3.1138 \(\int e^{3 \tanh ^{-1}(a x)} x^2 (c-a^2 c x^2) \, dx\)

Optimal. Leaf size=111 \[ -\frac{1}{5} a c x^4 \sqrt{1-a^2 x^2}-\frac{3}{4} c x^3 \sqrt{1-a^2 x^2}-\frac{19 c x^2 \sqrt{1-a^2 x^2}}{15 a}-\frac{c (195 a x+304) \sqrt{1-a^2 x^2}}{120 a^3}+\frac{13 c \sin ^{-1}(a x)}{8 a^3} \]

[Out]

(-19*c*x^2*Sqrt[1 - a^2*x^2])/(15*a) - (3*c*x^3*Sqrt[1 - a^2*x^2])/4 - (a*c*x^4*Sqrt[1 - a^2*x^2])/5 - (c*(304
+ 195*a*x)*Sqrt[1 - a^2*x^2])/(120*a^3) + (13*c*ArcSin[a*x])/(8*a^3)

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Rubi [A]  time = 0.24661, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {6148, 1809, 833, 780, 216} \[ -\frac{1}{5} a c x^4 \sqrt{1-a^2 x^2}-\frac{3}{4} c x^3 \sqrt{1-a^2 x^2}-\frac{19 c x^2 \sqrt{1-a^2 x^2}}{15 a}-\frac{c (195 a x+304) \sqrt{1-a^2 x^2}}{120 a^3}+\frac{13 c \sin ^{-1}(a x)}{8 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-19*c*x^2*Sqrt[1 - a^2*x^2])/(15*a) - (3*c*x^3*Sqrt[1 - a^2*x^2])/4 - (a*c*x^4*Sqrt[1 - a^2*x^2])/5 - (c*(304
+ 195*a*x)*Sqrt[1 - a^2*x^2])/(120*a^3) + (13*c*ArcSin[a*x])/(8*a^3)

Rule 6148

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

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 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)} x^2 \left (c-a^2 c x^2\right ) \, dx &=c \int \frac{x^2 (1+a x)^3}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{1}{5} a c x^4 \sqrt{1-a^2 x^2}-\frac{c \int \frac{x^2 \left (-5 a^2-19 a^3 x-15 a^4 x^2\right )}{\sqrt{1-a^2 x^2}} \, dx}{5 a^2}\\ &=-\frac{3}{4} c x^3 \sqrt{1-a^2 x^2}-\frac{1}{5} a c x^4 \sqrt{1-a^2 x^2}+\frac{c \int \frac{x^2 \left (65 a^4+76 a^5 x\right )}{\sqrt{1-a^2 x^2}} \, dx}{20 a^4}\\ &=-\frac{19 c x^2 \sqrt{1-a^2 x^2}}{15 a}-\frac{3}{4} c x^3 \sqrt{1-a^2 x^2}-\frac{1}{5} a c x^4 \sqrt{1-a^2 x^2}-\frac{c \int \frac{x \left (-152 a^5-195 a^6 x\right )}{\sqrt{1-a^2 x^2}} \, dx}{60 a^6}\\ &=-\frac{19 c x^2 \sqrt{1-a^2 x^2}}{15 a}-\frac{3}{4} c x^3 \sqrt{1-a^2 x^2}-\frac{1}{5} a c x^4 \sqrt{1-a^2 x^2}-\frac{c (304+195 a x) \sqrt{1-a^2 x^2}}{120 a^3}+\frac{(13 c) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{8 a^2}\\ &=-\frac{19 c x^2 \sqrt{1-a^2 x^2}}{15 a}-\frac{3}{4} c x^3 \sqrt{1-a^2 x^2}-\frac{1}{5} a c x^4 \sqrt{1-a^2 x^2}-\frac{c (304+195 a x) \sqrt{1-a^2 x^2}}{120 a^3}+\frac{13 c \sin ^{-1}(a x)}{8 a^3}\\ \end{align*}

Mathematica [A]  time = 0.0861077, size = 62, normalized size = 0.56 \[ \frac{195 c \sin ^{-1}(a x)-c \sqrt{1-a^2 x^2} \left (24 a^4 x^4+90 a^3 x^3+152 a^2 x^2+195 a x+304\right )}{120 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(c*Sqrt[1 - a^2*x^2]*(304 + 195*a*x + 152*a^2*x^2 + 90*a^3*x^3 + 24*a^4*x^4)) + 195*c*ArcSin[a*x])/(120*a^3)

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Maple [A]  time = 0.065, size = 170, normalized size = 1.5 \begin{align*}{\frac{{a}^{3}c{x}^{6}}{5}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{16\,ac{x}^{4}}{15}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{19\,c{x}^{2}}{15\,a}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{38\,c}{15\,{a}^{3}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{3\,{a}^{2}c{x}^{5}}{4}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{7\,c{x}^{3}}{8}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{13\,cx}{8\,{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{13\,c}{8\,{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)^3/(-a^2*x^2+1)^(3/2)*x^2*(-a^2*c*x^2+c),x)

[Out]

1/5*c*a^3*x^6/(-a^2*x^2+1)^(1/2)+16/15*c*a*x^4/(-a^2*x^2+1)^(1/2)+19/15*c/a*x^2/(-a^2*x^2+1)^(1/2)-38/15*c/a^3
/(-a^2*x^2+1)^(1/2)+3/4*c*a^2*x^5/(-a^2*x^2+1)^(1/2)+7/8*c*x^3/(-a^2*x^2+1)^(1/2)-13/8*c*x/a^2/(-a^2*x^2+1)^(1
/2)+13/8*c/a^2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))

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Maxima [A]  time = 1.45622, size = 216, normalized size = 1.95 \begin{align*} \frac{a^{3} c x^{6}}{5 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{3 \, a^{2} c x^{5}}{4 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{16 \, a c x^{4}}{15 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{7 \, c x^{3}}{8 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{19 \, c x^{2}}{15 \, \sqrt{-a^{2} x^{2} + 1} a} - \frac{13 \, c x}{8 \, \sqrt{-a^{2} x^{2} + 1} a^{2}} + \frac{13 \, c \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{8 \, \sqrt{a^{2}} a^{2}} - \frac{38 \, c}{15 \, \sqrt{-a^{2} x^{2} + 1} a^{3}} \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)*x^2*(-a^2*c*x^2+c),x, algorithm="maxima")

[Out]

1/5*a^3*c*x^6/sqrt(-a^2*x^2 + 1) + 3/4*a^2*c*x^5/sqrt(-a^2*x^2 + 1) + 16/15*a*c*x^4/sqrt(-a^2*x^2 + 1) + 7/8*c
*x^3/sqrt(-a^2*x^2 + 1) + 19/15*c*x^2/(sqrt(-a^2*x^2 + 1)*a) - 13/8*c*x/(sqrt(-a^2*x^2 + 1)*a^2) + 13/8*c*arcs
in(a^2*x/sqrt(a^2))/(sqrt(a^2)*a^2) - 38/15*c/(sqrt(-a^2*x^2 + 1)*a^3)

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Fricas [A]  time = 2.67267, size = 197, normalized size = 1.77 \begin{align*} -\frac{390 \, c \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (24 \, a^{4} c x^{4} + 90 \, a^{3} c x^{3} + 152 \, a^{2} c x^{2} + 195 \, a c x + 304 \, c\right )} \sqrt{-a^{2} x^{2} + 1}}{120 \, a^{3}} \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)*x^2*(-a^2*c*x^2+c),x, algorithm="fricas")

[Out]

-1/120*(390*c*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (24*a^4*c*x^4 + 90*a^3*c*x^3 + 152*a^2*c*x^2 + 195*a*c*
x + 304*c)*sqrt(-a^2*x^2 + 1))/a^3

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Sympy [C]  time = 16.7749, size = 371, normalized size = 3.34 \begin{align*} a^{3} c \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 ) + 3 a^{2} c \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 ) + 3 a c \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 \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)**3/(-a**2*x**2+1)**(3/2)*x**2*(-a**2*c*x**2+c),x)

[Out]

a**3*c*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**2*c*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)) + 3*a*c*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*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), Tr
ue))

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Giac [A]  time = 1.177, size = 93, normalized size = 0.84 \begin{align*} -\frac{1}{120} \, \sqrt{-a^{2} x^{2} + 1}{\left ({\left (2 \,{\left (3 \,{\left (4 \, a c x + 15 \, c\right )} x + \frac{76 \, c}{a}\right )} x + \frac{195 \, c}{a^{2}}\right )} x + \frac{304 \, c}{a^{3}}\right )} + \frac{13 \, c \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{8 \, a^{2}{\left | a \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)*x^2*(-a^2*c*x^2+c),x, algorithm="giac")

[Out]

-1/120*sqrt(-a^2*x^2 + 1)*((2*(3*(4*a*c*x + 15*c)*x + 76*c/a)*x + 195*c/a^2)*x + 304*c/a^3) + 13/8*c*arcsin(a*
x)*sgn(a)/(a^2*abs(a))