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

Optimal. Leaf size=58 \[ \frac{1}{4} c x^3 \sqrt{1-a^2 x^2}-\frac{c x \sqrt{1-a^2 x^2}}{8 a^2}+\frac{c \sin ^{-1}(a x)}{8 a^3} \]

[Out]

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

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Rubi [A]  time = 0.0568359, antiderivative size = 58, 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, 279, 321, 216} \[ \frac{1}{4} c x^3 \sqrt{1-a^2 x^2}-\frac{c x \sqrt{1-a^2 x^2}}{8 a^2}+\frac{c \sin ^{-1}(a x)}{8 a^3} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]*x^2*(c - a*c*x),x]

[Out]

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

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 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
n*p + 1)), x] + Dist[(a*n*p)/(m + n*p + 1), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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

Mathematica [A]  time = 0.0412941, size = 40, normalized size = 0.69 \[ \frac{c \left (a x \sqrt{1-a^2 x^2} \left (2 a^2 x^2-1\right )+\sin ^{-1}(a x)\right )}{8 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.039, size = 70, normalized size = 1.2 \begin{align*}{\frac{c{x}^{3}}{4}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{cx}{8\,{a}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{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)/(-a^2*x^2+1)^(1/2)*x^2*(-a*c*x+c),x)

[Out]

1/4*c*x^3*(-a^2*x^2+1)^(1/2)-1/8*c*x*(-a^2*x^2+1)^(1/2)/a^2+1/8*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 = 1.43469, size = 81, normalized size = 1.4 \begin{align*} \frac{1}{4} \, \sqrt{-a^{2} x^{2} + 1} c x^{3} - \frac{\sqrt{-a^{2} x^{2} + 1} c x}{8 \, a^{2}} + \frac{c \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{8 \, \sqrt{a^{2}} 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^2*(-a*c*x+c),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.60507, size = 132, normalized size = 2.28 \begin{align*} -\frac{2 \, c \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) -{\left (2 \, a^{3} c x^{3} - a c x\right )} \sqrt{-a^{2} x^{2} + 1}}{8 \, a^{3}} \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^2*(-a*c*x+c),x, algorithm="fricas")

[Out]

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

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Sympy [C]  time = 7.4772, size = 245, normalized size = 4.22 \begin{align*} - 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 ) + 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)/(-a**2*x**2+1)**(1/2)*x**2*(-a*c*x+c),x)

[Out]

-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)) + c*Piecewise((-I*x*s
qrt(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))

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Giac [A]  time = 1.22126, size = 61, normalized size = 1.05 \begin{align*} \frac{1}{8} \, \sqrt{-a^{2} x^{2} + 1}{\left (2 \, c x^{2} - \frac{c}{a^{2}}\right )} x + \frac{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)/(-a^2*x^2+1)^(1/2)*x^2*(-a*c*x+c),x, algorithm="giac")

[Out]

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