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

Optimal. Leaf size=131 \[ -\frac{2 c^2 (1-a x)^4}{a \sqrt{1-a^2 x^2}}-\frac{7 c^2 \sqrt{1-a^2 x^2} (1-a x)^2}{3 a}-\frac{35 c^2 \sqrt{1-a^2 x^2} (1-a x)}{6 a}-\frac{35 c^2 \sqrt{1-a^2 x^2}}{2 a}-\frac{35 c^2 \sin ^{-1}(a x)}{2 a} \]

[Out]

(-2*c^2*(1 - a*x)^4)/(a*Sqrt[1 - a^2*x^2]) - (35*c^2*Sqrt[1 - a^2*x^2])/(2*a) - (35*c^2*(1 - a*x)*Sqrt[1 - a^2
*x^2])/(6*a) - (7*c^2*(1 - a*x)^2*Sqrt[1 - a^2*x^2])/(3*a) - (35*c^2*ArcSin[a*x])/(2*a)

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Rubi [A]  time = 0.0941339, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {6127, 669, 671, 641, 216} \[ -\frac{2 c^2 (1-a x)^4}{a \sqrt{1-a^2 x^2}}-\frac{7 c^2 \sqrt{1-a^2 x^2} (1-a x)^2}{3 a}-\frac{35 c^2 \sqrt{1-a^2 x^2} (1-a x)}{6 a}-\frac{35 c^2 \sqrt{1-a^2 x^2}}{2 a}-\frac{35 c^2 \sin ^{-1}(a x)}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*c^2*(1 - a*x)^4)/(a*Sqrt[1 - a^2*x^2]) - (35*c^2*Sqrt[1 - a^2*x^2])/(2*a) - (35*c^2*(1 - a*x)*Sqrt[1 - a^2
*x^2])/(6*a) - (7*c^2*(1 - a*x)^2*Sqrt[1 - a^2*x^2])/(3*a) - (35*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 669

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*(p + 1)), x] - Dist[(e^2*(m + p))/(c*(p + 1)), Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]

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 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 &=\frac{\int \frac{(c-a c x)^5}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=-\frac{2 c^2 (1-a x)^4}{a \sqrt{1-a^2 x^2}}-\frac{7 \int \frac{(c-a c x)^3}{\sqrt{1-a^2 x^2}} \, dx}{c}\\ &=-\frac{2 c^2 (1-a x)^4}{a \sqrt{1-a^2 x^2}}-\frac{7 c^2 (1-a x)^2 \sqrt{1-a^2 x^2}}{3 a}-\frac{35}{3} \int \frac{(c-a c x)^2}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{2 c^2 (1-a x)^4}{a \sqrt{1-a^2 x^2}}-\frac{35 c^2 (1-a x) \sqrt{1-a^2 x^2}}{6 a}-\frac{7 c^2 (1-a x)^2 \sqrt{1-a^2 x^2}}{3 a}-\frac{1}{2} (35 c) \int \frac{c-a c x}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{2 c^2 (1-a x)^4}{a \sqrt{1-a^2 x^2}}-\frac{35 c^2 \sqrt{1-a^2 x^2}}{2 a}-\frac{35 c^2 (1-a x) \sqrt{1-a^2 x^2}}{6 a}-\frac{7 c^2 (1-a x)^2 \sqrt{1-a^2 x^2}}{3 a}-\frac{1}{2} \left (35 c^2\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{2 c^2 (1-a x)^4}{a \sqrt{1-a^2 x^2}}-\frac{35 c^2 \sqrt{1-a^2 x^2}}{2 a}-\frac{35 c^2 (1-a x) \sqrt{1-a^2 x^2}}{6 a}-\frac{7 c^2 (1-a x)^2 \sqrt{1-a^2 x^2}}{3 a}-\frac{35 c^2 \sin ^{-1}(a x)}{2 a}\\ \end{align*}

Mathematica [C]  time = 0.0165767, size = 45, normalized size = 0.34 \[ -\frac{c^2 (1-a x)^{9/2} \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{9}{2},\frac{11}{2},\frac{1}{2} (1-a x)\right )}{9 \sqrt{2} a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(c^2*(1 - a*x)^(9/2)*Hypergeometric2F1[3/2, 9/2, 11/2, (1 - a*x)/2])/(9*Sqrt[2]*a)

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Maple [A]  time = 0.045, size = 179, normalized size = 1.4 \begin{align*} -12\,{\frac{{c}^{2} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{5/2}}{{a}^{3} \left ( x+{a}^{-1} \right ) ^{2}}}-{\frac{35\,{c}^{2}}{3\,a} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{35\,x{c}^{2}}{2}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}-{\frac{35\,{c}^{2}}{2}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-4\,{\frac{{c}^{2} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{5/2}}{{a}^{4} \left ( x+{a}^{-1} \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-12*c^2/a^3/(x+1/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)-35/3*c^2/a*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)-35/2*c^
2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)*x-35/2*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))-4*c^2/a^4/(x+1/a)^3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)

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Maxima [C]  time = 1.46234, size = 265, normalized size = 2.02 \begin{align*} \frac{1}{2} \, \sqrt{a^{2} x^{2} + 4 \, a x + 3} c^{2} x + \frac{4 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c^{2}}{a^{3} x^{2} + 2 \, a^{2} x + a} - \frac{2 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c^{2}}{a^{2} x + a} + \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c^{2}}{3 \, a} - \frac{i \, c^{2} \arcsin \left (a x + 2\right )}{2 \, a} - \frac{18 \, c^{2} \arcsin \left (a x\right )}{a} - \frac{24 \, \sqrt{-a^{2} x^{2} + 1} c^{2}}{a^{2} x + a} + \frac{\sqrt{a^{2} x^{2} + 4 \, a x + 3} c^{2}}{a} - \frac{6 \, \sqrt{-a^{2} x^{2} + 1} c^{2}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(a^2*x^2 + 4*a*x + 3)*c^2*x + 4*(-a^2*x^2 + 1)^(3/2)*c^2/(a^3*x^2 + 2*a^2*x + a) - 2*(-a^2*x^2 + 1)^(3
/2)*c^2/(a^2*x + a) + 1/3*(-a^2*x^2 + 1)^(3/2)*c^2/a - 1/2*I*c^2*arcsin(a*x + 2)/a - 18*c^2*arcsin(a*x)/a - 24
*sqrt(-a^2*x^2 + 1)*c^2/(a^2*x + a) + sqrt(a^2*x^2 + 4*a*x + 3)*c^2/a - 6*sqrt(-a^2*x^2 + 1)*c^2/a

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Fricas [A]  time = 1.68004, size = 243, normalized size = 1.85 \begin{align*} -\frac{166 \, a c^{2} x + 166 \, c^{2} - 210 \,{\left (a c^{2} x + c^{2}\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (2 \, a^{3} c^{2} x^{3} - 13 \, a^{2} c^{2} x^{2} + 55 \, a c^{2} x + 166 \, c^{2}\right )} \sqrt{-a^{2} x^{2} + 1}}{6 \,{\left (a^{2} x + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(166*a*c^2*x + 166*c^2 - 210*(a*c^2*x + c^2)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (2*a^3*c^2*x^3 - 13
*a^2*c^2*x^2 + 55*a*c^2*x + 166*c^2)*sqrt(-a^2*x^2 + 1))/(a^2*x + a)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} c^{2} \left (\int \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int - \frac{2 a x \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int \frac{2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int - \frac{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**2*(Integral(sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(-2*a*x*sqrt(-a**2*x**
2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(2*a**3*x**3*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a*
*2*x**2 + 3*a*x + 1), x) + Integral(-a**4*x**4*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x))

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Giac [A]  time = 1.29119, size = 123, normalized size = 0.94 \begin{align*} -\frac{35 \, 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 - 15 \, c^{2}\right )} x + \frac{70 \, c^{2}}{a}\right )} + \frac{32 \, c^{2}}{{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} + 1\right )}{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-35/2*c^2*arcsin(a*x)*sgn(a)/abs(a) - 1/6*sqrt(-a^2*x^2 + 1)*((2*a*c^2*x - 15*c^2)*x + 70*c^2/a) + 32*c^2/(((s
qrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) + 1)*abs(a))