3.1148 \(\int e^{3 \tanh ^{-1}(a x)} (c-a^2 c x^2)^3 \, dx\)

Optimal. Leaf size=143 \[ -\frac{c^3 (a x+1)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}-\frac{3 c^3 (a x+1) \left (1-a^2 x^2\right )^{5/2}}{14 a}-\frac{3 c^3 \left (1-a^2 x^2\right )^{5/2}}{10 a}+\frac{3}{8} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac{9}{16} c^3 x \sqrt{1-a^2 x^2}+\frac{9 c^3 \sin ^{-1}(a x)}{16 a} \]

[Out]

(9*c^3*x*Sqrt[1 - a^2*x^2])/16 + (3*c^3*x*(1 - a^2*x^2)^(3/2))/8 - (3*c^3*(1 - a^2*x^2)^(5/2))/(10*a) - (3*c^3
*(1 + a*x)*(1 - a^2*x^2)^(5/2))/(14*a) - (c^3*(1 + a*x)^2*(1 - a^2*x^2)^(5/2))/(7*a) + (9*c^3*ArcSin[a*x])/(16
*a)

________________________________________________________________________________________

Rubi [A]  time = 0.081671, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {6138, 671, 641, 195, 216} \[ -\frac{c^3 (a x+1)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}-\frac{3 c^3 (a x+1) \left (1-a^2 x^2\right )^{5/2}}{14 a}-\frac{3 c^3 \left (1-a^2 x^2\right )^{5/2}}{10 a}+\frac{3}{8} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac{9}{16} c^3 x \sqrt{1-a^2 x^2}+\frac{9 c^3 \sin ^{-1}(a x)}{16 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9*c^3*x*Sqrt[1 - a^2*x^2])/16 + (3*c^3*x*(1 - a^2*x^2)^(3/2))/8 - (3*c^3*(1 - a^2*x^2)^(5/2))/(10*a) - (3*c^3
*(1 + a*x)*(1 - a^2*x^2)^(5/2))/(14*a) - (c^3*(1 + a*x)^2*(1 - a^2*x^2)^(5/2))/(7*a) + (9*c^3*ArcSin[a*x])/(16
*a)

Rule 6138

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] && IGtQ[(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 )^3 \, dx &=c^3 \int (1+a x)^3 \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=-\frac{c^3 (1+a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac{1}{7} \left (9 c^3\right ) \int (1+a x)^2 \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=-\frac{3 c^3 (1+a x) \left (1-a^2 x^2\right )^{5/2}}{14 a}-\frac{c^3 (1+a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac{1}{2} \left (3 c^3\right ) \int (1+a x) \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=-\frac{3 c^3 \left (1-a^2 x^2\right )^{5/2}}{10 a}-\frac{3 c^3 (1+a x) \left (1-a^2 x^2\right )^{5/2}}{14 a}-\frac{c^3 (1+a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac{1}{2} \left (3 c^3\right ) \int \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac{3}{8} c^3 x \left (1-a^2 x^2\right )^{3/2}-\frac{3 c^3 \left (1-a^2 x^2\right )^{5/2}}{10 a}-\frac{3 c^3 (1+a x) \left (1-a^2 x^2\right )^{5/2}}{14 a}-\frac{c^3 (1+a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac{1}{8} \left (9 c^3\right ) \int \sqrt{1-a^2 x^2} \, dx\\ &=\frac{9}{16} c^3 x \sqrt{1-a^2 x^2}+\frac{3}{8} c^3 x \left (1-a^2 x^2\right )^{3/2}-\frac{3 c^3 \left (1-a^2 x^2\right )^{5/2}}{10 a}-\frac{3 c^3 (1+a x) \left (1-a^2 x^2\right )^{5/2}}{14 a}-\frac{c^3 (1+a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac{1}{16} \left (9 c^3\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{9}{16} c^3 x \sqrt{1-a^2 x^2}+\frac{3}{8} c^3 x \left (1-a^2 x^2\right )^{3/2}-\frac{3 c^3 \left (1-a^2 x^2\right )^{5/2}}{10 a}-\frac{3 c^3 (1+a x) \left (1-a^2 x^2\right )^{5/2}}{14 a}-\frac{c^3 (1+a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac{9 c^3 \sin ^{-1}(a x)}{16 a}\\ \end{align*}

Mathematica [A]  time = 0.11944, size = 91, normalized size = 0.64 \[ -\frac{c^3 \left (\sqrt{1-a^2 x^2} \left (80 a^6 x^6+280 a^5 x^5+208 a^4 x^4-350 a^3 x^3-656 a^2 x^2-245 a x+368\right )+630 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{560 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(c^3*(Sqrt[1 - a^2*x^2]*(368 - 245*a*x - 656*a^2*x^2 - 350*a^3*x^3 + 208*a^4*x^4 + 280*a^5*x^5 + 80*a^6*x^6)
+ 630*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(560*a)

________________________________________________________________________________________

Maple [A]  time = 0.104, size = 229, normalized size = 1.6 \begin{align*}{\frac{{c}^{3}{a}^{7}{x}^{8}}{7}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{8\,{c}^{3}{a}^{5}{x}^{6}}{35}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{54\,{c}^{3}{a}^{3}{x}^{4}}{35}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{64\,{c}^{3}a{x}^{2}}{35}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{23\,{c}^{3}}{35\,a}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{{a}^{6}{c}^{3}{x}^{7}}{2}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{9\,{a}^{4}{c}^{3}{x}^{5}}{8}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{3\,{c}^{3}{a}^{2}{x}^{3}}{16}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{7\,{c}^{3}x}{16}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{9\,{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*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^3,x)

[Out]

1/7*c^3*a^7*x^8/(-a^2*x^2+1)^(1/2)+8/35*c^3*a^5*x^6/(-a^2*x^2+1)^(1/2)-54/35*c^3*a^3*x^4/(-a^2*x^2+1)^(1/2)+64
/35*c^3*a*x^2/(-a^2*x^2+1)^(1/2)-23/35*c^3/a/(-a^2*x^2+1)^(1/2)+1/2*c^3*a^6*x^7/(-a^2*x^2+1)^(1/2)-9/8*c^3*a^4
*x^5/(-a^2*x^2+1)^(1/2)+3/16*c^3*a^2*x^3/(-a^2*x^2+1)^(1/2)+7/16*c^3*x/(-a^2*x^2+1)^(1/2)+9/16*c^3/(a^2)^(1/2)
*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))

________________________________________________________________________________________

Maxima [A]  time = 1.46271, size = 296, normalized size = 2.07 \begin{align*} \frac{a^{7} c^{3} x^{8}}{7 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{a^{6} c^{3} x^{7}}{2 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{8 \, a^{5} c^{3} x^{6}}{35 \, \sqrt{-a^{2} x^{2} + 1}} - \frac{9 \, a^{4} c^{3} x^{5}}{8 \, \sqrt{-a^{2} x^{2} + 1}} - \frac{54 \, a^{3} c^{3} x^{4}}{35 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{3 \, a^{2} c^{3} x^{3}}{16 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{64 \, a c^{3} x^{2}}{35 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{7 \, c^{3} x}{16 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{9 \, c^{3} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{16 \, \sqrt{a^{2}}} - \frac{23 \, c^{3}}{35 \, \sqrt{-a^{2} x^{2} + 1} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a^7*c^3*x^8/sqrt(-a^2*x^2 + 1) + 1/2*a^6*c^3*x^7/sqrt(-a^2*x^2 + 1) + 8/35*a^5*c^3*x^6/sqrt(-a^2*x^2 + 1)
- 9/8*a^4*c^3*x^5/sqrt(-a^2*x^2 + 1) - 54/35*a^3*c^3*x^4/sqrt(-a^2*x^2 + 1) + 3/16*a^2*c^3*x^3/sqrt(-a^2*x^2 +
1) + 64/35*a*c^3*x^2/sqrt(-a^2*x^2 + 1) + 7/16*c^3*x/sqrt(-a^2*x^2 + 1) + 9/16*c^3*arcsin(a^2*x/sqrt(a^2))/sq
rt(a^2) - 23/35*c^3/(sqrt(-a^2*x^2 + 1)*a)

________________________________________________________________________________________

Fricas [A]  time = 2.65055, size = 261, normalized size = 1.83 \begin{align*} -\frac{630 \, c^{3} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (80 \, a^{6} c^{3} x^{6} + 280 \, a^{5} c^{3} x^{5} + 208 \, a^{4} c^{3} x^{4} - 350 \, a^{3} c^{3} x^{3} - 656 \, a^{2} c^{3} x^{2} - 245 \, a c^{3} x + 368 \, c^{3}\right )} \sqrt{-a^{2} x^{2} + 1}}{560 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/560*(630*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (80*a^6*c^3*x^6 + 280*a^5*c^3*x^5 + 208*a^4*c^3*x^4 -
350*a^3*c^3*x^3 - 656*a^2*c^3*x^2 - 245*a*c^3*x + 368*c^3)*sqrt(-a^2*x^2 + 1))/a

________________________________________________________________________________________

Sympy [C]  time = 22.7718, size = 632, normalized size = 4.42 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Giac [A]  time = 1.17549, size = 138, normalized size = 0.97 \begin{align*} \frac{9 \, c^{3} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{16 \,{\left | a \right |}} - \frac{1}{560} \, \sqrt{-a^{2} x^{2} + 1}{\left (\frac{368 \, c^{3}}{a} -{\left (245 \, c^{3} + 2 \,{\left (328 \, a c^{3} +{\left (175 \, a^{2} c^{3} - 4 \,{\left (26 \, a^{3} c^{3} + 5 \,{\left (2 \, 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*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

9/16*c^3*arcsin(a*x)*sgn(a)/abs(a) - 1/560*sqrt(-a^2*x^2 + 1)*(368*c^3/a - (245*c^3 + 2*(328*a*c^3 + (175*a^2*
c^3 - 4*(26*a^3*c^3 + 5*(2*a^5*c^3*x + 7*a^4*c^3)*x)*x)*x)*x)*x)