3.316 \(\int e^{\tanh ^{-1}(a x)} x^3 (c-a c x)^4 \, dx\)

Optimal. Leaf size=173 \[ \frac{1}{8} a c^4 x^5 \left (1-a^2 x^2\right )^{3/2}-\frac{3}{7} c^4 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac{29 c^4 x^3 \left (1-a^2 x^2\right )^{3/2}}{48 a}-\frac{19 c^4 x^2 \left (1-a^2 x^2\right )^{3/2}}{35 a^2}-\frac{29 c^4 x \sqrt{1-a^2 x^2}}{128 a^3}-\frac{c^4 (2432-3045 a x) \left (1-a^2 x^2\right )^{3/2}}{6720 a^4}-\frac{29 c^4 \sin ^{-1}(a x)}{128 a^4} \]

[Out]

(-29*c^4*x*Sqrt[1 - a^2*x^2])/(128*a^3) - (19*c^4*x^2*(1 - a^2*x^2)^(3/2))/(35*a^2) + (29*c^4*x^3*(1 - a^2*x^2
)^(3/2))/(48*a) - (3*c^4*x^4*(1 - a^2*x^2)^(3/2))/7 + (a*c^4*x^5*(1 - a^2*x^2)^(3/2))/8 - (c^4*(2432 - 3045*a*
x)*(1 - a^2*x^2)^(3/2))/(6720*a^4) - (29*c^4*ArcSin[a*x])/(128*a^4)

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Rubi [A]  time = 0.333121, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{1}{8} a c^4 x^5 \left (1-a^2 x^2\right )^{3/2}-\frac{3}{7} c^4 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac{29 c^4 x^3 \left (1-a^2 x^2\right )^{3/2}}{48 a}-\frac{19 c^4 x^2 \left (1-a^2 x^2\right )^{3/2}}{35 a^2}-\frac{29 c^4 x \sqrt{1-a^2 x^2}}{128 a^3}-\frac{c^4 (2432-3045 a x) \left (1-a^2 x^2\right )^{3/2}}{6720 a^4}-\frac{29 c^4 \sin ^{-1}(a x)}{128 a^4} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]*x^3*(c - a*c*x)^4,x]

[Out]

(-29*c^4*x*Sqrt[1 - a^2*x^2])/(128*a^3) - (19*c^4*x^2*(1 - a^2*x^2)^(3/2))/(35*a^2) + (29*c^4*x^3*(1 - a^2*x^2
)^(3/2))/(48*a) - (3*c^4*x^4*(1 - a^2*x^2)^(3/2))/7 + (a*c^4*x^5*(1 - a^2*x^2)^(3/2))/8 - (c^4*(2432 - 3045*a*
x)*(1 - a^2*x^2)^(3/2))/(6720*a^4) - (29*c^4*ArcSin[a*x])/(128*a^4)

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^3 (c-a c x)^4 \, dx &=c \int x^3 (c-a c x)^3 \sqrt{1-a^2 x^2} \, dx\\ &=\frac{1}{8} a c^4 x^5 \left (1-a^2 x^2\right )^{3/2}-\frac{c \int x^3 \sqrt{1-a^2 x^2} \left (-8 a^2 c^3+29 a^3 c^3 x-24 a^4 c^3 x^2\right ) \, dx}{8 a^2}\\ &=-\frac{3}{7} c^4 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac{1}{8} a c^4 x^5 \left (1-a^2 x^2\right )^{3/2}+\frac{c \int x^3 \left (152 a^4 c^3-203 a^5 c^3 x\right ) \sqrt{1-a^2 x^2} \, dx}{56 a^4}\\ &=\frac{29 c^4 x^3 \left (1-a^2 x^2\right )^{3/2}}{48 a}-\frac{3}{7} c^4 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac{1}{8} a c^4 x^5 \left (1-a^2 x^2\right )^{3/2}-\frac{c \int x^2 \left (609 a^5 c^3-912 a^6 c^3 x\right ) \sqrt{1-a^2 x^2} \, dx}{336 a^6}\\ &=-\frac{19 c^4 x^2 \left (1-a^2 x^2\right )^{3/2}}{35 a^2}+\frac{29 c^4 x^3 \left (1-a^2 x^2\right )^{3/2}}{48 a}-\frac{3}{7} c^4 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac{1}{8} a c^4 x^5 \left (1-a^2 x^2\right )^{3/2}+\frac{c \int x \left (1824 a^6 c^3-3045 a^7 c^3 x\right ) \sqrt{1-a^2 x^2} \, dx}{1680 a^8}\\ &=-\frac{19 c^4 x^2 \left (1-a^2 x^2\right )^{3/2}}{35 a^2}+\frac{29 c^4 x^3 \left (1-a^2 x^2\right )^{3/2}}{48 a}-\frac{3}{7} c^4 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac{1}{8} a c^4 x^5 \left (1-a^2 x^2\right )^{3/2}-\frac{c^4 (2432-3045 a x) \left (1-a^2 x^2\right )^{3/2}}{6720 a^4}-\frac{\left (29 c^4\right ) \int \sqrt{1-a^2 x^2} \, dx}{64 a^3}\\ &=-\frac{29 c^4 x \sqrt{1-a^2 x^2}}{128 a^3}-\frac{19 c^4 x^2 \left (1-a^2 x^2\right )^{3/2}}{35 a^2}+\frac{29 c^4 x^3 \left (1-a^2 x^2\right )^{3/2}}{48 a}-\frac{3}{7} c^4 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac{1}{8} a c^4 x^5 \left (1-a^2 x^2\right )^{3/2}-\frac{c^4 (2432-3045 a x) \left (1-a^2 x^2\right )^{3/2}}{6720 a^4}-\frac{\left (29 c^4\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{128 a^3}\\ &=-\frac{29 c^4 x \sqrt{1-a^2 x^2}}{128 a^3}-\frac{19 c^4 x^2 \left (1-a^2 x^2\right )^{3/2}}{35 a^2}+\frac{29 c^4 x^3 \left (1-a^2 x^2\right )^{3/2}}{48 a}-\frac{3}{7} c^4 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac{1}{8} a c^4 x^5 \left (1-a^2 x^2\right )^{3/2}-\frac{c^4 (2432-3045 a x) \left (1-a^2 x^2\right )^{3/2}}{6720 a^4}-\frac{29 c^4 \sin ^{-1}(a x)}{128 a^4}\\ \end{align*}

Mathematica [A]  time = 0.221875, size = 99, normalized size = 0.57 \[ -\frac{c^4 \left (\sqrt{1-a^2 x^2} \left (1680 a^7 x^7-5760 a^6 x^6+6440 a^5 x^5-1536 a^4 x^4-2030 a^3 x^3+2432 a^2 x^2-3045 a x+4864\right )-6090 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{13440 a^4} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcTanh[a*x]*x^3*(c - a*c*x)^4,x]

[Out]

-(c^4*(Sqrt[1 - a^2*x^2]*(4864 - 3045*a*x + 2432*a^2*x^2 - 2030*a^3*x^3 - 1536*a^4*x^4 + 6440*a^5*x^5 - 5760*a
^6*x^6 + 1680*a^7*x^7) - 6090*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(13440*a^4)

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Maple [A]  time = 0.071, size = 209, normalized size = 1.2 \begin{align*} -{\frac{{c}^{4}{a}^{3}{x}^{7}}{8}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{23\,{c}^{4}a{x}^{5}}{48}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{29\,{c}^{4}{x}^{3}}{192\,a}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{29\,{c}^{4}x}{128\,{a}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{29\,{c}^{4}}{128\,{a}^{3}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{3\,{c}^{4}{a}^{2}{x}^{6}}{7}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{4\,{c}^{4}{x}^{4}}{35}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{19\,{c}^{4}{x}^{2}}{105\,{a}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{38\,{c}^{4}}{105\,{a}^{4}}\sqrt{-{a}^{2}{x}^{2}+1}} \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^3*(-a*c*x+c)^4,x)

[Out]

-1/8*c^4*a^3*x^7*(-a^2*x^2+1)^(1/2)-23/48*c^4*a*x^5*(-a^2*x^2+1)^(1/2)+29/192*c^4/a*x^3*(-a^2*x^2+1)^(1/2)+29/
128*c^4*x*(-a^2*x^2+1)^(1/2)/a^3-29/128*c^4/a^3/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+3/7*c^4*a
^2*x^6*(-a^2*x^2+1)^(1/2)+4/35*c^4*x^4*(-a^2*x^2+1)^(1/2)-19/105*c^4*x^2/a^2*(-a^2*x^2+1)^(1/2)-38/105*c^4/a^4
*(-a^2*x^2+1)^(1/2)

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Maxima [A]  time = 1.44966, size = 269, normalized size = 1.55 \begin{align*} -\frac{1}{8} \, \sqrt{-a^{2} x^{2} + 1} a^{3} c^{4} x^{7} + \frac{3}{7} \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{4} x^{6} - \frac{23}{48} \, \sqrt{-a^{2} x^{2} + 1} a c^{4} x^{5} + \frac{4}{35} \, \sqrt{-a^{2} x^{2} + 1} c^{4} x^{4} + \frac{29 \, \sqrt{-a^{2} x^{2} + 1} c^{4} x^{3}}{192 \, a} - \frac{19 \, \sqrt{-a^{2} x^{2} + 1} c^{4} x^{2}}{105 \, a^{2}} + \frac{29 \, \sqrt{-a^{2} x^{2} + 1} c^{4} x}{128 \, a^{3}} - \frac{29 \, c^{4} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{128 \, \sqrt{a^{2}} a^{3}} - \frac{38 \, \sqrt{-a^{2} x^{2} + 1} c^{4}}{105 \, a^{4}} \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^3*(-a*c*x+c)^4,x, algorithm="maxima")

[Out]

-1/8*sqrt(-a^2*x^2 + 1)*a^3*c^4*x^7 + 3/7*sqrt(-a^2*x^2 + 1)*a^2*c^4*x^6 - 23/48*sqrt(-a^2*x^2 + 1)*a*c^4*x^5
+ 4/35*sqrt(-a^2*x^2 + 1)*c^4*x^4 + 29/192*sqrt(-a^2*x^2 + 1)*c^4*x^3/a - 19/105*sqrt(-a^2*x^2 + 1)*c^4*x^2/a^
2 + 29/128*sqrt(-a^2*x^2 + 1)*c^4*x/a^3 - 29/128*c^4*arcsin(a^2*x/sqrt(a^2))/(sqrt(a^2)*a^3) - 38/105*sqrt(-a^
2*x^2 + 1)*c^4/a^4

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Fricas [A]  time = 1.58328, size = 302, normalized size = 1.75 \begin{align*} \frac{6090 \, c^{4} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) -{\left (1680 \, a^{7} c^{4} x^{7} - 5760 \, a^{6} c^{4} x^{6} + 6440 \, a^{5} c^{4} x^{5} - 1536 \, a^{4} c^{4} x^{4} - 2030 \, a^{3} c^{4} x^{3} + 2432 \, a^{2} c^{4} x^{2} - 3045 \, a c^{4} x + 4864 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1}}{13440 \, a^{4}} \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^3*(-a*c*x+c)^4,x, algorithm="fricas")

[Out]

1/13440*(6090*c^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (1680*a^7*c^4*x^7 - 5760*a^6*c^4*x^6 + 6440*a^5*c^4
*x^5 - 1536*a^4*c^4*x^4 - 2030*a^3*c^4*x^3 + 2432*a^2*c^4*x^2 - 3045*a*c^4*x + 4864*c^4)*sqrt(-a^2*x^2 + 1))/a
^4

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Sympy [A]  time = 23.7803, size = 842, normalized size = 4.87 \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**3*(-a*c*x+c)**4,x)

[Out]

a**5*c**4*Piecewise((-I*x**9/(8*sqrt(a**2*x**2 - 1)) - I*x**7/(48*a**2*sqrt(a**2*x**2 - 1)) - 7*I*x**5/(192*a*
*4*sqrt(a**2*x**2 - 1)) - 35*I*x**3/(384*a**6*sqrt(a**2*x**2 - 1)) + 35*I*x/(128*a**8*sqrt(a**2*x**2 - 1)) - 3
5*I*acosh(a*x)/(128*a**9), Abs(a**2*x**2) > 1), (x**9/(8*sqrt(-a**2*x**2 + 1)) + x**7/(48*a**2*sqrt(-a**2*x**2
+ 1)) + 7*x**5/(192*a**4*sqrt(-a**2*x**2 + 1)) + 35*x**3/(384*a**6*sqrt(-a**2*x**2 + 1)) - 35*x/(128*a**8*sqr
t(-a**2*x**2 + 1)) + 35*asin(a*x)/(128*a**9), True)) - 3*a**4*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)) + 2*a**3*c**4*Piecewise((-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*a
cosh(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*sqrt(-a**2*x**2 + 1)) + 5*asin(a*x)/(16*a**7), True))
+ 2*a**2*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)) - 3*a*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*sqrt(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)) + c**4*Piecewise((-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))

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Giac [A]  time = 1.19732, size = 158, normalized size = 0.91 \begin{align*} -\frac{29 \, c^{4} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{128 \, a^{3}{\left | a \right |}} - \frac{1}{13440} \, \sqrt{-a^{2} x^{2} + 1}{\left ({\left (2 \,{\left (\frac{1216 \, c^{4}}{a^{2}} -{\left (\frac{1015 \, c^{4}}{a} + 4 \,{\left (192 \, c^{4} - 5 \,{\left (161 \, a c^{4} + 6 \,{\left (7 \, a^{3} c^{4} x - 24 \, a^{2} c^{4}\right )} x\right )} x\right )} x\right )} x\right )} x - \frac{3045 \, c^{4}}{a^{3}}\right )} x + \frac{4864 \, c^{4}}{a^{4}}\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^3*(-a*c*x+c)^4,x, algorithm="giac")

[Out]

-29/128*c^4*arcsin(a*x)*sgn(a)/(a^3*abs(a)) - 1/13440*sqrt(-a^2*x^2 + 1)*((2*(1216*c^4/a^2 - (1015*c^4/a + 4*(
192*c^4 - 5*(161*a*c^4 + 6*(7*a^3*c^4*x - 24*a^2*c^4)*x)*x)*x)*x)*x - 3045*c^4/a^3)*x + 4864*c^4/a^4)