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

Optimal. Leaf size=124 \[ \frac{c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac{c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a^2}-\frac{c^2 x \sqrt{1-a^2 x^2}}{16 a^3}-\frac{c^2 (16-15 a x) \left (1-a^2 x^2\right )^{3/2}}{120 a^4}-\frac{c^2 \sin ^{-1}(a x)}{16 a^4} \]

[Out]

-(c^2*x*Sqrt[1 - a^2*x^2])/(16*a^3) - (c^2*x^2*(1 - a^2*x^2)^(3/2))/(5*a^2) + (c^2*x^3*(1 - a^2*x^2)^(3/2))/(6
*a) - (c^2*(16 - 15*a*x)*(1 - a^2*x^2)^(3/2))/(120*a^4) - (c^2*ArcSin[a*x])/(16*a^4)

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Rubi [A]  time = 0.135755, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {6128, 833, 780, 195, 216} \[ \frac{c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac{c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a^2}-\frac{c^2 x \sqrt{1-a^2 x^2}}{16 a^3}-\frac{c^2 (16-15 a x) \left (1-a^2 x^2\right )^{3/2}}{120 a^4}-\frac{c^2 \sin ^{-1}(a x)}{16 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^2*x*Sqrt[1 - a^2*x^2])/(16*a^3) - (c^2*x^2*(1 - a^2*x^2)^(3/2))/(5*a^2) + (c^2*x^3*(1 - a^2*x^2)^(3/2))/(6
*a) - (c^2*(16 - 15*a*x)*(1 - a^2*x^2)^(3/2))/(120*a^4) - (c^2*ArcSin[a*x])/(16*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 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)^2 \, dx &=c \int x^3 (c-a c x) \sqrt{1-a^2 x^2} \, dx\\ &=\frac{c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac{c \int x^2 \left (3 a c-6 a^2 c x\right ) \sqrt{1-a^2 x^2} \, dx}{6 a^2}\\ &=-\frac{c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a^2}+\frac{c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}+\frac{c \int x \left (12 a^2 c-15 a^3 c x\right ) \sqrt{1-a^2 x^2} \, dx}{30 a^4}\\ &=-\frac{c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a^2}+\frac{c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac{c^2 (16-15 a x) \left (1-a^2 x^2\right )^{3/2}}{120 a^4}-\frac{c^2 \int \sqrt{1-a^2 x^2} \, dx}{8 a^3}\\ &=-\frac{c^2 x \sqrt{1-a^2 x^2}}{16 a^3}-\frac{c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a^2}+\frac{c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac{c^2 (16-15 a x) \left (1-a^2 x^2\right )^{3/2}}{120 a^4}-\frac{c^2 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{16 a^3}\\ &=-\frac{c^2 x \sqrt{1-a^2 x^2}}{16 a^3}-\frac{c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a^2}+\frac{c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac{c^2 (16-15 a x) \left (1-a^2 x^2\right )^{3/2}}{120 a^4}-\frac{c^2 \sin ^{-1}(a x)}{16 a^4}\\ \end{align*}

Mathematica [A]  time = 0.19242, size = 89, normalized size = 0.72 \[ -\frac{c^2 \left (\sqrt{1-a^2 x^2} \left (40 a^5 x^5-48 a^4 x^4-10 a^3 x^3+16 a^2 x^2-15 a x+32\right )-60 \sin ^{-1}(a x)-150 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{240 a^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(c^2*(Sqrt[1 - a^2*x^2]*(32 - 15*a*x + 16*a^2*x^2 - 10*a^3*x^3 - 48*a^4*x^4 + 40*a^5*x^5) - 60*ArcSin[a*x] -
150*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(240*a^4)

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Maple [A]  time = 0.05, size = 163, normalized size = 1.3 \begin{align*} -{\frac{a{c}^{2}{x}^{5}}{6}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{{c}^{2}{x}^{3}}{24\,a}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{x{c}^{2}}{16\,{a}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{{c}^{2}}{16\,{a}^{3}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{{c}^{2}{x}^{4}}{5}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{{c}^{2}{x}^{2}}{15\,{a}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{2\,{c}^{2}}{15\,{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)^2,x)

[Out]

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

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Maxima [A]  time = 1.43174, size = 207, normalized size = 1.67 \begin{align*} -\frac{1}{6} \, \sqrt{-a^{2} x^{2} + 1} a c^{2} x^{5} + \frac{1}{5} \, \sqrt{-a^{2} x^{2} + 1} c^{2} x^{4} + \frac{\sqrt{-a^{2} x^{2} + 1} c^{2} x^{3}}{24 \, a} - \frac{\sqrt{-a^{2} x^{2} + 1} c^{2} x^{2}}{15 \, a^{2}} + \frac{\sqrt{-a^{2} x^{2} + 1} c^{2} x}{16 \, a^{3}} - \frac{c^{2} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{16 \, \sqrt{a^{2}} a^{3}} - \frac{2 \, \sqrt{-a^{2} x^{2} + 1} c^{2}}{15 \, 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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.61169, size = 230, normalized size = 1.85 \begin{align*} \frac{30 \, c^{2} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) -{\left (40 \, a^{5} c^{2} x^{5} - 48 \, a^{4} c^{2} x^{4} - 10 \, a^{3} c^{2} x^{3} + 16 \, a^{2} c^{2} x^{2} - 15 \, a c^{2} x + 32 \, c^{2}\right )} \sqrt{-a^{2} x^{2} + 1}}{240 \, 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)^2,x, algorithm="fricas")

[Out]

1/240*(30*c^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (40*a^5*c^2*x^5 - 48*a^4*c^2*x^4 - 10*a^3*c^2*x^3 + 16*
a^2*c^2*x^2 - 15*a*c^2*x + 32*c^2)*sqrt(-a^2*x^2 + 1))/a^4

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Sympy [A]  time = 14.6914, size = 486, normalized size = 3.92 \begin{align*} a^{3} c^{2} \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 ) - a^{2} c^{2} \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 ) - a c^{2} \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^{2} \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)**2,x)

[Out]

a**3*c**2*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)) - a**2*c**2*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)) - a*c**2*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**2*Pie
cewise((-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 = 1.36787, size = 124, normalized size = 1. \begin{align*} -\frac{1}{240} \, \sqrt{-a^{2} x^{2} + 1}{\left ({\left (2 \,{\left ({\left (4 \,{\left (5 \, a c^{2} x - 6 \, c^{2}\right )} x - \frac{5 \, c^{2}}{a}\right )} x + \frac{8 \, c^{2}}{a^{2}}\right )} x - \frac{15 \, c^{2}}{a^{3}}\right )} x + \frac{32 \, c^{2}}{a^{4}}\right )} - \frac{c^{2} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{16 \, 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)^2,x, algorithm="giac")

[Out]

-1/240*sqrt(-a^2*x^2 + 1)*((2*((4*(5*a*c^2*x - 6*c^2)*x - 5*c^2/a)*x + 8*c^2/a^2)*x - 15*c^2/a^3)*x + 32*c^2/a
^4) - 1/16*c^2*arcsin(a*x)*sgn(a)/(a^3*abs(a))