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

Optimal. Leaf size=65 \[ \frac{\left (1-a^2 x^2\right )^{5/2}}{35 a c^3 (1-a x)^5}+\frac{\left (1-a^2 x^2\right )^{5/2}}{7 a c^3 (1-a x)^6} \]

[Out]

(1 - a^2*x^2)^(5/2)/(7*a*c^3*(1 - a*x)^6) + (1 - a^2*x^2)^(5/2)/(35*a*c^3*(1 - a*x)^5)

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Rubi [A]  time = 0.0514531, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6127, 659, 651} \[ \frac{\left (1-a^2 x^2\right )^{5/2}}{35 a c^3 (1-a x)^5}+\frac{\left (1-a^2 x^2\right )^{5/2}}{7 a c^3 (1-a x)^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 - a^2*x^2)^(5/2)/(7*a*c^3*(1 - a*x)^6) + (1 - a^2*x^2)^(5/2)/(35*a*c^3*(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)^3} \, dx &=c^3 \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}}{7 a c^3 (1-a x)^6}+\frac{1}{7} c^2 \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}}{7 a c^3 (1-a x)^6}+\frac{\left (1-a^2 x^2\right )^{5/2}}{35 a c^3 (1-a x)^5}\\ \end{align*}

Mathematica [A]  time = 0.0183458, size = 34, normalized size = 0.52 \[ -\frac{(a x-6) (a x+1)^{5/2}}{35 a c^3 (1-a x)^{7/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((-6 + a*x)*(1 + a*x)^(5/2))/(35*a*c^3*(1 - a*x)^(7/2))

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Maple [A]  time = 0.033, size = 40, normalized size = 0.6 \begin{align*} -{\frac{ \left ( ax-6 \right ) \left ( ax+1 \right ) ^{4}}{35\,{c}^{3} \left ( ax-1 \right ) ^{2}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)^3,x)

[Out]

-1/35*(a*x-6)*(a*x+1)^4/(a*x-1)^2/c^3/(-a^2*x^2+1)^(3/2)/a

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Maxima [B]  time = 0.994151, size = 292, normalized size = 4.49 \begin{align*} -\frac{8}{7 \,{\left (\sqrt{-a^{2} x^{2} + 1} a^{4} c^{3} x^{3} - 3 \, \sqrt{-a^{2} x^{2} + 1} a^{3} c^{3} x^{2} + 3 \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{3} x - \sqrt{-a^{2} x^{2} + 1} a c^{3}\right )}} - \frac{52}{35 \,{\left (\sqrt{-a^{2} x^{2} + 1} a^{3} c^{3} x^{2} - 2 \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{3} x + \sqrt{-a^{2} x^{2} + 1} a c^{3}\right )}} - \frac{18}{35 \,{\left (\sqrt{-a^{2} x^{2} + 1} a^{2} c^{3} x - \sqrt{-a^{2} x^{2} + 1} a c^{3}\right )}} + \frac{x}{35 \, \sqrt{-a^{2} x^{2} + 1} c^{3}} \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)^3,x, algorithm="maxima")

[Out]

-8/7/(sqrt(-a^2*x^2 + 1)*a^4*c^3*x^3 - 3*sqrt(-a^2*x^2 + 1)*a^3*c^3*x^2 + 3*sqrt(-a^2*x^2 + 1)*a^2*c^3*x - sqr
t(-a^2*x^2 + 1)*a*c^3) - 52/35/(sqrt(-a^2*x^2 + 1)*a^3*c^3*x^2 - 2*sqrt(-a^2*x^2 + 1)*a^2*c^3*x + sqrt(-a^2*x^
2 + 1)*a*c^3) - 18/35/(sqrt(-a^2*x^2 + 1)*a^2*c^3*x - sqrt(-a^2*x^2 + 1)*a*c^3) + 1/35*x/(sqrt(-a^2*x^2 + 1)*c
^3)

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Fricas [B]  time = 1.81045, size = 244, normalized size = 3.75 \begin{align*} \frac{6 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 36 \, a^{2} x^{2} - 24 \, a x -{\left (a^{3} x^{3} - 4 \, a^{2} x^{2} - 11 \, a x - 6\right )} \sqrt{-a^{2} x^{2} + 1} + 6}{35 \,{\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\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)^3,x, algorithm="fricas")

[Out]

1/35*(6*a^4*x^4 - 24*a^3*x^3 + 36*a^2*x^2 - 24*a*x - (a^3*x^3 - 4*a^2*x^2 - 11*a*x - 6)*sqrt(-a^2*x^2 + 1) + 6
)/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^3)

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

[Out]

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

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Giac [B]  time = 1.27725, size = 269, normalized size = 4.14 \begin{align*} -\frac{2 \,{\left (\frac{7 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{a^{2} x} - \frac{91 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac{70 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac{140 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} + \frac{35 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5}}{a^{10} x^{5}} - \frac{35 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{6}}{a^{12} x^{6}} - 6\right )}}{35 \, c^{3}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}^{7}{\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)^3,x, algorithm="giac")

[Out]

-2/35*(7*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 91*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/(a^4*x^2) + 70*(sqrt(-
a^2*x^2 + 1)*abs(a) + a)^3/(a^6*x^3) - 140*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4/(a^8*x^4) + 35*(sqrt(-a^2*x^2 + 1
)*abs(a) + a)^5/(a^10*x^5) - 35*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^6/(a^12*x^6) - 6)/(c^3*((sqrt(-a^2*x^2 + 1)*ab
s(a) + a)/(a^2*x) - 1)^7*abs(a))