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

Optimal. Leaf size=118 \[ \frac{128 x}{315 c^5 \sqrt{1-a^2 x^2}}+\frac{64 x}{315 c^5 \left (1-a^2 x^2\right )^{3/2}}+\frac{16 x}{105 c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac{8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}+\frac{a x+1}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}} \]

[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 = 0.0641603, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6138, 639, 192, 191} \[ \frac{128 x}{315 c^5 \sqrt{1-a^2 x^2}}+\frac{64 x}{315 c^5 \left (1-a^2 x^2\right )^{3/2}}+\frac{16 x}{105 c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac{8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}+\frac{a x+1}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[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 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 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 = 0.0314079, size = 91, normalized size = 0.77 \[ \frac{128 a^8 x^8-128 a^7 x^7-448 a^6 x^6+448 a^5 x^5+560 a^4 x^4-560 a^3 x^3-280 a^2 x^2+280 a x+35}{315 a c^5 (1-a x)^{9/2} (a x+1)^{7/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[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)^(9/2)*(1 + a*x)^(7/2))

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

[Out]

-1/315*(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^2*x^2+1)^(7/2)/a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.02386, size = 554, normalized size = 4.69 \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((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{a x}{- a^{10} x^{10} \sqrt{- a^{2} x^{2} + 1} + 5 a^{8} x^{8} \sqrt{- a^{2} x^{2} + 1} - 10 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 10 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 5 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{10} x^{10} \sqrt{- a^{2} x^{2} + 1} + 5 a^{8} x^{8} \sqrt{- a^{2} x^{2} + 1} - 10 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 10 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 5 a^{2} x^{2} \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((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a**2*c*x**2+c)**5,x)

[Out]

(Integral(a*x/(-a**10*x**10*sqrt(-a**2*x**2 + 1) + 5*a**8*x**8*sqrt(-a**2*x**2 + 1) - 10*a**6*x**6*sqrt(-a**2*
x**2 + 1) + 10*a**4*x**4*sqrt(-a**2*x**2 + 1) - 5*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) +
Integral(1/(-a**10*x**10*sqrt(-a**2*x**2 + 1) + 5*a**8*x**8*sqrt(-a**2*x**2 + 1) - 10*a**6*x**6*sqrt(-a**2*x*
*2 + 1) + 10*a**4*x**4*sqrt(-a**2*x**2 + 1) - 5*a**2*x**2*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{a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{5} \sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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