3.1140 \(\int e^{3 \tanh ^{-1}(a x)} (c-a^2 c x^2) \, dx\)

Optimal. Leaf size=91 \[ -\frac{c \sqrt{1-a^2 x^2} (a x+1)^2}{3 a}-\frac{5 c \sqrt{1-a^2 x^2} (a x+1)}{6 a}-\frac{5 c \sqrt{1-a^2 x^2}}{2 a}+\frac{5 c \sin ^{-1}(a x)}{2 a} \]

[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 = 0.0568851, antiderivative size = 91, 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 = {6138, 671, 641, 216} \[ -\frac{c \sqrt{1-a^2 x^2} (a x+1)^2}{3 a}-\frac{5 c \sqrt{1-a^2 x^2} (a x+1)}{6 a}-\frac{5 c \sqrt{1-a^2 x^2}}{2 a}+\frac{5 c \sin ^{-1}(a x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])*(c - a^2*c*x^2),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 6138

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] && IGtQ[(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 = 0.0476712, size = 57, normalized size = 0.63 \[ -\frac{c \left (\sqrt{1-a^2 x^2} \left (2 a^2 x^2+9 a x+22\right )+30 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{6 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(c*(Sqrt[1 - a^2*x^2]*(22 + 9*a*x + 2*a^2*x^2) + 30*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(6*a)

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Maple [A]  time = 0.049, size = 125, normalized size = 1.4 \begin{align*}{\frac{{a}^{3}c{x}^{4}}{3}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{10\,ac{x}^{2}}{3}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{11\,c}{3\,a}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{3\,{a}^{2}c{x}^{3}}{2}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{3\,cx}{2}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{5\,c}{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)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c),x)

[Out]

1/3*c*a^3*x^4/(-a^2*x^2+1)^(1/2)+10/3*c*a*x^2/(-a^2*x^2+1)^(1/2)-11/3*c/a/(-a^2*x^2+1)^(1/2)+3/2*c*a^2*x^3/(-a
^2*x^2+1)^(1/2)-3/2*c*x/(-a^2*x^2+1)^(1/2)+5/2*c/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))

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Maxima [A]  time = 1.46553, size = 155, normalized size = 1.7 \begin{align*} \frac{a^{3} c x^{4}}{3 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{3 \, a^{2} c x^{3}}{2 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{10 \, a c x^{2}}{3 \, \sqrt{-a^{2} x^{2} + 1}} - \frac{3 \, c x}{2 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{5 \, c \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{2 \, \sqrt{a^{2}}} - \frac{11 \, c}{3 \, \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^2*c*x^2+c),x, algorithm="maxima")

[Out]

1/3*a^3*c*x^4/sqrt(-a^2*x^2 + 1) + 3/2*a^2*c*x^3/sqrt(-a^2*x^2 + 1) + 10/3*a*c*x^2/sqrt(-a^2*x^2 + 1) - 3/2*c*
x/sqrt(-a^2*x^2 + 1) + 5/2*c*arcsin(a^2*x/sqrt(a^2))/sqrt(a^2) - 11/3*c/(sqrt(-a^2*x^2 + 1)*a)

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Fricas [A]  time = 2.63829, size = 143, normalized size = 1.57 \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*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c),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 [A]  time = 10.9856, size = 218, normalized size = 2.4 \begin{align*} a^{3} c \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 ) + 3 a^{2} 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 ) + 3 a c \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 \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**2*c*x**2+c),x)

[Out]

a**3*c*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, T
rue)) + 3*a**2*c*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)) + asin(a*x)/(2*a**3), True)) + 3*a*c*Piecewise(
(x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) + c*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 = 1.2289, size = 62, normalized size = 0.68 \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*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c),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)