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

Optimal. Leaf size=101 \[ \frac{c^2 (1-a x)^2 \sqrt{1-a^2 x^2}}{3 a}+\frac{5 c^2 (1-a x) \sqrt{1-a^2 x^2}}{6 a}+\frac{5 c^2 \sqrt{1-a^2 x^2}}{2 a}+\frac{5 c^2 \sin ^{-1}(a x)}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.0699396, antiderivative size = 101, 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, 671, 641, 216} \[ \frac{c^2 (1-a x)^2 \sqrt{1-a^2 x^2}}{3 a}+\frac{5 c^2 (1-a x) \sqrt{1-a^2 x^2}}{6 a}+\frac{5 c^2 \sqrt{1-a^2 x^2}}{2 a}+\frac{5 c^2 \sin ^{-1}(a x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[(c - a*c*x)^2/E^ArcTanh[a*x],x]

[Out]

(5*c^2*Sqrt[1 - a^2*x^2])/(2*a) + (5*c^2*(1 - a*x)*Sqrt[1 - a^2*x^2])/(6*a) + (c^2*(1 - a*x)^2*Sqrt[1 - a^2*x^
2])/(3*a) + (5*c^2*ArcSin[a*x])/(2*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 671

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(m + 2*p + 1)), x] + Dist[(2*c*d*(m + p))/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p, x]
, x] /; FreeQ[{a, c, d, e, p}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && 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 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)} (c-a c x)^2 \, dx &=\frac{\int \frac{(c-a c x)^3}{\sqrt{1-a^2 x^2}} \, dx}{c}\\ &=\frac{c^2 (1-a x)^2 \sqrt{1-a^2 x^2}}{3 a}+\frac{5}{3} \int \frac{(c-a c x)^2}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{5 c^2 (1-a x) \sqrt{1-a^2 x^2}}{6 a}+\frac{c^2 (1-a x)^2 \sqrt{1-a^2 x^2}}{3 a}+\frac{1}{2} (5 c) \int \frac{c-a c x}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{5 c^2 \sqrt{1-a^2 x^2}}{2 a}+\frac{5 c^2 (1-a x) \sqrt{1-a^2 x^2}}{6 a}+\frac{c^2 (1-a x)^2 \sqrt{1-a^2 x^2}}{3 a}+\frac{1}{2} \left (5 c^2\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{5 c^2 \sqrt{1-a^2 x^2}}{2 a}+\frac{5 c^2 (1-a x) \sqrt{1-a^2 x^2}}{6 a}+\frac{c^2 (1-a x)^2 \sqrt{1-a^2 x^2}}{3 a}+\frac{5 c^2 \sin ^{-1}(a x)}{2 a}\\ \end{align*}

Mathematica [A]  time = 0.0822894, size = 72, normalized size = 0.71 \[ \frac{c^2 \left (\frac{\sqrt{a x+1} \left (-2 a^3 x^3+11 a^2 x^2-31 a x+22\right )}{\sqrt{1-a x}}-30 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{6 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^2*((Sqrt[1 + a*x]*(22 - 31*a*x + 11*a^2*x^2 - 2*a^3*x^3))/Sqrt[1 - a*x] - 30*ArcSin[Sqrt[1 - a*x]/Sqrt[2]])
)/(6*a)

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Maple [A]  time = 0.041, size = 142, normalized size = 1.4 \begin{align*} -{\frac{{c}^{2}}{3\,a} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{3\,x{c}^{2}}{2}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{3\,{c}^{2}}{2}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+4\,{\frac{{c}^{2}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}{a}}+4\,{\frac{{c}^{2}}{\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a*c*x+c)^2/(a*x+1)*(-a^2*x^2+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.44836, size = 96, normalized size = 0.95 \begin{align*} -\frac{3}{2} \, \sqrt{-a^{2} x^{2} + 1} c^{2} x - \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c^{2}}{3 \, a} + \frac{5 \, c^{2} \arcsin \left (a x\right )}{2 \, a} + \frac{4 \, \sqrt{-a^{2} x^{2} + 1} c^{2}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^2/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.84439, size = 154, normalized size = 1.52 \begin{align*} -\frac{30 \, c^{2} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) -{\left (2 \, a^{2} c^{2} x^{2} - 9 \, a c^{2} x + 22 \, c^{2}\right )} \sqrt{-a^{2} x^{2} + 1}}{6 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^2/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

-1/6*(30*c^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (2*a^2*c^2*x^2 - 9*a*c^2*x + 22*c^2)*sqrt(-a^2*x^2 + 1))
/a

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} c^{2} \left (\int \frac{\sqrt{- a^{2} x^{2} + 1}}{a x + 1}\, dx + \int - \frac{2 a x \sqrt{- a^{2} x^{2} + 1}}{a x + 1}\, dx + \int \frac{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a x + 1}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)**2/(a*x+1)*(-a**2*x**2+1)**(1/2),x)

[Out]

c**2*(Integral(sqrt(-a**2*x**2 + 1)/(a*x + 1), x) + Integral(-2*a*x*sqrt(-a**2*x**2 + 1)/(a*x + 1), x) + Integ
ral(a**2*x**2*sqrt(-a**2*x**2 + 1)/(a*x + 1), x))

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Giac [A]  time = 1.20531, size = 73, normalized size = 0.72 \begin{align*} \frac{5 \, c^{2} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{2 \,{\left | a \right |}} + \frac{1}{6} \, \sqrt{-a^{2} x^{2} + 1}{\left ({\left (2 \, a c^{2} x - 9 \, c^{2}\right )} x + \frac{22 \, c^{2}}{a}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^2/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

5/2*c^2*arcsin(a*x)*sgn(a)/abs(a) + 1/6*sqrt(-a^2*x^2 + 1)*((2*a*c^2*x - 9*c^2)*x + 22*c^2/a)