3.178 \(\int e^{3 \tanh ^{-1}(a x)} (c-a c x)^4 \, dx\)

Optimal. Leaf size=83 \[ \frac{c^4 \left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac{1}{4} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac{3}{8} c^4 x \sqrt{1-a^2 x^2}+\frac{3 c^4 \sin ^{-1}(a x)}{8 a} \]

[Out]

(3*c^4*x*Sqrt[1 - a^2*x^2])/8 + (c^4*x*(1 - a^2*x^2)^(3/2))/4 + (c^4*(1 - a^2*x^2)^(5/2))/(5*a) + (3*c^4*ArcSi
n[a*x])/(8*a)

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Rubi [A]  time = 0.0485722, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6127, 641, 195, 216} \[ \frac{c^4 \left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac{1}{4} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac{3}{8} c^4 x \sqrt{1-a^2 x^2}+\frac{3 c^4 \sin ^{-1}(a x)}{8 a} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])*(c - a*c*x)^4,x]

[Out]

(3*c^4*x*Sqrt[1 - a^2*x^2])/8 + (c^4*x*(1 - a^2*x^2)^(3/2))/4 + (c^4*(1 - a^2*x^2)^(5/2))/(5*a) + (3*c^4*ArcSi
n[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 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[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^{3 \tanh ^{-1}(a x)} (c-a c x)^4 \, dx &=c^3 \int (c-a c x) \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac{c^4 \left (1-a^2 x^2\right )^{5/2}}{5 a}+c^4 \int \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac{1}{4} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac{c^4 \left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac{1}{4} \left (3 c^4\right ) \int \sqrt{1-a^2 x^2} \, dx\\ &=\frac{3}{8} c^4 x \sqrt{1-a^2 x^2}+\frac{1}{4} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac{c^4 \left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac{1}{8} \left (3 c^4\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{3}{8} c^4 x \sqrt{1-a^2 x^2}+\frac{1}{4} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac{c^4 \left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac{3 c^4 \sin ^{-1}(a x)}{8 a}\\ \end{align*}

Mathematica [A]  time = 0.0933112, size = 75, normalized size = 0.9 \[ \frac{c^4 \left (\sqrt{1-a^2 x^2} \left (8 a^4 x^4-10 a^3 x^3-16 a^2 x^2+25 a x+8\right )-30 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{40 a} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(3*ArcTanh[a*x])*(c - a*c*x)^4,x]

[Out]

(c^4*(Sqrt[1 - a^2*x^2]*(8 + 25*a*x - 16*a^2*x^2 - 10*a^3*x^3 + 8*a^4*x^4) - 30*ArcSin[Sqrt[1 - a*x]/Sqrt[2]])
)/(40*a)

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Maple [B]  time = 0.069, size = 183, normalized size = 2.2 \begin{align*} -{\frac{{c}^{4}{a}^{5}{x}^{6}}{5}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{3\,{c}^{4}{a}^{3}{x}^{4}}{5}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{3\,{c}^{4}a{x}^{2}}{5}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{{c}^{4}}{5\,a}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{{a}^{4}{c}^{4}{x}^{5}}{4}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{7\,{a}^{2}{c}^{4}{x}^{3}}{8}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{5\,{c}^{4}x}{8}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{3\,{c}^{4}}{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)^4,x)

[Out]

-1/5*c^4*a^5*x^6/(-a^2*x^2+1)^(1/2)+3/5*c^4*a^3*x^4/(-a^2*x^2+1)^(1/2)-3/5*c^4*a*x^2/(-a^2*x^2+1)^(1/2)+1/5*c^
4/a/(-a^2*x^2+1)^(1/2)+1/4*c^4*a^4*x^5/(-a^2*x^2+1)^(1/2)-7/8*c^4*a^2*x^3/(-a^2*x^2+1)^(1/2)+5/8*c^4*x/(-a^2*x
^2+1)^(1/2)+3/8*c^4/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))

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Maxima [B]  time = 1.45437, size = 234, normalized size = 2.82 \begin{align*} -\frac{a^{5} c^{4} x^{6}}{5 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{a^{4} c^{4} x^{5}}{4 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{3 \, a^{3} c^{4} x^{4}}{5 \, \sqrt{-a^{2} x^{2} + 1}} - \frac{7 \, a^{2} c^{4} x^{3}}{8 \, \sqrt{-a^{2} x^{2} + 1}} - \frac{3 \, a c^{4} x^{2}}{5 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{5 \, c^{4} x}{8 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{3 \, c^{4} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{8 \, \sqrt{a^{2}}} + \frac{c^{4}}{5 \, \sqrt{-a^{2} x^{2} + 1} 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)^4,x, algorithm="maxima")

[Out]

-1/5*a^5*c^4*x^6/sqrt(-a^2*x^2 + 1) + 1/4*a^4*c^4*x^5/sqrt(-a^2*x^2 + 1) + 3/5*a^3*c^4*x^4/sqrt(-a^2*x^2 + 1)
- 7/8*a^2*c^4*x^3/sqrt(-a^2*x^2 + 1) - 3/5*a*c^4*x^2/sqrt(-a^2*x^2 + 1) + 5/8*c^4*x/sqrt(-a^2*x^2 + 1) + 3/8*c
^4*arcsin(a^2*x/sqrt(a^2))/sqrt(a^2) + 1/5*c^4/(sqrt(-a^2*x^2 + 1)*a)

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Fricas [A]  time = 1.93317, size = 201, normalized size = 2.42 \begin{align*} -\frac{30 \, c^{4} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) -{\left (8 \, a^{4} c^{4} x^{4} - 10 \, a^{3} c^{4} x^{3} - 16 \, a^{2} c^{4} x^{2} + 25 \, a c^{4} x + 8 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1}}{40 \, 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)^4,x, algorithm="fricas")

[Out]

-1/40*(30*c^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (8*a^4*c^4*x^4 - 10*a^3*c^4*x^3 - 16*a^2*c^4*x^2 + 25*a
*c^4*x + 8*c^4)*sqrt(-a^2*x^2 + 1))/a

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Sympy [A]  time = 20.2233, size = 459, normalized size = 5.53 \begin{align*} - a^{5} c^{4} \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^{4} c^{4} \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^{3} c^{4} \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 ) - 2 a^{2} c^{4} \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 ) - a c^{4} \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 ) + c^{4} \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)**4,x)

[Out]

-a**5*c**4*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**4*c**4*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)) + 2*a**3*c**4*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)) - 2*a**2*c**4*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)) + a
sin(a*x)/(2*a**3), True)) - a*c**4*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) + c**4
*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0))

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Giac [A]  time = 1.28475, size = 105, normalized size = 1.27 \begin{align*} \frac{3 \, c^{4} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{8 \,{\left | a \right |}} + \frac{1}{40} \, \sqrt{-a^{2} x^{2} + 1}{\left (\frac{8 \, c^{4}}{a} +{\left (25 \, c^{4} - 2 \,{\left (8 \, a c^{4} -{\left (4 \, a^{3} c^{4} x - 5 \, a^{2} c^{4}\right )} x\right )} x\right )} x\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)^4,x, algorithm="giac")

[Out]

3/8*c^4*arcsin(a*x)*sgn(a)/abs(a) + 1/40*sqrt(-a^2*x^2 + 1)*(8*c^4/a + (25*c^4 - 2*(8*a*c^4 - (4*a^3*c^4*x - 5
*a^2*c^4)*x)*x)*x)