3.1193 \(\int e^{-\tanh ^{-1}(a x)} (c-a^2 c x^2)^3 \, dx\)

Optimal. Leaf size=105 \[ \frac{c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+\frac{1}{6} c^3 x \left (1-a^2 x^2\right )^{5/2}+\frac{5}{24} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac{5}{16} c^3 x \sqrt{1-a^2 x^2}+\frac{5 c^3 \sin ^{-1}(a x)}{16 a} \]

[Out]

(5*c^3*x*Sqrt[1 - a^2*x^2])/16 + (5*c^3*x*(1 - a^2*x^2)^(3/2))/24 + (c^3*x*(1 - a^2*x^2)^(5/2))/6 + (c^3*(1 -
a^2*x^2)^(7/2))/(7*a) + (5*c^3*ArcSin[a*x])/(16*a)

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Rubi [A]  time = 0.0590107, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6139, 641, 195, 216} \[ \frac{c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+\frac{1}{6} c^3 x \left (1-a^2 x^2\right )^{5/2}+\frac{5}{24} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac{5}{16} c^3 x \sqrt{1-a^2 x^2}+\frac{5 c^3 \sin ^{-1}(a x)}{16 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*c^3*x*Sqrt[1 - a^2*x^2])/16 + (5*c^3*x*(1 - a^2*x^2)^(3/2))/24 + (c^3*x*(1 - a^2*x^2)^(5/2))/6 + (c^3*(1 -
a^2*x^2)^(7/2))/(7*a) + (5*c^3*ArcSin[a*x])/(16*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 )^3 \, dx &=c^3 \int (1-a x) \left (1-a^2 x^2\right )^{5/2} \, dx\\ &=\frac{c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+c^3 \int \left (1-a^2 x^2\right )^{5/2} \, dx\\ &=\frac{1}{6} c^3 x \left (1-a^2 x^2\right )^{5/2}+\frac{c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+\frac{1}{6} \left (5 c^3\right ) \int \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac{5}{24} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac{1}{6} c^3 x \left (1-a^2 x^2\right )^{5/2}+\frac{c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+\frac{1}{8} \left (5 c^3\right ) \int \sqrt{1-a^2 x^2} \, dx\\ &=\frac{5}{16} c^3 x \sqrt{1-a^2 x^2}+\frac{5}{24} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac{1}{6} c^3 x \left (1-a^2 x^2\right )^{5/2}+\frac{c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+\frac{1}{16} \left (5 c^3\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{5}{16} c^3 x \sqrt{1-a^2 x^2}+\frac{5}{24} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac{1}{6} c^3 x \left (1-a^2 x^2\right )^{5/2}+\frac{c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+\frac{5 c^3 \sin ^{-1}(a x)}{16 a}\\ \end{align*}

Mathematica [A]  time = 0.121655, size = 91, normalized size = 0.87 \[ -\frac{c^3 \left (\sqrt{1-a^2 x^2} \left (48 a^6 x^6-56 a^5 x^5-144 a^4 x^4+182 a^3 x^3+144 a^2 x^2-231 a x-48\right )+210 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{336 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(c^3*(Sqrt[1 - a^2*x^2]*(-48 - 231*a*x + 144*a^2*x^2 + 182*a^3*x^3 - 144*a^4*x^4 - 56*a^5*x^5 + 48*a^6*x^6) +
210*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(336*a)

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Maple [A]  time = 0.046, size = 155, normalized size = 1.5 \begin{align*}{\frac{{c}^{3}{a}^{3}{x}^{4}}{7} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{2\,{c}^{3}a{x}^{2}}{7} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{{c}^{3}}{7\,a} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{{c}^{3}{a}^{2}{x}^{3}}{6} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{c}^{3}x}{8} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{c}^{3}x}{16}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{5\,{c}^{3}}{16}\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)^3/(a*x+1)*(-a^2*x^2+1)^(1/2),x)

[Out]

1/7*c^3*a^3*x^4*(-a^2*x^2+1)^(3/2)-2/7*c^3*a*x^2*(-a^2*x^2+1)^(3/2)+1/7*c^3*(-a^2*x^2+1)^(3/2)/a-1/6*c^3*a^2*x
^3*(-a^2*x^2+1)^(3/2)+3/8*c^3*x*(-a^2*x^2+1)^(3/2)+5/16*c^3*x*(-a^2*x^2+1)^(1/2)+5/16*c^3/(a^2)^(1/2)*arctan((
a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))

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Maxima [A]  time = 1.50055, size = 184, normalized size = 1.75 \begin{align*} \frac{1}{7} \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{3} c^{3} x^{4} - \frac{1}{6} \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{2} c^{3} x^{3} - \frac{2}{7} \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a c^{3} x^{2} + \frac{3}{8} \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c^{3} x + \frac{5}{16} \, \sqrt{-a^{2} x^{2} + 1} c^{3} x + \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c^{3}}{7 \, a} + \frac{5 \, c^{3} \arcsin \left (a x\right )}{16 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*(-a^2*x^2 + 1)^(3/2)*a^3*c^3*x^4 - 1/6*(-a^2*x^2 + 1)^(3/2)*a^2*c^3*x^3 - 2/7*(-a^2*x^2 + 1)^(3/2)*a*c^3*x
^2 + 3/8*(-a^2*x^2 + 1)^(3/2)*c^3*x + 5/16*sqrt(-a^2*x^2 + 1)*c^3*x + 1/7*(-a^2*x^2 + 1)^(3/2)*c^3/a + 5/16*c^
3*arcsin(a*x)/a

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Fricas [A]  time = 2.42659, size = 258, normalized size = 2.46 \begin{align*} -\frac{210 \, c^{3} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (48 \, a^{6} c^{3} x^{6} - 56 \, a^{5} c^{3} x^{5} - 144 \, a^{4} c^{3} x^{4} + 182 \, a^{3} c^{3} x^{3} + 144 \, a^{2} c^{3} x^{2} - 231 \, a c^{3} x - 48 \, c^{3}\right )} \sqrt{-a^{2} x^{2} + 1}}{336 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/336*(210*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (48*a^6*c^3*x^6 - 56*a^5*c^3*x^5 - 144*a^4*c^3*x^4 +
182*a^3*c^3*x^3 + 144*a^2*c^3*x^2 - 231*a*c^3*x - 48*c^3)*sqrt(-a^2*x^2 + 1))/a

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Sympy [C]  time = 11.5812, size = 629, normalized size = 5.99 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**5*c**3*Piecewise((x**6*sqrt(-a**2*x**2 + 1)/7 - x**4*sqrt(-a**2*x**2 + 1)/(35*a**2) - 4*x**2*sqrt(-a**2*x*
*2 + 1)/(105*a**4) - 8*sqrt(-a**2*x**2 + 1)/(105*a**6), Ne(a, 0)), (x**6/6, True)) + a**4*c**3*Piecewise((I*a*
*2*x**7/(6*sqrt(a**2*x**2 - 1)) - 5*I*x**5/(24*sqrt(a**2*x**2 - 1)) - I*x**3/(48*a**2*sqrt(a**2*x**2 - 1)) + I
*x/(16*a**4*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(16*a**5), Abs(a**2*x**2) > 1), (-a**2*x**7/(6*sqrt(-a**2*x**2
+ 1)) + 5*x**5/(24*sqrt(-a**2*x**2 + 1)) + x**3/(48*a**2*sqrt(-a**2*x**2 + 1)) - x/(16*a**4*sqrt(-a**2*x**2 +
1)) + asin(a*x)/(16*a**5), True)) + 2*a**3*c**3*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)) - 2*a**2*c**3*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**3*Piecewise((x**2/2, Eq(a**2, 0)), (-(-a**2
*x**2 + 1)**(3/2)/(3*a**2), True)) + c**3*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 = 1.22509, size = 136, normalized size = 1.3 \begin{align*} \frac{5 \, c^{3} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{16 \,{\left | a \right |}} + \frac{1}{336} \, \sqrt{-a^{2} x^{2} + 1}{\left (\frac{48 \, c^{3}}{a} +{\left (231 \, c^{3} - 2 \,{\left (72 \, a c^{3} +{\left (91 \, a^{2} c^{3} - 4 \,{\left (18 \, a^{3} c^{3} -{\left (6 \, a^{5} c^{3} x - 7 \, a^{4} c^{3}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/16*c^3*arcsin(a*x)*sgn(a)/abs(a) + 1/336*sqrt(-a^2*x^2 + 1)*(48*c^3/a + (231*c^3 - 2*(72*a*c^3 + (91*a^2*c^3
- 4*(18*a^3*c^3 - (6*a^5*c^3*x - 7*a^4*c^3)*x)*x)*x)*x)*x)