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

Optimal. Leaf size=97 \[ -\frac{\left (1-a^2 x^2\right )^{3/2}}{21 a^2 c^4 (1-a x)^3}-\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a^2 c^4 (1-a x)^4}+\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a^2 c^4 (1-a x)^5} \]

[Out]

(1 - a^2*x^2)^(3/2)/(7*a^2*c^4*(1 - a*x)^5) - (1 - a^2*x^2)^(3/2)/(7*a^2*c^4*(1 - a*x)^4) - (1 - a^2*x^2)^(3/2
)/(21*a^2*c^4*(1 - a*x)^3)

________________________________________________________________________________________

Rubi [A]  time = 0.100774, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {6128, 793, 659, 651} \[ -\frac{\left (1-a^2 x^2\right )^{3/2}}{21 a^2 c^4 (1-a x)^3}-\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a^2 c^4 (1-a x)^4}+\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a^2 c^4 (1-a x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 - a^2*x^2)^(3/2)/(7*a^2*c^4*(1 - a*x)^5) - (1 - a^2*x^2)^(3/2)/(7*a^2*c^4*(1 - a*x)^4) - (1 - a^2*x^2)^(3/2
)/(21*a^2*c^4*(1 - a*x)^3)

Rule 6128

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

Rule 793

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d*g - e*f)*(
d + e*x)^m*(a + c*x^2)^(p + 1))/(2*c*d*(m + p + 1)), x] + Dist[(m*(g*c*d + c*e*f) + 2*e*c*f*(p + 1))/(e*(2*c*d
)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2
+ a*e^2, 0] && ((LtQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) &&
NeQ[m + p + 1, 0]

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)} x}{(c-a c x)^4} \, dx &=c \int \frac{x \sqrt{1-a^2 x^2}}{(c-a c x)^5} \, dx\\ &=\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a^2 c^4 (1-a x)^5}-\frac{5 \int \frac{\sqrt{1-a^2 x^2}}{(c-a c x)^4} \, dx}{7 a}\\ &=\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a^2 c^4 (1-a x)^5}-\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a^2 c^4 (1-a x)^4}-\frac{\int \frac{\sqrt{1-a^2 x^2}}{(c-a c x)^3} \, dx}{7 a c}\\ &=\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a^2 c^4 (1-a x)^5}-\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a^2 c^4 (1-a x)^4}-\frac{\left (1-a^2 x^2\right )^{3/2}}{21 a^2 c^4 (1-a x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0192582, size = 42, normalized size = 0.43 \[ -\frac{(a x+1)^{3/2} \left (a^2 x^2-5 a x+1\right )}{21 a^2 c^4 (1-a x)^{7/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((1 + a*x)^(3/2)*(1 - 5*a*x + a^2*x^2))/(21*a^2*c^4*(1 - a*x)^(7/2))

________________________________________________________________________________________

Maple [A]  time = 0.038, size = 48, normalized size = 0.5 \begin{align*}{\frac{ \left ({a}^{2}{x}^{2}-5\,ax+1 \right ) \left ( ax+1 \right ) ^{2}}{21\,{c}^{4} \left ( ax-1 \right ) ^{3}{a}^{2}}{\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)*x/(-a*c*x+c)^4,x)

[Out]

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

________________________________________________________________________________________

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)*x/(-a*c*x+c)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.88299, size = 240, normalized size = 2.47 \begin{align*} -\frac{a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x +{\left (a^{3} x^{3} - 4 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt{-a^{2} x^{2} + 1} + 1}{21 \,{\left (a^{6} c^{4} x^{4} - 4 \, a^{5} c^{4} x^{3} + 6 \, a^{4} c^{4} x^{2} - 4 \, a^{3} c^{4} x + a^{2} 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)*x/(-a*c*x+c)^4,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

[Out]

(Integral(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(a*x**2/(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

________________________________________________________________________________________

Giac [A]  time = 1.19962, size = 200, normalized size = 2.06 \begin{align*} \frac{2 \,{\left (\frac{7 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{a^{2} x} + \frac{28 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac{7 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} + \frac{21 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5}}{a^{10} x^{5}} - 1\right )}}{21 \, a 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)*x/(-a*c*x+c)^4,x, algorithm="giac")

[Out]

2/21*(7*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) + 28*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3/(a^6*x^3) - 7*(sqrt(-a^
2*x^2 + 1)*abs(a) + a)^4/(a^8*x^4) + 21*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5/(a^10*x^5) - 1)/(a*c^4*((sqrt(-a^2*x
^2 + 1)*abs(a) + a)/(a^2*x) - 1)^7*abs(a))