3.1154 \(\int \frac{e^{3 \tanh ^{-1}(a x)}}{(c-a^2 c x^2)^3} \, dx\)

Optimal. Leaf size=119 \[ \frac{8 x}{35 c^3 \sqrt{1-a^2 x^2}}+\frac{4}{35 a c^3 (1-a x) \sqrt{1-a^2 x^2}}+\frac{4}{35 a c^3 (1-a x)^2 \sqrt{1-a^2 x^2}}+\frac{1}{7 a c^3 (1-a x)^3 \sqrt{1-a^2 x^2}} \]

[Out]

(8*x)/(35*c^3*Sqrt[1 - a^2*x^2]) + 1/(7*a*c^3*(1 - a*x)^3*Sqrt[1 - a^2*x^2]) + 4/(35*a*c^3*(1 - a*x)^2*Sqrt[1
- a^2*x^2]) + 4/(35*a*c^3*(1 - a*x)*Sqrt[1 - a^2*x^2])

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Rubi [A]  time = 0.0885734, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6138, 655, 659, 191} \[ \frac{8 x}{35 c^3 \sqrt{1-a^2 x^2}}+\frac{4}{35 a c^3 (1-a x) \sqrt{1-a^2 x^2}}+\frac{4}{35 a c^3 (1-a x)^2 \sqrt{1-a^2 x^2}}+\frac{1}{7 a c^3 (1-a x)^3 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*x)/(35*c^3*Sqrt[1 - a^2*x^2]) + 1/(7*a*c^3*(1 - a*x)^3*Sqrt[1 - a^2*x^2]) + 4/(35*a*c^3*(1 - a*x)^2*Sqrt[1
- a^2*x^2]) + 4/(35*a*c^3*(1 - a*x)*Sqrt[1 - a^2*x^2])

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 655

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^(2*m)/a^m, Int[(a + c*x^2)^(m + p
)/(d - e*x)^m, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && IntegerQ[m]
&& RationalQ[p] && (LtQ[0, -m, p] || LtQ[p, -m, 0]) && NeQ[m, 2] && NeQ[m, -1]

Rule 659

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1)
)/(2*c*d*(m + p + 1)), x] + Dist[Simplify[m + 2*p + 2]/(2*d*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p,
x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2
], 0]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{e^{3 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac{\int \frac{(1+a x)^3}{\left (1-a^2 x^2\right )^{9/2}} \, dx}{c^3}\\ &=\frac{\int \frac{1}{(1-a x)^3 \left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac{1}{7 a c^3 (1-a x)^3 \sqrt{1-a^2 x^2}}+\frac{4 \int \frac{1}{(1-a x)^2 \left (1-a^2 x^2\right )^{3/2}} \, dx}{7 c^3}\\ &=\frac{1}{7 a c^3 (1-a x)^3 \sqrt{1-a^2 x^2}}+\frac{4}{35 a c^3 (1-a x)^2 \sqrt{1-a^2 x^2}}+\frac{12 \int \frac{1}{(1-a x) \left (1-a^2 x^2\right )^{3/2}} \, dx}{35 c^3}\\ &=\frac{1}{7 a c^3 (1-a x)^3 \sqrt{1-a^2 x^2}}+\frac{4}{35 a c^3 (1-a x)^2 \sqrt{1-a^2 x^2}}+\frac{4}{35 a c^3 (1-a x) \sqrt{1-a^2 x^2}}+\frac{8 \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{35 c^3}\\ &=\frac{8 x}{35 c^3 \sqrt{1-a^2 x^2}}+\frac{1}{7 a c^3 (1-a x)^3 \sqrt{1-a^2 x^2}}+\frac{4}{35 a c^3 (1-a x)^2 \sqrt{1-a^2 x^2}}+\frac{4}{35 a c^3 (1-a x) \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0208613, size = 61, normalized size = 0.51 \[ \frac{8 a^4 x^4-24 a^3 x^3+20 a^2 x^2+4 a x-13}{35 a c^3 (a x-1)^3 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-13 + 4*a*x + 20*a^2*x^2 - 24*a^3*x^3 + 8*a^4*x^4)/(35*a*c^3*(-1 + a*x)^3*Sqrt[1 - a^2*x^2])

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Maple [A]  time = 0.033, size = 63, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 8\,{x}^{4}{a}^{4}-24\,{x}^{3}{a}^{3}+20\,{a}^{2}{x}^{2}+4\,ax-13 \right ) \left ( ax+1 \right ) }{35\,{c}^{3} \left ( ax-1 \right ) ^{2}a} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{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)^3,x)

[Out]

-1/35*(8*a^4*x^4-24*a^3*x^3+20*a^2*x^2+4*a*x-13)*(a*x+1)/(a*x-1)^2/c^3/(-a^2*x^2+1)^(3/2)/a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (a x + 1\right )}^{3}}{{\left (a^{2} c x^{2} - c\right )}^{3}{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.6794, size = 308, normalized size = 2.59 \begin{align*} \frac{13 \, a^{5} x^{5} - 39 \, a^{4} x^{4} + 26 \, a^{3} x^{3} + 26 \, a^{2} x^{2} - 39 \, a x -{\left (8 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 20 \, a^{2} x^{2} + 4 \, a x - 13\right )} \sqrt{-a^{2} x^{2} + 1} + 13}{35 \,{\left (a^{6} c^{3} x^{5} - 3 \, a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} + 2 \, a^{3} c^{3} x^{2} - 3 \, a^{2} c^{3} x + a c^{3}\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)^3,x, algorithm="fricas")

[Out]

1/35*(13*a^5*x^5 - 39*a^4*x^4 + 26*a^3*x^3 + 26*a^2*x^2 - 39*a*x - (8*a^4*x^4 - 24*a^3*x^3 + 20*a^2*x^2 + 4*a*
x - 13)*sqrt(-a^2*x^2 + 1) + 13)/(a^6*c^3*x^5 - 3*a^5*c^3*x^4 + 2*a^4*c^3*x^3 + 2*a^3*c^3*x^2 - 3*a^2*c^3*x +
a*c^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{3 a x}{a^{8} x^{8} \sqrt{- a^{2} x^{2} + 1} - 4 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 6 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a^{2} x^{2}}{a^{8} x^{8} \sqrt{- a^{2} x^{2} + 1} - 4 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 6 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a^{3} x^{3}}{a^{8} x^{8} \sqrt{- a^{2} x^{2} + 1} - 4 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 6 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{8} x^{8} \sqrt{- a^{2} x^{2} + 1} - 4 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 6 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}} \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)**3,x)

[Out]

(Integral(3*a*x/(a**8*x**8*sqrt(-a**2*x**2 + 1) - 4*a**6*x**6*sqrt(-a**2*x**2 + 1) + 6*a**4*x**4*sqrt(-a**2*x*
*2 + 1) - 4*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(3*a**2*x**2/(a**8*x**8*sqrt(
-a**2*x**2 + 1) - 4*a**6*x**6*sqrt(-a**2*x**2 + 1) + 6*a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**2*x**2*sqrt(-a**2
*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a**3*x**3/(a**8*x**8*sqrt(-a**2*x**2 + 1) - 4*a**6*x**6*sqrt
(-a**2*x**2 + 1) + 6*a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1))
, x) + Integral(1/(a**8*x**8*sqrt(-a**2*x**2 + 1) - 4*a**6*x**6*sqrt(-a**2*x**2 + 1) + 6*a**4*x**4*sqrt(-a**2*
x**2 + 1) - 4*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**3

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (a x + 1\right )}^{3}}{{\left (a^{2} c x^{2} - c\right )}^{3}{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \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)^3,x, algorithm="giac")

[Out]

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