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

Optimal. Leaf size=97 \[ \frac{2 \left (1-a^2 x^2\right )^{3/2}}{105 a c^4 (1-a x)^3}+\frac{2 \left (1-a^2 x^2\right )^{3/2}}{35 a c^4 (1-a x)^4}+\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a c^4 (1-a x)^5} \]

[Out]

(1 - a^2*x^2)^(3/2)/(7*a*c^4*(1 - a*x)^5) + (2*(1 - a^2*x^2)^(3/2))/(35*a*c^4*(1 - a*x)^4) + (2*(1 - a^2*x^2)^
(3/2))/(105*a*c^4*(1 - a*x)^3)

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Rubi [A]  time = 0.070342, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {6127, 659, 651} \[ \frac{2 \left (1-a^2 x^2\right )^{3/2}}{105 a c^4 (1-a x)^3}+\frac{2 \left (1-a^2 x^2\right )^{3/2}}{35 a c^4 (1-a x)^4}+\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a c^4 (1-a x)^5} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]/(c - a*c*x)^4,x]

[Out]

(1 - a^2*x^2)^(3/2)/(7*a*c^4*(1 - a*x)^5) + (2*(1 - a^2*x^2)^(3/2))/(35*a*c^4*(1 - a*x)^4) + (2*(1 - a^2*x^2)^
(3/2))/(105*a*c^4*(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)^4} \, dx &=c \int \frac{\sqrt{1-a^2 x^2}}{(c-a c x)^5} \, dx\\ &=\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a c^4 (1-a x)^5}+\frac{2}{7} \int \frac{\sqrt{1-a^2 x^2}}{(c-a c x)^4} \, dx\\ &=\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a c^4 (1-a x)^5}+\frac{2 \left (1-a^2 x^2\right )^{3/2}}{35 a c^4 (1-a x)^4}+\frac{2 \int \frac{\sqrt{1-a^2 x^2}}{(c-a c x)^3} \, dx}{35 c}\\ &=\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a c^4 (1-a x)^5}+\frac{2 \left (1-a^2 x^2\right )^{3/2}}{35 a c^4 (1-a x)^4}+\frac{2 \left (1-a^2 x^2\right )^{3/2}}{105 a c^4 (1-a x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0197159, size = 43, normalized size = 0.44 \[ -\frac{(a x+1)^{3/2} \left (-2 a^2 x^2+10 a x-23\right )}{105 a c^4 (1-a x)^{7/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((1 + a*x)^(3/2)*(-23 + 10*a*x - 2*a^2*x^2))/(105*a*c^4*(1 - a*x)^(7/2))

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Maple [A]  time = 0.035, size = 49, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,{a}^{2}{x}^{2}-10\,ax+23 \right ) \left ( ax+1 \right ) ^{2}}{105\, \left ( ax-1 \right ) ^{3}{c}^{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)^4,x)

[Out]

-1/105*(2*a^2*x^2-10*a*x+23)*(a*x+1)^2/(a*x-1)^3/c^4/(-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)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.88557, size = 254, normalized size = 2.62 \begin{align*} \frac{23 \, a^{4} x^{4} - 92 \, a^{3} x^{3} + 138 \, a^{2} x^{2} - 92 \, a x +{\left (2 \, a^{3} x^{3} - 8 \, a^{2} x^{2} + 13 \, a x + 23\right )} \sqrt{-a^{2} x^{2} + 1} + 23}{105 \,{\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, 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)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^4,x, algorithm="fricas")

[Out]

1/105*(23*a^4*x^4 - 92*a^3*x^3 + 138*a^2*x^2 - 92*a*x + (2*a^3*x^3 - 8*a^2*x^2 + 13*a*x + 23)*sqrt(-a^2*x^2 +
1) + 23)/(a^5*c^4*x^4 - 4*a^4*c^4*x^3 + 6*a^3*c^4*x^2 - 4*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{a x}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + 6 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^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + 6 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)/(-a**2*x**2+1)**(1/2)/(-a*c*x+c)**4,x)

[Out]

(Integral(a*x/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**3*x**3*sqrt(-a**2*x**2 + 1) + 6*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**4*x**4*sqrt(-a**2*x**2 + 1) -
4*a**3*x**3*sqrt(-a**2*x**2 + 1) + 6*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 = 1.29471, size = 269, normalized size = 2.77 \begin{align*} -\frac{2 \,{\left (\frac{56 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{a^{2} x} - \frac{273 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac{350 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac{455 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} + \frac{210 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5}}{a^{10} x^{5}} - \frac{105 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{6}}{a^{12} x^{6}} - 23\right )}}{105 \, c^{4}{\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)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^4,x, algorithm="giac")

[Out]

-2/105*(56*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 273*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/(a^4*x^2) + 350*(sq
rt(-a^2*x^2 + 1)*abs(a) + a)^3/(a^6*x^3) - 455*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4/(a^8*x^4) + 210*(sqrt(-a^2*x^
2 + 1)*abs(a) + a)^5/(a^10*x^5) - 105*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^6/(a^12*x^6) - 23)/(c^4*((sqrt(-a^2*x^2
+ 1)*abs(a) + a)/(a^2*x) - 1)^7*abs(a))