3.1124 \(\int \frac{e^{2 \tanh ^{-1}(a x)}}{(c-a^2 c x^2)^{7/2}} \, dx\)

Optimal. Leaf size=97 \[ \frac{8 x}{21 c^3 \sqrt{c-a^2 c x^2}}+\frac{4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac{x}{7 c \left (c-a^2 c x^2\right )^{5/2}}+\frac{2 (a x+1)}{7 a \left (c-a^2 c x^2\right )^{7/2}} \]

[Out]

(2*(1 + a*x))/(7*a*(c - a^2*c*x^2)^(7/2)) + x/(7*c*(c - a^2*c*x^2)^(5/2)) + (4*x)/(21*c^2*(c - a^2*c*x^2)^(3/2
)) + (8*x)/(21*c^3*Sqrt[c - a^2*c*x^2])

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Rubi [A]  time = 0.0800799, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6141, 653, 192, 191} \[ \frac{8 x}{21 c^3 \sqrt{c-a^2 c x^2}}+\frac{4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac{x}{7 c \left (c-a^2 c x^2\right )^{5/2}}+\frac{2 (a x+1)}{7 a \left (c-a^2 c x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^(7/2),x]

[Out]

(2*(1 + a*x))/(7*a*(c - a^2*c*x^2)^(7/2)) + x/(7*c*(c - a^2*c*x^2)^(5/2)) + (4*x)/(21*c^2*(c - a^2*c*x^2)^(3/2
)) + (8*x)/(21*c^3*Sqrt[c - a^2*c*x^2])

Rule 6141

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^(n/2), Int[(c + d*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] || GtQ[c, 0]) && IGt
Q[n/2, 0]

Rule 653

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

Rule 192

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

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^{2 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx &=c \int \frac{(1+a x)^2}{\left (c-a^2 c x^2\right )^{9/2}} \, dx\\ &=\frac{2 (1+a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}+\frac{5}{7} \int \frac{1}{\left (c-a^2 c x^2\right )^{7/2}} \, dx\\ &=\frac{2 (1+a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}+\frac{x}{7 c \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 \int \frac{1}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{7 c}\\ &=\frac{2 (1+a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}+\frac{x}{7 c \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac{8 \int \frac{1}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{21 c^2}\\ &=\frac{2 (1+a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}+\frac{x}{7 c \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac{8 x}{21 c^3 \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0498029, size = 96, normalized size = 0.99 \[ -\frac{\sqrt{1-a^2 x^2} \left (8 a^5 x^5-16 a^4 x^4-4 a^3 x^3+24 a^2 x^2-9 a x-6\right )}{21 a c^3 (1-a x)^{7/2} (a x+1)^{3/2} \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^(7/2),x]

[Out]

-(Sqrt[1 - a^2*x^2]*(-6 - 9*a*x + 24*a^2*x^2 - 4*a^3*x^3 - 16*a^4*x^4 + 8*a^5*x^5))/(21*a*c^3*(1 - a*x)^(7/2)*
(1 + a*x)^(3/2)*Sqrt[c - a^2*c*x^2])

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Maple [A]  time = 0.031, size = 64, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 8\,{x}^{5}{a}^{5}-16\,{x}^{4}{a}^{4}-4\,{x}^{3}{a}^{3}+24\,{a}^{2}{x}^{2}-9\,ax-6 \right ) \left ( ax+1 \right ) ^{2}}{21\,a} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^2/(-a^2*x^2+1)/(-a^2*c*x^2+c)^(7/2),x)

[Out]

-1/21*(8*a^5*x^5-16*a^4*x^4-4*a^3*x^3+24*a^2*x^2-9*a*x-6)*(a*x+1)^2/(-a^2*c*x^2+c)^(7/2)/a

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)/(-a^2*c*x^2+c)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.55198, size = 251, normalized size = 2.59 \begin{align*} -\frac{{\left (8 \, a^{5} x^{5} - 16 \, a^{4} x^{4} - 4 \, a^{3} x^{3} + 24 \, a^{2} x^{2} - 9 \, a x - 6\right )} \sqrt{-a^{2} c x^{2} + c}}{21 \,{\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)/(-a^2*c*x^2+c)^(7/2),x, algorithm="fricas")

[Out]

-1/21*(8*a^5*x^5 - 16*a^4*x^4 - 4*a^3*x^3 + 24*a^2*x^2 - 9*a*x - 6)*sqrt(-a^2*c*x^2 + c)/(a^7*c^4*x^6 - 2*a^6*
c^4*x^5 - a^5*c^4*x^4 + 4*a^4*c^4*x^3 - a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{a x}{- a^{7} c^{3} x^{7} \sqrt{- a^{2} c x^{2} + c} + a^{6} c^{3} x^{6} \sqrt{- a^{2} c x^{2} + c} + 3 a^{5} c^{3} x^{5} \sqrt{- a^{2} c x^{2} + c} - 3 a^{4} c^{3} x^{4} \sqrt{- a^{2} c x^{2} + c} - 3 a^{3} c^{3} x^{3} \sqrt{- a^{2} c x^{2} + c} + 3 a^{2} c^{3} x^{2} \sqrt{- a^{2} c x^{2} + c} + a c^{3} x \sqrt{- a^{2} c x^{2} + c} - c^{3} \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{1}{- a^{7} c^{3} x^{7} \sqrt{- a^{2} c x^{2} + c} + a^{6} c^{3} x^{6} \sqrt{- a^{2} c x^{2} + c} + 3 a^{5} c^{3} x^{5} \sqrt{- a^{2} c x^{2} + c} - 3 a^{4} c^{3} x^{4} \sqrt{- a^{2} c x^{2} + c} - 3 a^{3} c^{3} x^{3} \sqrt{- a^{2} c x^{2} + c} + 3 a^{2} c^{3} x^{2} \sqrt{- a^{2} c x^{2} + c} + a c^{3} x \sqrt{- a^{2} c x^{2} + c} - c^{3} \sqrt{- a^{2} c x^{2} + c}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**2/(-a**2*x**2+1)/(-a**2*c*x**2+c)**(7/2),x)

[Out]

-Integral(a*x/(-a**7*c**3*x**7*sqrt(-a**2*c*x**2 + c) + a**6*c**3*x**6*sqrt(-a**2*c*x**2 + c) + 3*a**5*c**3*x*
*5*sqrt(-a**2*c*x**2 + c) - 3*a**4*c**3*x**4*sqrt(-a**2*c*x**2 + c) - 3*a**3*c**3*x**3*sqrt(-a**2*c*x**2 + c)
+ 3*a**2*c**3*x**2*sqrt(-a**2*c*x**2 + c) + a*c**3*x*sqrt(-a**2*c*x**2 + c) - c**3*sqrt(-a**2*c*x**2 + c)), x)
- Integral(1/(-a**7*c**3*x**7*sqrt(-a**2*c*x**2 + c) + a**6*c**3*x**6*sqrt(-a**2*c*x**2 + c) + 3*a**5*c**3*x*
*5*sqrt(-a**2*c*x**2 + c) - 3*a**4*c**3*x**4*sqrt(-a**2*c*x**2 + c) - 3*a**3*c**3*x**3*sqrt(-a**2*c*x**2 + c)
+ 3*a**2*c**3*x**2*sqrt(-a**2*c*x**2 + c) + a*c**3*x*sqrt(-a**2*c*x**2 + c) - c**3*sqrt(-a**2*c*x**2 + c)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (a x + 1\right )}^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{7}{2}}{\left (a^{2} x^{2} - 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)/(-a^2*c*x^2+c)^(7/2),x, algorithm="giac")

[Out]

integrate(-(a*x + 1)^2/((-a^2*c*x^2 + c)^(7/2)*(a^2*x^2 - 1)), x)