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

Optimal. Leaf size=121 \[ -\frac{1}{6} c^3 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac{2 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{c^3 (32-45 a x) \left (1-a^2 x^2\right )^{3/2}}{120 a^3}+\frac{3 c^3 x \sqrt{1-a^2 x^2}}{16 a^2}+\frac{3 c^3 \sin ^{-1}(a x)}{16 a^3} \]

[Out]

(3*c^3*x*Sqrt[1 - a^2*x^2])/(16*a^2) + (2*c^3*x^2*(1 - a^2*x^2)^(3/2))/(5*a) - (c^3*x^3*(1 - a^2*x^2)^(3/2))/6
+ (c^3*(32 - 45*a*x)*(1 - a^2*x^2)^(3/2))/(120*a^3) + (3*c^3*ArcSin[a*x])/(16*a^3)

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Rubi [A]  time = 0.200639, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{1}{6} c^3 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac{2 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{c^3 (32-45 a x) \left (1-a^2 x^2\right )^{3/2}}{120 a^3}+\frac{3 c^3 x \sqrt{1-a^2 x^2}}{16 a^2}+\frac{3 c^3 \sin ^{-1}(a x)}{16 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*c^3*x*Sqrt[1 - a^2*x^2])/(16*a^2) + (2*c^3*x^2*(1 - a^2*x^2)^(3/2))/(5*a) - (c^3*x^3*(1 - a^2*x^2)^(3/2))/6
+ (c^3*(32 - 45*a*x)*(1 - a^2*x^2)^(3/2))/(120*a^3) + (3*c^3*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)^3 \, dx &=c \int x^2 (c-a c x)^2 \sqrt{1-a^2 x^2} \, dx\\ &=-\frac{1}{6} c^3 x^3 \left (1-a^2 x^2\right )^{3/2}-\frac{c \int x^2 \left (-9 a^2 c^2+12 a^3 c^2 x\right ) \sqrt{1-a^2 x^2} \, dx}{6 a^2}\\ &=\frac{2 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}-\frac{1}{6} c^3 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac{c \int x \left (-24 a^3 c^2+45 a^4 c^2 x\right ) \sqrt{1-a^2 x^2} \, dx}{30 a^4}\\ &=\frac{2 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}-\frac{1}{6} c^3 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac{c^3 (32-45 a x) \left (1-a^2 x^2\right )^{3/2}}{120 a^3}+\frac{\left (3 c^3\right ) \int \sqrt{1-a^2 x^2} \, dx}{8 a^2}\\ &=\frac{3 c^3 x \sqrt{1-a^2 x^2}}{16 a^2}+\frac{2 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}-\frac{1}{6} c^3 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac{c^3 (32-45 a x) \left (1-a^2 x^2\right )^{3/2}}{120 a^3}+\frac{\left (3 c^3\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{16 a^2}\\ &=\frac{3 c^3 x \sqrt{1-a^2 x^2}}{16 a^2}+\frac{2 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}-\frac{1}{6} c^3 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac{c^3 (32-45 a x) \left (1-a^2 x^2\right )^{3/2}}{120 a^3}+\frac{3 c^3 \sin ^{-1}(a x)}{16 a^3}\\ \end{align*}

Mathematica [A]  time = 0.0856895, size = 83, normalized size = 0.69 \[ \frac{c^3 \left (\sqrt{1-a^2 x^2} \left (40 a^5 x^5-96 a^4 x^4+50 a^3 x^3+32 a^2 x^2-45 a x+64\right )-90 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{240 a^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*(Sqrt[1 - a^2*x^2]*(64 - 45*a*x + 32*a^2*x^2 + 50*a^3*x^3 - 96*a^4*x^4 + 40*a^5*x^5) - 90*ArcSin[Sqrt[1 -
a*x]/Sqrt[2]]))/(240*a^3)

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Maple [A]  time = 0.046, size = 163, normalized size = 1.4 \begin{align*}{\frac{{c}^{3}{a}^{2}{x}^{5}}{6}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{5\,{c}^{3}{x}^{3}}{24}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{3\,{c}^{3}x}{16\,{a}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{3\,{c}^{3}}{16\,{a}^{2}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{\frac{2\,{c}^{3}a{x}^{4}}{5}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{2\,{c}^{3}{x}^{2}}{15\,a}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{4\,{c}^{3}}{15\,{a}^{3}}\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^2*(-a*c*x+c)^3,x)

[Out]

1/6*c^3*a^2*x^5*(-a^2*x^2+1)^(1/2)+5/24*c^3*x^3*(-a^2*x^2+1)^(1/2)-3/16*c^3*x*(-a^2*x^2+1)^(1/2)/a^2+3/16*c^3/
a^2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-2/5*c^3*a*x^4*(-a^2*x^2+1)^(1/2)+2/15*c^3/a*x^2*(-a^2
*x^2+1)^(1/2)+4/15*c^3/a^3*(-a^2*x^2+1)^(1/2)

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Maxima [A]  time = 1.4305, size = 207, normalized size = 1.71 \begin{align*} \frac{1}{6} \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{3} x^{5} - \frac{2}{5} \, \sqrt{-a^{2} x^{2} + 1} a c^{3} x^{4} + \frac{5}{24} \, \sqrt{-a^{2} x^{2} + 1} c^{3} x^{3} + \frac{2 \, \sqrt{-a^{2} x^{2} + 1} c^{3} x^{2}}{15 \, a} - \frac{3 \, \sqrt{-a^{2} x^{2} + 1} c^{3} x}{16 \, a^{2}} + \frac{3 \, c^{3} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{16 \, \sqrt{a^{2}} a^{2}} + \frac{4 \, \sqrt{-a^{2} x^{2} + 1} c^{3}}{15 \, 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)^3,x, algorithm="maxima")

[Out]

1/6*sqrt(-a^2*x^2 + 1)*a^2*c^3*x^5 - 2/5*sqrt(-a^2*x^2 + 1)*a*c^3*x^4 + 5/24*sqrt(-a^2*x^2 + 1)*c^3*x^3 + 2/15
*sqrt(-a^2*x^2 + 1)*c^3*x^2/a - 3/16*sqrt(-a^2*x^2 + 1)*c^3*x/a^2 + 3/16*c^3*arcsin(a^2*x/sqrt(a^2))/(sqrt(a^2
)*a^2) + 4/15*sqrt(-a^2*x^2 + 1)*c^3/a^3

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Fricas [A]  time = 1.64863, size = 231, normalized size = 1.91 \begin{align*} -\frac{90 \, c^{3} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) -{\left (40 \, a^{5} c^{3} x^{5} - 96 \, a^{4} c^{3} x^{4} + 50 \, a^{3} c^{3} x^{3} + 32 \, a^{2} c^{3} x^{2} - 45 \, a c^{3} x + 64 \, c^{3}\right )} \sqrt{-a^{2} x^{2} + 1}}{240 \, 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)^3,x, algorithm="fricas")

[Out]

-1/240*(90*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (40*a^5*c^3*x^5 - 96*a^4*c^3*x^4 + 50*a^3*c^3*x^3 + 32
*a^2*c^3*x^2 - 45*a*c^3*x + 64*c^3)*sqrt(-a^2*x^2 + 1))/a^3

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Sympy [C]  time = 12.1477, size = 423, normalized size = 3.5 \begin{align*} - a^{4} 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^{3} c^{3} \left (\begin{cases} - \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{5 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{4}} - \frac{8 \sqrt{- a^{2} x^{2} + 1}}{15 a^{6}} & \text{for}\: a \neq 0 \\\frac{x^{6}}{6} & \text{otherwise} \end{cases}\right ) - 2 a 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 ) + c^{3} \left (\begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left (a x \right )}}{2 a^{3}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left (a x \right )}}{2 a^{3}} & \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**2*(-a*c*x+c)**3,x)

[Out]

-a**4*c**3*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*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**3*c**3*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*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)) + c**3*Piecewise((-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))

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Giac [A]  time = 1.19496, size = 124, normalized size = 1.02 \begin{align*} \frac{3 \, c^{3} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{16 \, a^{2}{\left | a \right |}} + \frac{1}{240} \, \sqrt{-a^{2} x^{2} + 1}{\left ({\left (2 \,{\left (\frac{16 \, c^{3}}{a} +{\left (25 \, c^{3} + 4 \,{\left (5 \, a^{2} c^{3} x - 12 \, a c^{3}\right )} x\right )} x\right )} x - \frac{45 \, c^{3}}{a^{2}}\right )} x + \frac{64 \, c^{3}}{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)^3,x, algorithm="giac")

[Out]

3/16*c^3*arcsin(a*x)*sgn(a)/(a^2*abs(a)) + 1/240*sqrt(-a^2*x^2 + 1)*((2*(16*c^3/a + (25*c^3 + 4*(5*a^2*c^3*x -
12*a*c^3)*x)*x)*x - 45*c^3/a^2)*x + 64*c^3/a^3)