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

Optimal. Leaf size=148 \[ -\frac{1}{7} c^3 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac{c^3 x^3 \left (1-a^2 x^2\right )^{3/2}}{3 a}-\frac{11 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}{35 a^2}-\frac{c^3 x \sqrt{1-a^2 x^2}}{8 a^3}-\frac{c^3 (88-105 a x) \left (1-a^2 x^2\right )^{3/2}}{420 a^4}-\frac{c^3 \sin ^{-1}(a x)}{8 a^4} \]

[Out]

-(c^3*x*Sqrt[1 - a^2*x^2])/(8*a^3) - (11*c^3*x^2*(1 - a^2*x^2)^(3/2))/(35*a^2) + (c^3*x^3*(1 - a^2*x^2)^(3/2))
/(3*a) - (c^3*x^4*(1 - a^2*x^2)^(3/2))/7 - (c^3*(88 - 105*a*x)*(1 - a^2*x^2)^(3/2))/(420*a^4) - (c^3*ArcSin[a*
x])/(8*a^4)

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Rubi [A]  time = 0.237505, antiderivative size = 148, 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} \[ -\frac{1}{7} c^3 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac{c^3 x^3 \left (1-a^2 x^2\right )^{3/2}}{3 a}-\frac{11 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}{35 a^2}-\frac{c^3 x \sqrt{1-a^2 x^2}}{8 a^3}-\frac{c^3 (88-105 a x) \left (1-a^2 x^2\right )^{3/2}}{420 a^4}-\frac{c^3 \sin ^{-1}(a x)}{8 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^3*x*Sqrt[1 - a^2*x^2])/(8*a^3) - (11*c^3*x^2*(1 - a^2*x^2)^(3/2))/(35*a^2) + (c^3*x^3*(1 - a^2*x^2)^(3/2))
/(3*a) - (c^3*x^4*(1 - a^2*x^2)^(3/2))/7 - (c^3*(88 - 105*a*x)*(1 - a^2*x^2)^(3/2))/(420*a^4) - (c^3*ArcSin[a*
x])/(8*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)^3 \, dx &=c \int x^3 (c-a c x)^2 \sqrt{1-a^2 x^2} \, dx\\ &=-\frac{1}{7} c^3 x^4 \left (1-a^2 x^2\right )^{3/2}-\frac{c \int x^3 \left (-11 a^2 c^2+14 a^3 c^2 x\right ) \sqrt{1-a^2 x^2} \, dx}{7 a^2}\\ &=\frac{c^3 x^3 \left (1-a^2 x^2\right )^{3/2}}{3 a}-\frac{1}{7} c^3 x^4 \left (1-a^2 x^2\right )^{3/2}+\frac{c \int x^2 \left (-42 a^3 c^2+66 a^4 c^2 x\right ) \sqrt{1-a^2 x^2} \, dx}{42 a^4}\\ &=-\frac{11 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}{35 a^2}+\frac{c^3 x^3 \left (1-a^2 x^2\right )^{3/2}}{3 a}-\frac{1}{7} c^3 x^4 \left (1-a^2 x^2\right )^{3/2}-\frac{c \int x \left (-132 a^4 c^2+210 a^5 c^2 x\right ) \sqrt{1-a^2 x^2} \, dx}{210 a^6}\\ &=-\frac{11 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}{35 a^2}+\frac{c^3 x^3 \left (1-a^2 x^2\right )^{3/2}}{3 a}-\frac{1}{7} c^3 x^4 \left (1-a^2 x^2\right )^{3/2}-\frac{c^3 (88-105 a x) \left (1-a^2 x^2\right )^{3/2}}{420 a^4}-\frac{c^3 \int \sqrt{1-a^2 x^2} \, dx}{4 a^3}\\ &=-\frac{c^3 x \sqrt{1-a^2 x^2}}{8 a^3}-\frac{11 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}{35 a^2}+\frac{c^3 x^3 \left (1-a^2 x^2\right )^{3/2}}{3 a}-\frac{1}{7} c^3 x^4 \left (1-a^2 x^2\right )^{3/2}-\frac{c^3 (88-105 a x) \left (1-a^2 x^2\right )^{3/2}}{420 a^4}-\frac{c^3 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{8 a^3}\\ &=-\frac{c^3 x \sqrt{1-a^2 x^2}}{8 a^3}-\frac{11 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}{35 a^2}+\frac{c^3 x^3 \left (1-a^2 x^2\right )^{3/2}}{3 a}-\frac{1}{7} c^3 x^4 \left (1-a^2 x^2\right )^{3/2}-\frac{c^3 (88-105 a x) \left (1-a^2 x^2\right )^{3/2}}{420 a^4}-\frac{c^3 \sin ^{-1}(a x)}{8 a^4}\\ \end{align*}

Mathematica [A]  time = 0.123037, size = 91, normalized size = 0.61 \[ \frac{c^3 \left (\sqrt{1-a^2 x^2} \left (120 a^6 x^6-280 a^5 x^5+144 a^4 x^4+70 a^3 x^3-88 a^2 x^2+105 a x-176\right )+210 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{840 a^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*(Sqrt[1 - a^2*x^2]*(-176 + 105*a*x - 88*a^2*x^2 + 70*a^3*x^3 + 144*a^4*x^4 - 280*a^5*x^5 + 120*a^6*x^6) +
210*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(840*a^4)

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Maple [A]  time = 0.055, size = 186, normalized size = 1.3 \begin{align*}{\frac{{a}^{2}{c}^{3}{x}^{6}}{7}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{6\,{c}^{3}{x}^{4}}{35}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{11\,{c}^{3}{x}^{2}}{105\,{a}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{22\,{c}^{3}}{105\,{a}^{4}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{a{c}^{3}{x}^{5}}{3}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{{c}^{3}{x}^{3}}{12\,a}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{{c}^{3}x}{8\,{a}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{{c}^{3}}{8\,{a}^{3}}\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^3*(-a*c*x+c)^3,x)

[Out]

1/7*c^3*a^2*x^6*(-a^2*x^2+1)^(1/2)+6/35*c^3*x^4*(-a^2*x^2+1)^(1/2)-11/105*c^3*x^2/a^2*(-a^2*x^2+1)^(1/2)-22/10
5*c^3/a^4*(-a^2*x^2+1)^(1/2)-1/3*c^3*a*x^5*(-a^2*x^2+1)^(1/2)+1/12*c^3/a*x^3*(-a^2*x^2+1)^(1/2)+1/8*c^3*x*(-a^
2*x^2+1)^(1/2)/a^3-1/8*c^3/a^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 = 1.44087, size = 238, normalized size = 1.61 \begin{align*} \frac{1}{7} \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{3} x^{6} - \frac{1}{3} \, \sqrt{-a^{2} x^{2} + 1} a c^{3} x^{5} + \frac{6}{35} \, \sqrt{-a^{2} x^{2} + 1} c^{3} x^{4} + \frac{\sqrt{-a^{2} x^{2} + 1} c^{3} x^{3}}{12 \, a} - \frac{11 \, \sqrt{-a^{2} x^{2} + 1} c^{3} x^{2}}{105 \, a^{2}} + \frac{\sqrt{-a^{2} x^{2} + 1} c^{3} x}{8 \, a^{3}} - \frac{c^{3} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{8 \, \sqrt{a^{2}} a^{3}} - \frac{22 \, \sqrt{-a^{2} x^{2} + 1} c^{3}}{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)^3,x, algorithm="maxima")

[Out]

1/7*sqrt(-a^2*x^2 + 1)*a^2*c^3*x^6 - 1/3*sqrt(-a^2*x^2 + 1)*a*c^3*x^5 + 6/35*sqrt(-a^2*x^2 + 1)*c^3*x^4 + 1/12
*sqrt(-a^2*x^2 + 1)*c^3*x^3/a - 11/105*sqrt(-a^2*x^2 + 1)*c^3*x^2/a^2 + 1/8*sqrt(-a^2*x^2 + 1)*c^3*x/a^3 - 1/8
*c^3*arcsin(a^2*x/sqrt(a^2))/(sqrt(a^2)*a^3) - 22/105*sqrt(-a^2*x^2 + 1)*c^3/a^4

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Fricas [A]  time = 1.67375, size = 261, normalized size = 1.76 \begin{align*} \frac{210 \, c^{3} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (120 \, a^{6} c^{3} x^{6} - 280 \, a^{5} c^{3} x^{5} + 144 \, a^{4} c^{3} x^{4} + 70 \, a^{3} c^{3} x^{3} - 88 \, a^{2} c^{3} x^{2} + 105 \, a c^{3} x - 176 \, c^{3}\right )} \sqrt{-a^{2} x^{2} + 1}}{840 \, 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)^3,x, algorithm="fricas")

[Out]

1/840*(210*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (120*a^6*c^3*x^6 - 280*a^5*c^3*x^5 + 144*a^4*c^3*x^4 +
70*a^3*c^3*x^3 - 88*a^2*c^3*x^2 + 105*a*c^3*x - 176*c^3)*sqrt(-a^2*x^2 + 1))/a^4

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Sympy [A]  time = 14.4931, size = 512, normalized size = 3.46 \begin{align*} - a^{4} c^{3} \left (\begin{cases} - \frac{x^{6} \sqrt{- a^{2} x^{2} + 1}}{7 a^{2}} - \frac{6 x^{4} \sqrt{- a^{2} x^{2} + 1}}{35 a^{4}} - \frac{8 x^{2} \sqrt{- a^{2} x^{2} + 1}}{35 a^{6}} - \frac{16 \sqrt{- a^{2} x^{2} + 1}}{35 a^{8}} & \text{for}\: a \neq 0 \\\frac{x^{8}}{8} & \text{otherwise} \end{cases}\right ) + 2 a^{3} c^{3} \left (\begin{cases} - \frac{i x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{5}}{24 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i x^{3}}{48 a^{4} \sqrt{a^{2} x^{2} - 1}} + \frac{5 i x}{16 a^{6} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \operatorname{acosh}{\left (a x \right )}}{16 a^{7}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{5}}{24 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 x^{3}}{48 a^{4} \sqrt{- a^{2} x^{2} + 1}} - \frac{5 x}{16 a^{6} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \operatorname{asin}{\left (a x \right )}}{16 a^{7}} & \text{otherwise} \end{cases}\right ) - 2 a c^{3} \left (\begin{cases} - \frac{i x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{3 i x}{8 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \operatorname{acosh}{\left (a x \right )}}{8 a^{5}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{3 x}{8 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \operatorname{asin}{\left (a x \right )}}{8 a^{5}} & \text{otherwise} \end{cases}\right ) + c^{3} \left (\begin{cases} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{3 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{3 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\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)**3,x)

[Out]

-a**4*c**3*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*sqr
t(-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**3*Piec
ewise((-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*
sqrt(-a**2*x**2 + 1)) + 5*asin(a*x)/(16*a**7), True)) - 2*a*c**3*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**3*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 = 1.2967, size = 140, normalized size = 0.95 \begin{align*} \frac{1}{840} \, \sqrt{-a^{2} x^{2} + 1}{\left ({\left (2 \,{\left ({\left (\frac{35 \, c^{3}}{a} + 4 \,{\left (18 \, c^{3} + 5 \,{\left (3 \, a^{2} c^{3} x - 7 \, a c^{3}\right )} x\right )} x\right )} x - \frac{44 \, c^{3}}{a^{2}}\right )} x + \frac{105 \, c^{3}}{a^{3}}\right )} x - \frac{176 \, c^{3}}{a^{4}}\right )} - \frac{c^{3} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{8 \, a^{3}{\left | a \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)^3,x, algorithm="giac")

[Out]

1/840*sqrt(-a^2*x^2 + 1)*((2*((35*c^3/a + 4*(18*c^3 + 5*(3*a^2*c^3*x - 7*a*c^3)*x)*x)*x - 44*c^3/a^2)*x + 105*
c^3/a^3)*x - 176*c^3/a^4) - 1/8*c^3*arcsin(a*x)*sgn(a)/(a^3*abs(a))