3.1194 \(\int e^{-\tanh ^{-1}(a x)} (c-a^2 c x^2)^2 \, dx\)

Optimal. Leaf size=83 \[ \frac{c^2 \left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac{1}{4} c^2 x \left (1-a^2 x^2\right )^{3/2}+\frac{3}{8} c^2 x \sqrt{1-a^2 x^2}+\frac{3 c^2 \sin ^{-1}(a x)}{8 a} \]

[Out]

(3*c^2*x*Sqrt[1 - a^2*x^2])/8 + (c^2*x*(1 - a^2*x^2)^(3/2))/4 + (c^2*(1 - a^2*x^2)^(5/2))/(5*a) + (3*c^2*ArcSi
n[a*x])/(8*a)

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Rubi [A]  time = 0.0507057, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6139, 641, 195, 216} \[ \frac{c^2 \left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac{1}{4} c^2 x \left (1-a^2 x^2\right )^{3/2}+\frac{3}{8} c^2 x \sqrt{1-a^2 x^2}+\frac{3 c^2 \sin ^{-1}(a x)}{8 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*c^2*x*Sqrt[1 - a^2*x^2])/8 + (c^2*x*(1 - a^2*x^2)^(3/2))/4 + (c^2*(1 - a^2*x^2)^(5/2))/(5*a) + (3*c^2*ArcSi
n[a*x])/(8*a)

Rule 6139

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

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)} \left (c-a^2 c x^2\right )^2 \, dx &=c^2 \int (1-a x) \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac{c^2 \left (1-a^2 x^2\right )^{5/2}}{5 a}+c^2 \int \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac{1}{4} c^2 x \left (1-a^2 x^2\right )^{3/2}+\frac{c^2 \left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac{1}{4} \left (3 c^2\right ) \int \sqrt{1-a^2 x^2} \, dx\\ &=\frac{3}{8} c^2 x \sqrt{1-a^2 x^2}+\frac{1}{4} c^2 x \left (1-a^2 x^2\right )^{3/2}+\frac{c^2 \left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac{1}{8} \left (3 c^2\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{3}{8} c^2 x \sqrt{1-a^2 x^2}+\frac{1}{4} c^2 x \left (1-a^2 x^2\right )^{3/2}+\frac{c^2 \left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac{3 c^2 \sin ^{-1}(a x)}{8 a}\\ \end{align*}

Mathematica [A]  time = 0.0948982, size = 75, normalized size = 0.9 \[ \frac{c^2 \left (\sqrt{1-a^2 x^2} \left (8 a^4 x^4-10 a^3 x^3-16 a^2 x^2+25 a x+8\right )-30 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{40 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^2*(Sqrt[1 - a^2*x^2]*(8 + 25*a*x - 16*a^2*x^2 - 10*a^3*x^3 + 8*a^4*x^4) - 30*ArcSin[Sqrt[1 - a*x]/Sqrt[2]])
)/(40*a)

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Maple [A]  time = 0.037, size = 109, normalized size = 1.3 \begin{align*} -{\frac{a{c}^{2}{x}^{2}}{5} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{{c}^{2}}{5\,a} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{x{c}^{2}}{4} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{3\,x{c}^{2}}{8}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{3\,{c}^{2}}{8}\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^2*c*x^2+c)^2/(a*x+1)*(-a^2*x^2+1)^(1/2),x)

[Out]

-1/5*c^2*a*x^2*(-a^2*x^2+1)^(3/2)+1/5*c^2*(-a^2*x^2+1)^(3/2)/a+1/4*c^2*x*(-a^2*x^2+1)^(3/2)+3/8*c^2*x*(-a^2*x^
2+1)^(1/2)+3/8*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 = 1.45793, size = 122, normalized size = 1.47 \begin{align*} -\frac{1}{5} \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a c^{2} x^{2} + \frac{1}{4} \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c^{2} x + \frac{3}{8} \, \sqrt{-a^{2} x^{2} + 1} c^{2} x + \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c^{2}}{5 \, a} + \frac{3 \, c^{2} \arcsin \left (a x\right )}{8 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*(-a^2*x^2 + 1)^(3/2)*a*c^2*x^2 + 1/4*(-a^2*x^2 + 1)^(3/2)*c^2*x + 3/8*sqrt(-a^2*x^2 + 1)*c^2*x + 1/5*(-a^
2*x^2 + 1)^(3/2)*c^2/a + 3/8*c^2*arcsin(a*x)/a

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Fricas [A]  time = 2.40067, size = 201, normalized size = 2.42 \begin{align*} -\frac{30 \, c^{2} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) -{\left (8 \, a^{4} c^{2} x^{4} - 10 \, a^{3} c^{2} x^{3} - 16 \, a^{2} c^{2} x^{2} + 25 \, a c^{2} x + 8 \, c^{2}\right )} \sqrt{-a^{2} x^{2} + 1}}{40 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/40*(30*c^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (8*a^4*c^2*x^4 - 10*a^3*c^2*x^3 - 16*a^2*c^2*x^2 + 25*a
*c^2*x + 8*c^2)*sqrt(-a^2*x^2 + 1))/a

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Sympy [C]  time = 6.20297, size = 337, normalized size = 4.06 \begin{align*} a^{3} c^{2} \left (\begin{cases} \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{5} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{15 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 a^{2} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left (a x \right )}}{8 a^{3}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{a^{2} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left (a x \right )}}{8 a^{3}} & \text{otherwise} \end{cases}\right ) - a c^{2} \left (\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left (- a^{2} x^{2} + 1\right )^{\frac{3}{2}}}{3 a^{2}} & \text{otherwise} \end{cases}\right ) + c^{2} \left (\begin{cases} \frac{i a^{2} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left (a x \right )}}{2 a} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left (a x \right )}}{2 a} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*c**2*Piecewise((x**4*sqrt(-a**2*x**2 + 1)/5 - x**2*sqrt(-a**2*x**2 + 1)/(15*a**2) - 2*sqrt(-a**2*x**2 + 1
)/(15*a**4), Ne(a, 0)), (x**4/4, True)) - a**2*c**2*Piecewise((I*a**2*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*x**3/
(8*sqrt(a**2*x**2 - 1)) + I*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**
2*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*x**3/(8*sqrt(-a**2*x**2 + 1)) - x/(8*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x
)/(8*a**3), True)) - a*c**2*Piecewise((x**2/2, Eq(a**2, 0)), (-(-a**2*x**2 + 1)**(3/2)/(3*a**2), True)) + c**2
*Piecewise((I*a**2*x**3/(2*sqrt(a**2*x**2 - 1)) - I*x/(2*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(2*a), Abs(a**2*x
**2) > 1), (x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/(2*a), True))

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Giac [A]  time = 1.21018, size = 105, normalized size = 1.27 \begin{align*} \frac{3 \, c^{2} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{8 \,{\left | a \right |}} + \frac{1}{40} \, \sqrt{-a^{2} x^{2} + 1}{\left ({\left (25 \, c^{2} - 2 \,{\left (8 \, a c^{2} -{\left (4 \, a^{3} c^{2} x - 5 \, a^{2} c^{2}\right )} x\right )} x\right )} x + \frac{8 \, c^{2}}{a}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*c^2*arcsin(a*x)*sgn(a)/abs(a) + 1/40*sqrt(-a^2*x^2 + 1)*((25*c^2 - 2*(8*a*c^2 - (4*a^3*c^2*x - 5*a^2*c^2)*
x)*x)*x + 8*c^2/a)