3.1257 \(\int e^{-3 \tanh ^{-1}(a x)} (c-a^2 c x^2) \, dx\)

Optimal. Leaf size=93 \[ \frac{c \sqrt{1-a^2 x^2} (1-a x)^2}{3 a}+\frac{5 c \sqrt{1-a^2 x^2} (1-a x)}{6 a}+\frac{5 c \sqrt{1-a^2 x^2}}{2 a}+\frac{5 c \sin ^{-1}(a x)}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.0567689, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6139, 671, 641, 216} \[ \frac{c \sqrt{1-a^2 x^2} (1-a x)^2}{3 a}+\frac{5 c \sqrt{1-a^2 x^2} (1-a x)}{6 a}+\frac{5 c \sqrt{1-a^2 x^2}}{2 a}+\frac{5 c \sin ^{-1}(a x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[(c - a^2*c*x^2)/E^(3*ArcTanh[a*x]),x]

[Out]

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

Rule 6139

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

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^{-3 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx &=c \int \frac{(1-a x)^3}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{c (1-a x)^2 \sqrt{1-a^2 x^2}}{3 a}+\frac{1}{3} (5 c) \int \frac{(1-a x)^2}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{5 c (1-a x) \sqrt{1-a^2 x^2}}{6 a}+\frac{c (1-a x)^2 \sqrt{1-a^2 x^2}}{3 a}+\frac{1}{2} (5 c) \int \frac{1-a x}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{5 c \sqrt{1-a^2 x^2}}{2 a}+\frac{5 c (1-a x) \sqrt{1-a^2 x^2}}{6 a}+\frac{c (1-a x)^2 \sqrt{1-a^2 x^2}}{3 a}+\frac{1}{2} (5 c) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{5 c \sqrt{1-a^2 x^2}}{2 a}+\frac{5 c (1-a x) \sqrt{1-a^2 x^2}}{6 a}+\frac{c (1-a x)^2 \sqrt{1-a^2 x^2}}{3 a}+\frac{5 c \sin ^{-1}(a x)}{2 a}\\ \end{align*}

Mathematica [A]  time = 0.0856578, size = 70, normalized size = 0.75 \[ \frac{c \left (\frac{\sqrt{a x+1} \left (-2 a^3 x^3+11 a^2 x^2-31 a x+22\right )}{\sqrt{1-a x}}-30 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{6 a} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c - a^2*c*x^2)/E^(3*ArcTanh[a*x]),x]

[Out]

(c*((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 = 0.04, size = 133, normalized size = 1.4 \begin{align*} 2\,{\frac{c \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{5/2}}{{a}^{3} \left ( x+{a}^{-1} \right ) ^{2}}}+{\frac{5\,c}{3\,a} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{5\,cx}{2}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}+{\frac{5\,c}{2}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x)

[Out]

2*c/a^3/(x+1/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)+5/3*c/a*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)+5/2*c*(-a^2*(x
+1/a)^2+2*a*(x+1/a))^(1/2)*x+5/2*c/(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 [C]  time = 1.46424, size = 165, normalized size = 1.77 \begin{align*} -\frac{1}{2} \, \sqrt{a^{2} x^{2} + 4 \, a x + 3} c x + \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c}{a^{2} x + a} - \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c}{3 \, a} + \frac{i \, c \arcsin \left (a x + 2\right )}{2 \, a} + \frac{3 \, c \arcsin \left (a x\right )}{a} - \frac{\sqrt{a^{2} x^{2} + 4 \, a x + 3} c}{a} + \frac{3 \, \sqrt{-a^{2} x^{2} + 1} c}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.71979, size = 143, normalized size = 1.54 \begin{align*} -\frac{30 \, c \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) -{\left (2 \, a^{2} c x^{2} - 9 \, a c x + 22 \, c\right )} \sqrt{-a^{2} x^{2} + 1}}{6 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} c \left (\int \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int - \frac{2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int \frac{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*c*x**2+c)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)

[Out]

c*(Integral(sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(-2*a**2*x**2*sqrt(-a**2*
x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(a**4*x**4*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a
**2*x**2 + 3*a*x + 1), x))

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Giac [A]  time = 1.16713, size = 62, normalized size = 0.67 \begin{align*} \frac{5 \, c \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 x - 9 \, c\right )} x + \frac{22 \, c}{a}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="giac")

[Out]

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