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

Optimal. Leaf size=59 \[ \frac{1}{4} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac{3}{8} c^3 x \sqrt{1-a^2 x^2}+\frac{3 c^3 \sin ^{-1}(a x)}{8 a} \]

[Out]

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

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Rubi [A]  time = 0.0356936, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6127, 195, 216} \[ \frac{1}{4} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac{3}{8} c^3 x \sqrt{1-a^2 x^2}+\frac{3 c^3 \sin ^{-1}(a x)}{8 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6127

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

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

Mathematica [A]  time = 0.0364763, size = 44, normalized size = 0.75 \[ \frac{c^3 \left (a x \sqrt{1-a^2 x^2} \left (5-2 a^2 x^2\right )+3 \sin ^{-1}(a x)\right )}{8 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^3*(a*x*(5 - 2*a^2*x^2)*Sqrt[1 - a^2*x^2] + 3*ArcSin[a*x]))/(8*a)

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Maple [A]  time = 0.044, size = 96, normalized size = 1.6 \begin{align*}{\frac{{a}^{4}{c}^{3}{x}^{5}}{4}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{7\,{c}^{3}{a}^{2}{x}^{3}}{8}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{5\,{c}^{3}x}{8}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{3\,{c}^{3}}{8}\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)*(-a*c*x+c)^3,x)

[Out]

1/4*c^3*a^4*x^5/(-a^2*x^2+1)^(1/2)-7/8*c^3*a^2*x^3/(-a^2*x^2+1)^(1/2)+5/8*c^3*x/(-a^2*x^2+1)^(1/2)+3/8*c^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 = 1.47573, size = 116, normalized size = 1.97 \begin{align*} \frac{a^{4} c^{3} x^{5}}{4 \, \sqrt{-a^{2} x^{2} + 1}} - \frac{7 \, a^{2} c^{3} x^{3}}{8 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{5 \, c^{3} x}{8 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{3 \, c^{3} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{8 \, \sqrt{a^{2}}} \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)*(-a*c*x+c)^3,x, algorithm="maxima")

[Out]

1/4*a^4*c^3*x^5/sqrt(-a^2*x^2 + 1) - 7/8*a^2*c^3*x^3/sqrt(-a^2*x^2 + 1) + 5/8*c^3*x/sqrt(-a^2*x^2 + 1) + 3/8*c
^3*arcsin(a^2*x/sqrt(a^2))/sqrt(a^2)

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Fricas [A]  time = 1.89182, size = 140, normalized size = 2.37 \begin{align*} -\frac{6 \, c^{3} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (2 \, a^{3} c^{3} x^{3} - 5 \, a c^{3} x\right )} \sqrt{-a^{2} x^{2} + 1}}{8 \, a} \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)*(-a*c*x+c)^3,x, algorithm="fricas")

[Out]

-1/8*(6*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (2*a^3*c^3*x^3 - 5*a*c^3*x)*sqrt(-a^2*x^2 + 1))/a

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Sympy [A]  time = 8.58381, size = 301, normalized size = 5.1 \begin{align*} a^{4} 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 ) - 2 a^{2} 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 ) + c^{3} \left (\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left (x \sqrt{a^{2}} \right )} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left (x \sqrt{- a^{2}} \right )} & \text{for}\: a^{2} < 0 \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)*(-a*c*x+c)**3,x)

[Out]

a**4*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*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)) - 2*a**2*c**3*Piece
wise((-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**3*Piecewise((sqrt(a**(-2))*asin(x*s
qrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0))

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Giac [A]  time = 1.2034, size = 65, normalized size = 1.1 \begin{align*} \frac{3 \, c^{3} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{8 \,{\left | a \right |}} - \frac{1}{8} \,{\left (2 \, a^{2} c^{3} x^{2} - 5 \, c^{3}\right )} \sqrt{-a^{2} x^{2} + 1} x \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)*(-a*c*x+c)^3,x, algorithm="giac")

[Out]

3/8*c^3*arcsin(a*x)*sgn(a)/abs(a) - 1/8*(2*a^2*c^3*x^2 - 5*c^3)*sqrt(-a^2*x^2 + 1)*x