3.1200 \(\int \frac{e^{-\tanh ^{-1}(a x)}}{(c-a^2 c x^2)^5} \, dx\)

Optimal. Leaf size=119 \[ \frac{128 x}{315 c^5 \sqrt{1-a^2 x^2}}+\frac{64 x}{315 c^5 \left (1-a^2 x^2\right )^{3/2}}+\frac{16 x}{105 c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac{8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}-\frac{1-a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}} \]

[Out]

-(1 - a*x)/(9*a*c^5*(1 - a^2*x^2)^(9/2)) + (8*x)/(63*c^5*(1 - a^2*x^2)^(7/2)) + (16*x)/(105*c^5*(1 - a^2*x^2)^
(5/2)) + (64*x)/(315*c^5*(1 - a^2*x^2)^(3/2)) + (128*x)/(315*c^5*Sqrt[1 - a^2*x^2])

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Rubi [A]  time = 0.0659426, 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 = {6139, 639, 192, 191} \[ \frac{128 x}{315 c^5 \sqrt{1-a^2 x^2}}+\frac{64 x}{315 c^5 \left (1-a^2 x^2\right )^{3/2}}+\frac{16 x}{105 c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac{8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}-\frac{1-a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^ArcTanh[a*x]*(c - a^2*c*x^2)^5),x]

[Out]

-(1 - a*x)/(9*a*c^5*(1 - a^2*x^2)^(9/2)) + (8*x)/(63*c^5*(1 - a^2*x^2)^(7/2)) + (16*x)/(105*c^5*(1 - a^2*x^2)^
(5/2)) + (64*x)/(315*c^5*(1 - a^2*x^2)^(3/2)) + (128*x)/(315*c^5*Sqrt[1 - a^2*x^2])

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 639

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)*(a + c*x^2)^(p + 1))/(2*a
*c*(p + 1)), x] + Dist[(d*(2*p + 3))/(2*a*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

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^{-\tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^5} \, dx &=\frac{\int \frac{1-a x}{\left (1-a^2 x^2\right )^{11/2}} \, dx}{c^5}\\ &=-\frac{1-a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}}+\frac{8 \int \frac{1}{\left (1-a^2 x^2\right )^{9/2}} \, dx}{9 c^5}\\ &=-\frac{1-a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}}+\frac{8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}+\frac{16 \int \frac{1}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{21 c^5}\\ &=-\frac{1-a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}}+\frac{8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}+\frac{16 x}{105 c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac{64 \int \frac{1}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{105 c^5}\\ &=-\frac{1-a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}}+\frac{8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}+\frac{16 x}{105 c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac{64 x}{315 c^5 \left (1-a^2 x^2\right )^{3/2}}+\frac{128 \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{315 c^5}\\ &=-\frac{1-a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}}+\frac{8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}+\frac{16 x}{105 c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac{64 x}{315 c^5 \left (1-a^2 x^2\right )^{3/2}}+\frac{128 x}{315 c^5 \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0395975, size = 91, normalized size = 0.76 \[ -\frac{128 a^8 x^8+128 a^7 x^7-448 a^6 x^6-448 a^5 x^5+560 a^4 x^4+560 a^3 x^3-280 a^2 x^2-280 a x+35}{315 a c^5 (1-a x)^{7/2} (a x+1)^{9/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^ArcTanh[a*x]*(c - a^2*c*x^2)^5),x]

[Out]

-(35 - 280*a*x - 280*a^2*x^2 + 560*a^3*x^3 + 560*a^4*x^4 - 448*a^5*x^5 - 448*a^6*x^6 + 128*a^7*x^7 + 128*a^8*x
^8)/(315*a*c^5*(1 - a*x)^(7/2)*(1 + a*x)^(9/2))

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Maple [A]  time = 0.032, size = 90, normalized size = 0.8 \begin{align*} -{\frac{128\,{a}^{8}{x}^{8}+128\,{a}^{7}{x}^{7}-448\,{x}^{6}{a}^{6}-448\,{x}^{5}{a}^{5}+560\,{x}^{4}{a}^{4}+560\,{x}^{3}{a}^{3}-280\,{a}^{2}{x}^{2}-280\,ax+35}{ \left ( 315\,ax+315 \right ){c}^{5}a} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^5,x)

[Out]

-1/315/(-a^2*x^2+1)^(7/2)*(128*a^8*x^8+128*a^7*x^7-448*a^6*x^6-448*a^5*x^5+560*a^4*x^4+560*a^3*x^3-280*a^2*x^2
-280*a*x+35)/(a*x+1)/c^5/a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{\sqrt{-a^{2} x^{2} + 1}}{{\left (a^{2} c x^{2} - c\right )}^{5}{\left (a x + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^5,x, algorithm="maxima")

[Out]

-integrate(sqrt(-a^2*x^2 + 1)/((a^2*c*x^2 - c)^5*(a*x + 1)), x)

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Fricas [B]  time = 3.50518, size = 555, normalized size = 4.66 \begin{align*} -\frac{35 \, a^{9} x^{9} + 35 \, a^{8} x^{8} - 140 \, a^{7} x^{7} - 140 \, a^{6} x^{6} + 210 \, a^{5} x^{5} + 210 \, a^{4} x^{4} - 140 \, a^{3} x^{3} - 140 \, a^{2} x^{2} + 35 \, a x +{\left (128 \, a^{8} x^{8} + 128 \, a^{7} x^{7} - 448 \, a^{6} x^{6} - 448 \, a^{5} x^{5} + 560 \, a^{4} x^{4} + 560 \, a^{3} x^{3} - 280 \, a^{2} x^{2} - 280 \, a x + 35\right )} \sqrt{-a^{2} x^{2} + 1} + 35}{315 \,{\left (a^{10} c^{5} x^{9} + a^{9} c^{5} x^{8} - 4 \, a^{8} c^{5} x^{7} - 4 \, a^{7} c^{5} x^{6} + 6 \, a^{6} c^{5} x^{5} + 6 \, a^{5} c^{5} x^{4} - 4 \, a^{4} c^{5} x^{3} - 4 \, a^{3} c^{5} x^{2} + a^{2} c^{5} x + a c^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^5,x, algorithm="fricas")

[Out]

-1/315*(35*a^9*x^9 + 35*a^8*x^8 - 140*a^7*x^7 - 140*a^6*x^6 + 210*a^5*x^5 + 210*a^4*x^4 - 140*a^3*x^3 - 140*a^
2*x^2 + 35*a*x + (128*a^8*x^8 + 128*a^7*x^7 - 448*a^6*x^6 - 448*a^5*x^5 + 560*a^4*x^4 + 560*a^3*x^3 - 280*a^2*
x^2 - 280*a*x + 35)*sqrt(-a^2*x^2 + 1) + 35)/(a^10*c^5*x^9 + a^9*c^5*x^8 - 4*a^8*c^5*x^7 - 4*a^7*c^5*x^6 + 6*a
^6*c^5*x^5 + 6*a^5*c^5*x^4 - 4*a^4*c^5*x^3 - 4*a^3*c^5*x^2 + a^2*c^5*x + a*c^5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{a^{9} x^{9} \sqrt{- a^{2} x^{2} + 1} + a^{8} x^{8} \sqrt{- a^{2} x^{2} + 1} - 4 a^{7} x^{7} \sqrt{- a^{2} x^{2} + 1} - 4 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 6 a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + 6 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 4 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(-a**2*c*x**2+c)**5,x)

[Out]

Integral(1/(a**9*x**9*sqrt(-a**2*x**2 + 1) + a**8*x**8*sqrt(-a**2*x**2 + 1) - 4*a**7*x**7*sqrt(-a**2*x**2 + 1)
- 4*a**6*x**6*sqrt(-a**2*x**2 + 1) + 6*a**5*x**5*sqrt(-a**2*x**2 + 1) + 6*a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*
a**3*x**3*sqrt(-a**2*x**2 + 1) - 4*a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2
+ 1)), x)/c**5

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{\sqrt{-a^{2} x^{2} + 1}}{{\left (a^{2} c x^{2} - c\right )}^{5}{\left (a x + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^5,x, algorithm="giac")

[Out]

integrate(-sqrt(-a^2*x^2 + 1)/((a^2*c*x^2 - c)^5*(a*x + 1)), x)