3.317 \(\int e^{\tanh ^{-1}(a x)} x^2 (c-a c x)^4 \, dx\)

Optimal. Leaf size=146 \[ \frac{1}{7} a c^4 x^4 \left (1-a^2 x^2\right )^{3/2}-\frac{1}{2} c^4 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac{5 c^4 x^2 \left (1-a^2 x^2\right )^{3/2}}{7 a}+\frac{5 c^4 (16-21 a x) \left (1-a^2 x^2\right )^{3/2}}{168 a^3}+\frac{5 c^4 x \sqrt{1-a^2 x^2}}{16 a^2}+\frac{5 c^4 \sin ^{-1}(a x)}{16 a^3} \]

[Out]

(5*c^4*x*Sqrt[1 - a^2*x^2])/(16*a^2) + (5*c^4*x^2*(1 - a^2*x^2)^(3/2))/(7*a) - (c^4*x^3*(1 - a^2*x^2)^(3/2))/2
+ (a*c^4*x^4*(1 - a^2*x^2)^(3/2))/7 + (5*c^4*(16 - 21*a*x)*(1 - a^2*x^2)^(3/2))/(168*a^3) + (5*c^4*ArcSin[a*x
])/(16*a^3)

________________________________________________________________________________________

Rubi [A]  time = 0.304134, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {6128, 1809, 833, 780, 195, 216} \[ \frac{1}{7} a c^4 x^4 \left (1-a^2 x^2\right )^{3/2}-\frac{1}{2} c^4 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac{5 c^4 x^2 \left (1-a^2 x^2\right )^{3/2}}{7 a}+\frac{5 c^4 (16-21 a x) \left (1-a^2 x^2\right )^{3/2}}{168 a^3}+\frac{5 c^4 x \sqrt{1-a^2 x^2}}{16 a^2}+\frac{5 c^4 \sin ^{-1}(a x)}{16 a^3} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]*x^2*(c - a*c*x)^4,x]

[Out]

(5*c^4*x*Sqrt[1 - a^2*x^2])/(16*a^2) + (5*c^4*x^2*(1 - a^2*x^2)^(3/2))/(7*a) - (c^4*x^3*(1 - a^2*x^2)^(3/2))/2
+ (a*c^4*x^4*(1 - a^2*x^2)^(3/2))/7 + (5*c^4*(16 - 21*a*x)*(1 - a^2*x^2)^(3/2))/(168*a^3) + (5*c^4*ArcSin[a*x
])/(16*a^3)

Rule 6128

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

Rule 1809

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[(f*(c*x)^(m + q - 1)*(a + b*x^2)^(p + 1))/(b*c^(q - 1)*(m + q + 2*p + 1)), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 833

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(g*(d + e*x)
^m*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[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)} x^2 (c-a c x)^4 \, dx &=c \int x^2 (c-a c x)^3 \sqrt{1-a^2 x^2} \, dx\\ &=\frac{1}{7} a c^4 x^4 \left (1-a^2 x^2\right )^{3/2}-\frac{c \int x^2 \sqrt{1-a^2 x^2} \left (-7 a^2 c^3+25 a^3 c^3 x-21 a^4 c^3 x^2\right ) \, dx}{7 a^2}\\ &=-\frac{1}{2} c^4 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac{1}{7} a c^4 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac{c \int x^2 \left (105 a^4 c^3-150 a^5 c^3 x\right ) \sqrt{1-a^2 x^2} \, dx}{42 a^4}\\ &=\frac{5 c^4 x^2 \left (1-a^2 x^2\right )^{3/2}}{7 a}-\frac{1}{2} c^4 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac{1}{7} a c^4 x^4 \left (1-a^2 x^2\right )^{3/2}-\frac{c \int x \left (300 a^5 c^3-525 a^6 c^3 x\right ) \sqrt{1-a^2 x^2} \, dx}{210 a^6}\\ &=\frac{5 c^4 x^2 \left (1-a^2 x^2\right )^{3/2}}{7 a}-\frac{1}{2} c^4 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac{1}{7} a c^4 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac{5 c^4 (16-21 a x) \left (1-a^2 x^2\right )^{3/2}}{168 a^3}+\frac{\left (5 c^4\right ) \int \sqrt{1-a^2 x^2} \, dx}{8 a^2}\\ &=\frac{5 c^4 x \sqrt{1-a^2 x^2}}{16 a^2}+\frac{5 c^4 x^2 \left (1-a^2 x^2\right )^{3/2}}{7 a}-\frac{1}{2} c^4 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac{1}{7} a c^4 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac{5 c^4 (16-21 a x) \left (1-a^2 x^2\right )^{3/2}}{168 a^3}+\frac{\left (5 c^4\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{16 a^2}\\ &=\frac{5 c^4 x \sqrt{1-a^2 x^2}}{16 a^2}+\frac{5 c^4 x^2 \left (1-a^2 x^2\right )^{3/2}}{7 a}-\frac{1}{2} c^4 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac{1}{7} a c^4 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac{5 c^4 (16-21 a x) \left (1-a^2 x^2\right )^{3/2}}{168 a^3}+\frac{5 c^4 \sin ^{-1}(a x)}{16 a^3}\\ \end{align*}

Mathematica [A]  time = 0.103453, size = 91, normalized size = 0.62 \[ -\frac{c^4 \left (\sqrt{1-a^2 x^2} \left (48 a^6 x^6-168 a^5 x^5+192 a^4 x^4-42 a^3 x^3-80 a^2 x^2+105 a x-160\right )+210 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{336 a^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcTanh[a*x]*x^2*(c - a*c*x)^4,x]

[Out]

-(c^4*(Sqrt[1 - a^2*x^2]*(-160 + 105*a*x - 80*a^2*x^2 - 42*a^3*x^3 + 192*a^4*x^4 - 168*a^5*x^5 + 48*a^6*x^6) +
210*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(336*a^3)

________________________________________________________________________________________

Maple [A]  time = 0.062, size = 186, normalized size = 1.3 \begin{align*} -{\frac{{c}^{4}{a}^{3}{x}^{6}}{7}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{4\,{c}^{4}a{x}^{4}}{7}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{5\,{c}^{4}{x}^{2}}{21\,a}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{10\,{c}^{4}}{21\,{a}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{{c}^{4}{a}^{2}{x}^{5}}{2}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{{c}^{4}{x}^{3}}{8}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{5\,{c}^{4}x}{16\,{a}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{5\,{c}^{4}}{16\,{a}^{2}}\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)/(-a^2*x^2+1)^(1/2)*x^2*(-a*c*x+c)^4,x)

[Out]

-1/7*c^4*a^3*x^6*(-a^2*x^2+1)^(1/2)-4/7*c^4*a*x^4*(-a^2*x^2+1)^(1/2)+5/21*c^4/a*x^2*(-a^2*x^2+1)^(1/2)+10/21*c
^4/a^3*(-a^2*x^2+1)^(1/2)+1/2*c^4*a^2*x^5*(-a^2*x^2+1)^(1/2)+1/8*c^4*x^3*(-a^2*x^2+1)^(1/2)-5/16*c^4*x*(-a^2*x
^2+1)^(1/2)/a^2+5/16*c^4/a^2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))

________________________________________________________________________________________

Maxima [A]  time = 1.45246, size = 238, normalized size = 1.63 \begin{align*} -\frac{1}{7} \, \sqrt{-a^{2} x^{2} + 1} a^{3} c^{4} x^{6} + \frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{4} x^{5} - \frac{4}{7} \, \sqrt{-a^{2} x^{2} + 1} a c^{4} x^{4} + \frac{1}{8} \, \sqrt{-a^{2} x^{2} + 1} c^{4} x^{3} + \frac{5 \, \sqrt{-a^{2} x^{2} + 1} c^{4} x^{2}}{21 \, a} - \frac{5 \, \sqrt{-a^{2} x^{2} + 1} c^{4} x}{16 \, a^{2}} + \frac{5 \, c^{4} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{16 \, \sqrt{a^{2}} a^{2}} + \frac{10 \, \sqrt{-a^{2} x^{2} + 1} c^{4}}{21 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2*(-a*c*x+c)^4,x, algorithm="maxima")

[Out]

-1/7*sqrt(-a^2*x^2 + 1)*a^3*c^4*x^6 + 1/2*sqrt(-a^2*x^2 + 1)*a^2*c^4*x^5 - 4/7*sqrt(-a^2*x^2 + 1)*a*c^4*x^4 +
1/8*sqrt(-a^2*x^2 + 1)*c^4*x^3 + 5/21*sqrt(-a^2*x^2 + 1)*c^4*x^2/a - 5/16*sqrt(-a^2*x^2 + 1)*c^4*x/a^2 + 5/16*
c^4*arcsin(a^2*x/sqrt(a^2))/(sqrt(a^2)*a^2) + 10/21*sqrt(-a^2*x^2 + 1)*c^4/a^3

________________________________________________________________________________________

Fricas [A]  time = 1.61897, size = 261, normalized size = 1.79 \begin{align*} -\frac{210 \, c^{4} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (48 \, a^{6} c^{4} x^{6} - 168 \, a^{5} c^{4} x^{5} + 192 \, a^{4} c^{4} x^{4} - 42 \, a^{3} c^{4} x^{3} - 80 \, a^{2} c^{4} x^{2} + 105 \, a c^{4} x - 160 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1}}{336 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2*(-a*c*x+c)^4,x, algorithm="fricas")

[Out]

-1/336*(210*c^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (48*a^6*c^4*x^6 - 168*a^5*c^4*x^5 + 192*a^4*c^4*x^4 -
42*a^3*c^4*x^3 - 80*a^2*c^4*x^2 + 105*a*c^4*x - 160*c^4)*sqrt(-a^2*x^2 + 1))/a^3

________________________________________________________________________________________

Sympy [C]  time = 16.7034, size = 683, normalized size = 4.68 \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)/(-a**2*x**2+1)**(1/2)*x**2*(-a*c*x+c)**4,x)

[Out]

a**5*c**4*Piecewise((-x**6*sqrt(-a**2*x**2 + 1)/(7*a**2) - 6*x**4*sqrt(-a**2*x**2 + 1)/(35*a**4) - 8*x**2*sqrt
(-a**2*x**2 + 1)/(35*a**6) - 16*sqrt(-a**2*x**2 + 1)/(35*a**8), Ne(a, 0)), (x**8/8, True)) - 3*a**4*c**4*Piece
wise((-I*x**7/(6*sqrt(a**2*x**2 - 1)) - I*x**5/(24*a**2*sqrt(a**2*x**2 - 1)) - 5*I*x**3/(48*a**4*sqrt(a**2*x**
2 - 1)) + 5*I*x/(16*a**6*sqrt(a**2*x**2 - 1)) - 5*I*acosh(a*x)/(16*a**7), Abs(a**2*x**2) > 1), (x**7/(6*sqrt(-
a**2*x**2 + 1)) + x**5/(24*a**2*sqrt(-a**2*x**2 + 1)) + 5*x**3/(48*a**4*sqrt(-a**2*x**2 + 1)) - 5*x/(16*a**6*s
qrt(-a**2*x**2 + 1)) + 5*asin(a*x)/(16*a**7), True)) + 2*a**3*c**4*Piecewise((-x**4*sqrt(-a**2*x**2 + 1)/(5*a*
*2) - 4*x**2*sqrt(-a**2*x**2 + 1)/(15*a**4) - 8*sqrt(-a**2*x**2 + 1)/(15*a**6), Ne(a, 0)), (x**6/6, True)) + 2
*a**2*c**4*Piecewise((-I*x**5/(4*sqrt(a**2*x**2 - 1)) - I*x**3/(8*a**2*sqrt(a**2*x**2 - 1)) + 3*I*x/(8*a**4*sq
rt(a**2*x**2 - 1)) - 3*I*acosh(a*x)/(8*a**5), Abs(a**2*x**2) > 1), (x**5/(4*sqrt(-a**2*x**2 + 1)) + x**3/(8*a*
*2*sqrt(-a**2*x**2 + 1)) - 3*x/(8*a**4*sqrt(-a**2*x**2 + 1)) + 3*asin(a*x)/(8*a**5), True)) - 3*a*c**4*Piecewi
se((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)/(3*a**4), Ne(a, 0)), (x**4/4, True)) + c**4*P
iecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) > 1), (x**3/(2*sqrt(-a**2*
x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True))

________________________________________________________________________________________

Giac [A]  time = 1.20694, size = 140, normalized size = 0.96 \begin{align*} \frac{5 \, c^{4} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{16 \, a^{2}{\left | a \right |}} - \frac{1}{336} \, \sqrt{-a^{2} x^{2} + 1}{\left ({\left (\frac{105 \, c^{4}}{a^{2}} - 2 \,{\left (\frac{40 \, c^{4}}{a} + 3 \,{\left (7 \, c^{4} - 4 \,{\left (8 \, a c^{4} +{\left (2 \, a^{3} c^{4} x - 7 \, a^{2} c^{4}\right )} x\right )} x\right )} x\right )} x\right )} x - \frac{160 \, c^{4}}{a^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2*(-a*c*x+c)^4,x, algorithm="giac")

[Out]

5/16*c^4*arcsin(a*x)*sgn(a)/(a^2*abs(a)) - 1/336*sqrt(-a^2*x^2 + 1)*((105*c^4/a^2 - 2*(40*c^4/a + 3*(7*c^4 - 4
*(8*a*c^4 + (2*a^3*c^4*x - 7*a^2*c^4)*x)*x)*x)*x)*x - 160*c^4/a^3)