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

Optimal. Leaf size=129 \[ \frac{2 \left (1-a^2 x^2\right )^{3/2}}{315 a c^5 (1-a x)^3}+\frac{2 \left (1-a^2 x^2\right )^{3/2}}{105 a c^5 (1-a x)^4}+\frac{\left (1-a^2 x^2\right )^{3/2}}{21 a c^5 (1-a x)^5}+\frac{\left (1-a^2 x^2\right )^{3/2}}{9 a c^5 (1-a x)^6} \]

[Out]

(1 - a^2*x^2)^(3/2)/(9*a*c^5*(1 - a*x)^6) + (1 - a^2*x^2)^(3/2)/(21*a*c^5*(1 - a*x)^5) + (2*(1 - a^2*x^2)^(3/2
))/(105*a*c^5*(1 - a*x)^4) + (2*(1 - a^2*x^2)^(3/2))/(315*a*c^5*(1 - a*x)^3)

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Rubi [A]  time = 0.0910225, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {6127, 659, 651} \[ \frac{2 \left (1-a^2 x^2\right )^{3/2}}{315 a c^5 (1-a x)^3}+\frac{2 \left (1-a^2 x^2\right )^{3/2}}{105 a c^5 (1-a x)^4}+\frac{\left (1-a^2 x^2\right )^{3/2}}{21 a c^5 (1-a x)^5}+\frac{\left (1-a^2 x^2\right )^{3/2}}{9 a c^5 (1-a x)^6} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]/(c - a*c*x)^5,x]

[Out]

(1 - a^2*x^2)^(3/2)/(9*a*c^5*(1 - a*x)^6) + (1 - a^2*x^2)^(3/2)/(21*a*c^5*(1 - a*x)^5) + (2*(1 - a^2*x^2)^(3/2
))/(105*a*c^5*(1 - a*x)^4) + (2*(1 - a^2*x^2)^(3/2))/(315*a*c^5*(1 - a*x)^3)

Rule 6127

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

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 651

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*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

Rubi steps

\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{(c-a c x)^5} \, dx &=c \int \frac{\sqrt{1-a^2 x^2}}{(c-a c x)^6} \, dx\\ &=\frac{\left (1-a^2 x^2\right )^{3/2}}{9 a c^5 (1-a x)^6}+\frac{1}{3} \int \frac{\sqrt{1-a^2 x^2}}{(c-a c x)^5} \, dx\\ &=\frac{\left (1-a^2 x^2\right )^{3/2}}{9 a c^5 (1-a x)^6}+\frac{\left (1-a^2 x^2\right )^{3/2}}{21 a c^5 (1-a x)^5}+\frac{2 \int \frac{\sqrt{1-a^2 x^2}}{(c-a c x)^4} \, dx}{21 c}\\ &=\frac{\left (1-a^2 x^2\right )^{3/2}}{9 a c^5 (1-a x)^6}+\frac{\left (1-a^2 x^2\right )^{3/2}}{21 a c^5 (1-a x)^5}+\frac{2 \left (1-a^2 x^2\right )^{3/2}}{105 a c^5 (1-a x)^4}+\frac{2 \int \frac{\sqrt{1-a^2 x^2}}{(c-a c x)^3} \, dx}{105 c^2}\\ &=\frac{\left (1-a^2 x^2\right )^{3/2}}{9 a c^5 (1-a x)^6}+\frac{\left (1-a^2 x^2\right )^{3/2}}{21 a c^5 (1-a x)^5}+\frac{2 \left (1-a^2 x^2\right )^{3/2}}{105 a c^5 (1-a x)^4}+\frac{2 \left (1-a^2 x^2\right )^{3/2}}{315 a c^5 (1-a x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0246279, size = 51, normalized size = 0.4 \[ \frac{(a x+1)^{3/2} \left (-2 a^3 x^3+12 a^2 x^2-33 a x+58\right )}{315 a c^5 (1-a x)^{9/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((1 + a*x)^(3/2)*(58 - 33*a*x + 12*a^2*x^2 - 2*a^3*x^3))/(315*a*c^5*(1 - a*x)^(9/2))

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Maple [A]  time = 0.035, size = 57, normalized size = 0.4 \begin{align*} -{\frac{ \left ( 2\,{x}^{3}{a}^{3}-12\,{a}^{2}{x}^{2}+33\,ax-58 \right ) \left ( ax+1 \right ) ^{2}}{315\,{c}^{5} \left ( ax-1 \right ) ^{4}a}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \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*c*x+c)^5,x)

[Out]

-1/315*(2*a^3*x^3-12*a^2*x^2+33*a*x-58)*(a*x+1)^2/(a*x-1)^4/c^5/(-a^2*x^2+1)^(1/2)/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)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.70595, size = 319, normalized size = 2.47 \begin{align*} \frac{58 \, a^{5} x^{5} - 290 \, a^{4} x^{4} + 580 \, a^{3} x^{3} - 580 \, a^{2} x^{2} + 290 \, a x +{\left (2 \, a^{4} x^{4} - 10 \, a^{3} x^{3} + 21 \, a^{2} x^{2} - 25 \, a x - 58\right )} \sqrt{-a^{2} x^{2} + 1} - 58}{315 \,{\left (a^{6} c^{5} x^{5} - 5 \, a^{5} c^{5} x^{4} + 10 \, a^{4} c^{5} x^{3} - 10 \, a^{3} c^{5} x^{2} + 5 \, 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*c*x+c)^5,x, algorithm="fricas")

[Out]

1/315*(58*a^5*x^5 - 290*a^4*x^4 + 580*a^3*x^3 - 580*a^2*x^2 + 290*a*x + (2*a^4*x^4 - 10*a^3*x^3 + 21*a^2*x^2 -
25*a*x - 58)*sqrt(-a^2*x^2 + 1) - 58)/(a^6*c^5*x^5 - 5*a^5*c^5*x^4 + 10*a^4*c^5*x^3 - 10*a^3*c^5*x^2 + 5*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^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 10 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 10 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 5 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 10 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 10 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 5 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((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a*c*x+c)**5,x)

[Out]

-(Integral(a*x/(a**5*x**5*sqrt(-a**2*x**2 + 1) - 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 10*a**3*x**3*sqrt(-a**2*x*
*2 + 1) - 10*a**2*x**2*sqrt(-a**2*x**2 + 1) + 5*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integra
l(1/(a**5*x**5*sqrt(-a**2*x**2 + 1) - 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 10*a**3*x**3*sqrt(-a**2*x**2 + 1) - 1
0*a**2*x**2*sqrt(-a**2*x**2 + 1) + 5*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{a x + 1}{\sqrt{-a^{2} x^{2} + 1}{\left (a c x - c\right )}^{5}}\,{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*c*x+c)^5,x, algorithm="giac")

[Out]

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