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

Optimal. Leaf size=70 \[ -\frac{c^2 (4-3 a x) \left (1-a^2 x^2\right )^{3/2}}{12 a^2}-\frac{c^2 x \sqrt{1-a^2 x^2}}{8 a}-\frac{c^2 \sin ^{-1}(a x)}{8 a^2} \]

[Out]

-(c^2*x*Sqrt[1 - a^2*x^2])/(8*a) - (c^2*(4 - 3*a*x)*(1 - a^2*x^2)^(3/2))/(12*a^2) - (c^2*ArcSin[a*x])/(8*a^2)

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Rubi [A]  time = 0.0620288, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {6128, 780, 195, 216} \[ -\frac{c^2 (4-3 a x) \left (1-a^2 x^2\right )^{3/2}}{12 a^2}-\frac{c^2 x \sqrt{1-a^2 x^2}}{8 a}-\frac{c^2 \sin ^{-1}(a x)}{8 a^2} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]*x*(c - a*c*x)^2,x]

[Out]

-(c^2*x*Sqrt[1 - a^2*x^2])/(8*a) - (c^2*(4 - 3*a*x)*(1 - a^2*x^2)^(3/2))/(12*a^2) - (c^2*ArcSin[a*x])/(8*a^2)

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 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 (c-a c x)^2 \, dx &=c \int x (c-a c x) \sqrt{1-a^2 x^2} \, dx\\ &=-\frac{c^2 (4-3 a x) \left (1-a^2 x^2\right )^{3/2}}{12 a^2}-\frac{c^2 \int \sqrt{1-a^2 x^2} \, dx}{4 a}\\ &=-\frac{c^2 x \sqrt{1-a^2 x^2}}{8 a}-\frac{c^2 (4-3 a x) \left (1-a^2 x^2\right )^{3/2}}{12 a^2}-\frac{c^2 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{8 a}\\ &=-\frac{c^2 x \sqrt{1-a^2 x^2}}{8 a}-\frac{c^2 (4-3 a x) \left (1-a^2 x^2\right )^{3/2}}{12 a^2}-\frac{c^2 \sin ^{-1}(a x)}{8 a^2}\\ \end{align*}

Mathematica [A]  time = 0.0962106, size = 67, normalized size = 0.96 \[ -\frac{c^2 \left (\sqrt{1-a^2 x^2} \left (6 a^3 x^3-8 a^2 x^2-3 a x+8\right )-6 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{24 a^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(c^2*(Sqrt[1 - a^2*x^2]*(8 - 3*a*x - 8*a^2*x^2 + 6*a^3*x^3) - 6*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(24*a^2)

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Maple [A]  time = 0.041, size = 117, normalized size = 1.7 \begin{align*} -{\frac{a{c}^{2}{x}^{3}}{4}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{x{c}^{2}}{8\,a}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{{c}^{2}}{8\,a}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{{c}^{2}{x}^{2}}{3}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{{c}^{2}}{3\,{a}^{2}}\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*(-a*c*x+c)^2,x)

[Out]

-1/4*c^2*a*x^3*(-a^2*x^2+1)^(1/2)+1/8*c^2*x*(-a^2*x^2+1)^(1/2)/a-1/8*c^2/a/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-
a^2*x^2+1)^(1/2))+1/3*c^2*x^2*(-a^2*x^2+1)^(1/2)-1/3*c^2/a^2*(-a^2*x^2+1)^(1/2)

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Maxima [A]  time = 1.43219, size = 144, normalized size = 2.06 \begin{align*} -\frac{1}{4} \, \sqrt{-a^{2} x^{2} + 1} a c^{2} x^{3} + \frac{1}{3} \, \sqrt{-a^{2} x^{2} + 1} c^{2} x^{2} + \frac{\sqrt{-a^{2} x^{2} + 1} c^{2} x}{8 \, a} - \frac{c^{2} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{8 \, \sqrt{a^{2}} a} - \frac{\sqrt{-a^{2} x^{2} + 1} c^{2}}{3 \, a^{2}} \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*(-a*c*x+c)^2,x, algorithm="maxima")

[Out]

-1/4*sqrt(-a^2*x^2 + 1)*a*c^2*x^3 + 1/3*sqrt(-a^2*x^2 + 1)*c^2*x^2 + 1/8*sqrt(-a^2*x^2 + 1)*c^2*x/a - 1/8*c^2*
arcsin(a^2*x/sqrt(a^2))/(sqrt(a^2)*a) - 1/3*sqrt(-a^2*x^2 + 1)*c^2/a^2

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Fricas [A]  time = 1.59973, size = 176, normalized size = 2.51 \begin{align*} \frac{6 \, c^{2} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) -{\left (6 \, a^{3} c^{2} x^{3} - 8 \, a^{2} c^{2} x^{2} - 3 \, a c^{2} x + 8 \, c^{2}\right )} \sqrt{-a^{2} x^{2} + 1}}{24 \, a^{2}} \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*(-a*c*x+c)^2,x, algorithm="fricas")

[Out]

1/24*(6*c^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (6*a^3*c^2*x^3 - 8*a^2*c^2*x^2 - 3*a*c^2*x + 8*c^2)*sqrt(
-a^2*x^2 + 1))/a^2

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Sympy [A]  time = 9.61017, size = 330, normalized size = 4.71 \begin{align*} a^{3} 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 ) - a^{2} 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 ) - a c^{2} \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 ) + c^{2} \left (\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \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*(-a*c*x+c)**2,x)

[Out]

a**3*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*sqr
t(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)) - a**2*c**2*Piecewi
se((-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)) - a*c**2
*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)) + c**2*Piecewise((x**2/2, Eq(a**2,
0)), (-sqrt(-a**2*x**2 + 1)/a**2, True))

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Giac [A]  time = 1.35527, size = 93, normalized size = 1.33 \begin{align*} -\frac{c^{2} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{8 \, a{\left | a \right |}} - \frac{1}{24} \, \sqrt{-a^{2} x^{2} + 1}{\left ({\left (2 \,{\left (3 \, a c^{2} x - 4 \, c^{2}\right )} x - \frac{3 \, c^{2}}{a}\right )} x + \frac{8 \, c^{2}}{a^{2}}\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*(-a*c*x+c)^2,x, algorithm="giac")

[Out]

-1/8*c^2*arcsin(a*x)*sgn(a)/(a*abs(a)) - 1/24*sqrt(-a^2*x^2 + 1)*((2*(3*a*c^2*x - 4*c^2)*x - 3*c^2/a)*x + 8*c^
2/a^2)