac{1}{2} c^2 x \sqrt{1-a^2 x^2}-\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac{1}{2} c^2 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{1}{2} c^2 x \sqrt{1-a^2 x^2}-\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac{c^2 \sin ^{-1}(a x)}{2 a}\\ \end{align*}

Mathematica [A]  time = 0.0735729, size = 59, normalized size = 0.97 \[ \frac{c^2 \left (\sqrt{1-a^2 x^2} \left (2 a^2 x^2+3 a x-2\right )-6 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{6 a} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(3*ArcTanh[a*x])*(c - a*c*x)^2,x]

[Out]

(c^2*(Sqrt[1 - a^2*x^2]*(-2 + 3*a*x + 2*a^2*x^2) - 6*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(6*a)

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Maple [B]  time = 0.047, size = 137, normalized size = 2.3 \begin{align*} -{\frac{{c}^{2}{a}^{3}{x}^{4}}{3}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{2\,a{c}^{2}{x}^{2}}{3}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{{c}^{2}}{3\,a}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{{a}^{2}{c}^{2}{x}^{3}}{2}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{x{c}^{2}}{2}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{{c}^{2}}{2}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \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)^2,x)

[Out]

-1/3*c^2*a^3*x^4/(-a^2*x^2+1)^(1/2)+2/3*c^2*a*x^2/(-a^2*x^2+1)^(1/2)-1/3*c^2/a/(-a^2*x^2+1)^(1/2)-1/2*c^2*a^2*
x^3/(-a^2*x^2+1)^(1/2)+1/2*c^2*x/(-a^2*x^2+1)^(1/2)+1/2*c^2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2
))

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Maxima [B]  time = 1.44349, size = 171, normalized size = 2.8 \begin{align*} -\frac{a^{3} c^{2} x^{4}}{3 \, \sqrt{-a^{2} x^{2} + 1}} - \frac{a^{2} c^{2} x^{3}}{2 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{2 \, a c^{2} x^{2}}{3 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{c^{2} x}{2 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{c^{2} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{2 \, \sqrt{a^{2}}} - \frac{c^{2}}{3 \, \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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.82295, size = 151, normalized size = 2.48 \begin{align*} -\frac{6 \, c^{2} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) -{\left (2 \, a^{2} c^{2} x^{2} + 3 \, a c^{2} x - 2 \, c^{2}\right )} \sqrt{-a^{2} x^{2} + 1}}{6 \, 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)^2,x, algorithm="fricas")

[Out]

-1/6*(6*c^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (2*a^2*c^2*x^2 + 3*a*c^2*x - 2*c^2)*sqrt(-a^2*x^2 + 1))/a

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Sympy [A]  time = 12.4624, size = 221, normalized size = 3.62 \begin{align*} - a^{3} c^{2} \left (\begin{cases} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{3 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{3 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right ) - a^{2} c^{2} \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 ) + a c^{2} \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^{2} \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)**2,x)

[Out]

-a**3*c**2*Piecewise((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)/(3*a**4), Ne(a, 0)), (x**4/
4, True)) - a**2*c**2*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*sqrt(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True)) + a*c**2*Piec
ewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) + c**2*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 = 1.22594, size = 73, normalized size = 1.2 \begin{align*} \frac{c^{2} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{2 \,{\left | a \right |}} + \frac{1}{6} \, \sqrt{-a^{2} x^{2} + 1}{\left ({\left (2 \, a c^{2} x + 3 \, c^{2}\right )} x - \frac{2 \, c^{2}}{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)^2,x, algorithm="giac")

[Out]

1/2*c^2*arcsin(a*x)*sgn(a)/abs(a) + 1/6*sqrt(-a^2*x^2 + 1)*((2*a*c^2*x + 3*c^2)*x - 2*c^2/a)