3.185 \(\int \frac{e^{3 \tanh ^{-1}(a x)}}{(c-a c x)^4} \, dx\)

Optimal. Leaf size=97 \[ \frac{2 \left (1-a^2 x^2\right )^{5/2}}{315 a c^4 (1-a x)^5}+\frac{2 \left (1-a^2 x^2\right )^{5/2}}{63 a c^4 (1-a x)^6}+\frac{\left (1-a^2 x^2\right )^{5/2}}{9 a c^4 (1-a x)^7} \]

[Out]

(1 - a^2*x^2)^(5/2)/(9*a*c^4*(1 - a*x)^7) + (2*(1 - a^2*x^2)^(5/2))/(63*a*c^4*(1 - a*x)^6) + (2*(1 - a^2*x^2)^
(5/2))/(315*a*c^4*(1 - a*x)^5)

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Rubi [A]  time = 0.069434, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6127, 659, 651} \[ \frac{2 \left (1-a^2 x^2\right )^{5/2}}{315 a c^4 (1-a x)^5}+\frac{2 \left (1-a^2 x^2\right )^{5/2}}{63 a c^4 (1-a x)^6}+\frac{\left (1-a^2 x^2\right )^{5/2}}{9 a c^4 (1-a x)^7} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])/(c - a*c*x)^4,x]

[Out]

(1 - a^2*x^2)^(5/2)/(9*a*c^4*(1 - a*x)^7) + (2*(1 - a^2*x^2)^(5/2))/(63*a*c^4*(1 - a*x)^6) + (2*(1 - a^2*x^2)^
(5/2))/(315*a*c^4*(1 - a*x)^5)

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^{3 \tanh ^{-1}(a x)}}{(c-a c x)^4} \, dx &=c^3 \int \frac{\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^7} \, dx\\ &=\frac{\left (1-a^2 x^2\right )^{5/2}}{9 a c^4 (1-a x)^7}+\frac{1}{9} \left (2 c^2\right ) \int \frac{\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^6} \, dx\\ &=\frac{\left (1-a^2 x^2\right )^{5/2}}{9 a c^4 (1-a x)^7}+\frac{2 \left (1-a^2 x^2\right )^{5/2}}{63 a c^4 (1-a x)^6}+\frac{1}{63} (2 c) \int \frac{\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^5} \, dx\\ &=\frac{\left (1-a^2 x^2\right )^{5/2}}{9 a c^4 (1-a x)^7}+\frac{2 \left (1-a^2 x^2\right )^{5/2}}{63 a c^4 (1-a x)^6}+\frac{2 \left (1-a^2 x^2\right )^{5/2}}{315 a c^4 (1-a x)^5}\\ \end{align*}

Mathematica [A]  time = 0.0244797, size = 43, normalized size = 0.44 \[ \frac{(a x+1)^{5/2} \left (2 a^2 x^2-14 a x+47\right )}{315 a c^4 (1-a x)^{9/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(3*ArcTanh[a*x])/(c - a*c*x)^4,x]

[Out]

((1 + a*x)^(5/2)*(47 - 14*a*x + 2*a^2*x^2))/(315*a*c^4*(1 - a*x)^(9/2))

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Maple [A]  time = 0.033, size = 49, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,{a}^{2}{x}^{2}-14\,ax+47 \right ) \left ( ax+1 \right ) ^{4}}{315\,{c}^{4} \left ( ax-1 \right ) ^{3}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*c*x+c)^4,x)

[Out]

-1/315*(2*a^2*x^2-14*a*x+47)*(a*x+1)^4/(a*x-1)^3/c^4/(-a^2*x^2+1)^(3/2)/a

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Maxima [B]  time = 1.0056, size = 441, normalized size = 4.55 \begin{align*} \frac{8}{9 \,{\left (\sqrt{-a^{2} x^{2} + 1} a^{5} c^{4} x^{4} - 4 \, \sqrt{-a^{2} x^{2} + 1} a^{4} c^{4} x^{3} + 6 \, \sqrt{-a^{2} x^{2} + 1} a^{3} c^{4} x^{2} - 4 \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{4} x + \sqrt{-a^{2} x^{2} + 1} a c^{4}\right )}} + \frac{68}{63 \,{\left (\sqrt{-a^{2} x^{2} + 1} a^{4} c^{4} x^{3} - 3 \, \sqrt{-a^{2} x^{2} + 1} a^{3} c^{4} x^{2} + 3 \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{4} x - \sqrt{-a^{2} x^{2} + 1} a c^{4}\right )}} + \frac{106}{315 \,{\left (\sqrt{-a^{2} x^{2} + 1} a^{3} c^{4} x^{2} - 2 \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{4} x + \sqrt{-a^{2} x^{2} + 1} a c^{4}\right )}} - \frac{1}{315 \,{\left (\sqrt{-a^{2} x^{2} + 1} a^{2} c^{4} x - \sqrt{-a^{2} x^{2} + 1} a c^{4}\right )}} + \frac{2 \, x}{315 \, \sqrt{-a^{2} x^{2} + 1} c^{4}} \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*c*x+c)^4,x, algorithm="maxima")

[Out]

8/9/(sqrt(-a^2*x^2 + 1)*a^5*c^4*x^4 - 4*sqrt(-a^2*x^2 + 1)*a^4*c^4*x^3 + 6*sqrt(-a^2*x^2 + 1)*a^3*c^4*x^2 - 4*
sqrt(-a^2*x^2 + 1)*a^2*c^4*x + sqrt(-a^2*x^2 + 1)*a*c^4) + 68/63/(sqrt(-a^2*x^2 + 1)*a^4*c^4*x^3 - 3*sqrt(-a^2
*x^2 + 1)*a^3*c^4*x^2 + 3*sqrt(-a^2*x^2 + 1)*a^2*c^4*x - sqrt(-a^2*x^2 + 1)*a*c^4) + 106/315/(sqrt(-a^2*x^2 +
1)*a^3*c^4*x^2 - 2*sqrt(-a^2*x^2 + 1)*a^2*c^4*x + sqrt(-a^2*x^2 + 1)*a*c^4) - 1/315/(sqrt(-a^2*x^2 + 1)*a^2*c^
4*x - sqrt(-a^2*x^2 + 1)*a*c^4) + 2/315*x/(sqrt(-a^2*x^2 + 1)*c^4)

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Fricas [A]  time = 1.76256, size = 319, normalized size = 3.29 \begin{align*} \frac{47 \, a^{5} x^{5} - 235 \, a^{4} x^{4} + 470 \, a^{3} x^{3} - 470 \, a^{2} x^{2} + 235 \, a x -{\left (2 \, a^{4} x^{4} - 10 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 80 \, a x + 47\right )} \sqrt{-a^{2} x^{2} + 1} - 47}{315 \,{\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, 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)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^4,x, algorithm="fricas")

[Out]

1/315*(47*a^5*x^5 - 235*a^4*x^4 + 470*a^3*x^3 - 470*a^2*x^2 + 235*a*x - (2*a^4*x^4 - 10*a^3*x^3 + 21*a^2*x^2 +
80*a*x + 47)*sqrt(-a^2*x^2 + 1) - 47)/(a^6*c^4*x^5 - 5*a^5*c^4*x^4 + 10*a^4*c^4*x^3 - 10*a^3*c^4*x^2 + 5*a^2*
c^4*x - a*c^4)

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

[Out]

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

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Giac [B]  time = 1.20365, size = 342, normalized size = 3.53 \begin{align*} -\frac{2 \,{\left (\frac{108 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{a^{2} x} - \frac{1062 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac{1638 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac{3402 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} + \frac{2520 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5}}{a^{10} x^{5}} - \frac{2310 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{6}}{a^{12} x^{6}} + \frac{630 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{7}}{a^{14} x^{7}} - \frac{315 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{8}}{a^{16} x^{8}} - 47\right )}}{315 \, c^{4}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}^{9}{\left | a \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*c*x+c)^4,x, algorithm="giac")

[Out]

-2/315*(108*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1062*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/(a^4*x^2) + 1638*
(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3/(a^6*x^3) - 3402*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4/(a^8*x^4) + 2520*(sqrt(-a
^2*x^2 + 1)*abs(a) + a)^5/(a^10*x^5) - 2310*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^6/(a^12*x^6) + 630*(sqrt(-a^2*x^2
+ 1)*abs(a) + a)^7/(a^14*x^7) - 315*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^8/(a^16*x^8) - 47)/(c^4*((sqrt(-a^2*x^2 +
1)*abs(a) + a)/(a^2*x) - 1)^9*abs(a))