3.1256 \(\int e^{-3 \tanh ^{-1}(a x)} (c-a^2 c x^2)^2 \, dx\)

Optimal. Leaf size=123 \[ \frac{c^2 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{7 c^2 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac{7 c^2 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac{7}{8} c^2 x \sqrt{1-a^2 x^2}+\frac{7 c^2 \sin ^{-1}(a x)}{8 a} \]

[Out]

(7*c^2*x*Sqrt[1 - a^2*x^2])/8 + (7*c^2*(1 - a^2*x^2)^(3/2))/(12*a) + (7*c^2*(1 - a*x)*(1 - a^2*x^2)^(3/2))/(20
*a) + (c^2*(1 - a*x)^2*(1 - a^2*x^2)^(3/2))/(5*a) + (7*c^2*ArcSin[a*x])/(8*a)

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Rubi [A]  time = 0.0745975, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {6139, 671, 641, 195, 216} \[ \frac{c^2 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{7 c^2 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac{7 c^2 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac{7}{8} c^2 x \sqrt{1-a^2 x^2}+\frac{7 c^2 \sin ^{-1}(a x)}{8 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*c^2*x*Sqrt[1 - a^2*x^2])/8 + (7*c^2*(1 - a^2*x^2)^(3/2))/(12*a) + (7*c^2*(1 - a*x)*(1 - a^2*x^2)^(3/2))/(20
*a) + (c^2*(1 - a*x)^2*(1 - a^2*x^2)^(3/2))/(5*a) + (7*c^2*ArcSin[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 671

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(m + 2*p + 1)), x] + Dist[(2*c*d*(m + p))/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p, x]
, x] /; FreeQ[{a, c, d, e, p}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && 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^{-3 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx &=c^2 \int (1-a x)^3 \sqrt{1-a^2 x^2} \, dx\\ &=\frac{c^2 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{1}{5} \left (7 c^2\right ) \int (1-a x)^2 \sqrt{1-a^2 x^2} \, dx\\ &=\frac{7 c^2 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac{c^2 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{1}{4} \left (7 c^2\right ) \int (1-a x) \sqrt{1-a^2 x^2} \, dx\\ &=\frac{7 c^2 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac{7 c^2 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac{c^2 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{1}{4} \left (7 c^2\right ) \int \sqrt{1-a^2 x^2} \, dx\\ &=\frac{7}{8} c^2 x \sqrt{1-a^2 x^2}+\frac{7 c^2 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac{7 c^2 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac{c^2 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{1}{8} \left (7 c^2\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{7}{8} c^2 x \sqrt{1-a^2 x^2}+\frac{7 c^2 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac{7 c^2 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac{c^2 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{7 c^2 \sin ^{-1}(a x)}{8 a}\\ \end{align*}

Mathematica [A]  time = 0.0998186, size = 75, normalized size = 0.61 \[ -\frac{c^2 \left (\sqrt{1-a^2 x^2} \left (24 a^4 x^4-90 a^3 x^3+112 a^2 x^2-15 a x-136\right )+210 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{120 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(c^2*(Sqrt[1 - a^2*x^2]*(-136 - 15*a*x + 112*a^2*x^2 - 90*a^3*x^3 + 24*a^4*x^4) + 210*ArcSin[Sqrt[1 - a*x]/Sq
rt[2]]))/(120*a)

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Maple [A]  time = 0.034, size = 189, normalized size = 1.5 \begin{align*} -{\frac{{c}^{2}}{5\,a} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}-{\frac{3\,x{c}^{2}}{4} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{9\,x{c}^{2}}{8}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{9\,{c}^{2}}{8}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{4\,{c}^{2}}{3\,a} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}+2\,{c}^{2}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }x+2\,{\frac{{c}^{2}}{\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*c^2*(-a^2*x^2+1)^(5/2)/a-3/4*c^2*x*(-a^2*x^2+1)^(3/2)-9/8*c^2*x*(-a^2*x^2+1)^(1/2)-9/8*c^2/(a^2)^(1/2)*ar
ctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+4/3*c^2/a*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)+2*c^2*(-a^2*(x+1/a)^2+2*a*
(x+1/a))^(1/2)*x+2*c^2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))

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Maxima [C]  time = 1.45503, size = 198, normalized size = 1.61 \begin{align*} -\frac{3}{4} \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c^{2} x - \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}} c^{2}}{5 \, a} + 2 \, \sqrt{a^{2} x^{2} + 4 \, a x + 3} c^{2} x - \frac{9}{8} \, \sqrt{-a^{2} x^{2} + 1} c^{2} x + \frac{4 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c^{2}}{3 \, a} - \frac{2 i \, c^{2} \arcsin \left (a x + 2\right )}{a} - \frac{9 \, c^{2} \arcsin \left (a x\right )}{8 \, a} + \frac{4 \, \sqrt{a^{2} x^{2} + 4 \, a x + 3} c^{2}}{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)^3*(-a^2*x^2+1)^(3/2),x, algorithm="maxima")

[Out]

-3/4*(-a^2*x^2 + 1)^(3/2)*c^2*x - 1/5*(-a^2*x^2 + 1)^(5/2)*c^2/a + 2*sqrt(a^2*x^2 + 4*a*x + 3)*c^2*x - 9/8*sqr
t(-a^2*x^2 + 1)*c^2*x + 4/3*(-a^2*x^2 + 1)^(3/2)*c^2/a - 2*I*c^2*arcsin(a*x + 2)/a - 9/8*c^2*arcsin(a*x)/a + 4
*sqrt(a^2*x^2 + 4*a*x + 3)*c^2/a

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Fricas [A]  time = 2.57265, size = 209, normalized size = 1.7 \begin{align*} -\frac{210 \, c^{2} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (24 \, a^{4} c^{2} x^{4} - 90 \, a^{3} c^{2} x^{3} + 112 \, a^{2} c^{2} x^{2} - 15 \, a c^{2} x - 136 \, c^{2}\right )} \sqrt{-a^{2} x^{2} + 1}}{120 \, 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)^3*(-a^2*x^2+1)^(3/2),x, algorithm="fricas")

[Out]

-1/120*(210*c^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (24*a^4*c^2*x^4 - 90*a^3*c^2*x^3 + 112*a^2*c^2*x^2 -
15*a*c^2*x - 136*c^2)*sqrt(-a^2*x^2 + 1))/a

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Sympy [C]  time = 12.5228, size = 340, normalized size = 2.76 \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 ) + 3 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 ) - 3 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)**3*(-a**2*x**2+1)**(3/2),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)) + 3*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)) - 3*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 = 1.17667, size = 105, normalized size = 0.85 \begin{align*} \frac{7 \, c^{2} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{8 \,{\left | a \right |}} + \frac{1}{120} \, \sqrt{-a^{2} x^{2} + 1}{\left ({\left (15 \, c^{2} - 2 \,{\left (56 \, a c^{2} + 3 \,{\left (4 \, a^{3} c^{2} x - 15 \, a^{2} c^{2}\right )} x\right )} x\right )} x + \frac{136 \, 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)^3*(-a^2*x^2+1)^(3/2),x, algorithm="giac")

[Out]

7/8*c^2*arcsin(a*x)*sgn(a)/abs(a) + 1/120*sqrt(-a^2*x^2 + 1)*((15*c^2 - 2*(56*a*c^2 + 3*(4*a^3*c^2*x - 15*a^2*
c^2)*x)*x)*x + 136*c^2/a)