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

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 = 0.0372357, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6127, 641, 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^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 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

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^{\tanh ^{-1}(a x)} (c-a c x)^2 \, dx &=c \int (c-a c x) \sqrt{1-a^2 x^2} \, 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 = 0.0740159, 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^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 [A]  time = 0.039, size = 91, normalized size = 1.5 \begin{align*} -{\frac{{c}^{2}{x}^{2}a}{3}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{{c}^{2}}{3\,a}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{x{c}^{2}}{2}\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)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2,x)

[Out]

-1/3*c^2*a*x^2*(-a^2*x^2+1)^(1/2)+1/3*c^2*(-a^2*x^2+1)^(1/2)/a+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 [A]  time = 1.42586, size = 109, normalized size = 1.79 \begin{align*} -\frac{1}{3} \, \sqrt{-a^{2} x^{2} + 1} a c^{2} x^{2} + \frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1} c^{2} x + \frac{c^{2} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{2 \, \sqrt{a^{2}}} + \frac{\sqrt{-a^{2} x^{2} + 1} c^{2}}{3 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.62567, 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)/(-a^2*x^2+1)^(1/2)*(-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 = 5.30452, size = 102, normalized size = 1.67 \begin{align*} \begin{cases} \frac{c^{2} \sqrt{- a^{2} x^{2} + 1} - c^{2} \left (\begin{cases} - \frac{a x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left (a x \right )}}{2} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right ) + c^{2} \left (\begin{cases} \frac{\left (- a^{2} x^{2} + 1\right )^{\frac{3}{2}}}{3} - \sqrt{- a^{2} x^{2} + 1} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right ) + c^{2} \operatorname{asin}{\left (a x \right )}}{a} & \text{for}\: a \neq 0 \\c^{2} x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((c**2*sqrt(-a**2*x**2 + 1) - c**2*Piecewise((-a*x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/2, (a*x > -1)
& (a*x < 1))) + c**2*Piecewise(((-a**2*x**2 + 1)**(3/2)/3 - sqrt(-a**2*x**2 + 1), (a*x > -1) & (a*x < 1))) + c
**2*asin(a*x))/a, Ne(a, 0)), (c**2*x, True))

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Giac [A]  time = 1.31044, 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)/(-a^2*x^2+1)^(1/2)*(-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)