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

Optimal. Leaf size=133 $\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{9 c^5 \left (a-\frac{1}{x}\right )^6}-\frac{8 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{21 c^5 \left (a-\frac{1}{x}\right )^5}+\frac{47 a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{105 c^5 \left (a-\frac{1}{x}\right )^4}-\frac{58 a^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{315 c^5 \left (a-\frac{1}{x}\right )^3}$

[Out]

(a^5*(1 - 1/(a^2*x^2))^(3/2))/(9*c^5*(a - x^(-1))^6) - (8*a^4*(1 - 1/(a^2*x^2))^(3/2))/(21*c^5*(a - x^(-1))^5)
+ (47*a^3*(1 - 1/(a^2*x^2))^(3/2))/(105*c^5*(a - x^(-1))^4) - (58*a^2*(1 - 1/(a^2*x^2))^(3/2))/(315*c^5*(a -
x^(-1))^3)

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Rubi [A]  time = 0.34239, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 16, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.375, Rules used = {6175, 6178, 1639, 793, 659, 651} $\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{9 c^5 \left (a-\frac{1}{x}\right )^6}-\frac{8 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{21 c^5 \left (a-\frac{1}{x}\right )^5}+\frac{47 a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{105 c^5 \left (a-\frac{1}{x}\right )^4}-\frac{58 a^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{315 c^5 \left (a-\frac{1}{x}\right )^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^ArcCoth[a*x]/(c - a*c*x)^5,x]

[Out]

(a^5*(1 - 1/(a^2*x^2))^(3/2))/(9*c^5*(a - x^(-1))^6) - (8*a^4*(1 - 1/(a^2*x^2))^(3/2))/(21*c^5*(a - x^(-1))^5)
+ (47*a^3*(1 - 1/(a^2*x^2))^(3/2))/(105*c^5*(a - x^(-1))^4) - (58*a^2*(1 - 1/(a^2*x^2))^(3/2))/(315*c^5*(a -
x^(-1))^3)

Rule 6175

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

Rule 6178

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_.), x_Symbol] :> -Dist[c^n, Subst[Int[((c
+ d*x)^(p - n)*(1 - x^2/a^2)^(n/2))/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] &&
IntegerQ[(n - 1)/2] && IntegerQ[m] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p, n/2 + 1] || LtQ[-5, m, -1]) && In
tegerQ[2*p]

Rule 1639

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + c*x^2)^(p + 1))/(c*e^(q - 1)*(m + q + 2*p + 1)), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - 2*e*f*(m + p + q)*(d + e*x)^(q - 2)*(a*e - c*d*x), x], x], x] /; NeQ[m + q +
2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

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^{\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx &=-\frac{\int \frac{e^{\coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right )^5 x^5} \, dx}{a^5 c^5}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^3 \sqrt{1-\frac{x^2}{a^2}}}{\left (1-\frac{x}{a}\right )^6} \, dx,x,\frac{1}{x}\right )}{a^5 c^5}\\ &=\frac{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{c^5 \left (a-\frac{1}{x}\right )^4}-\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{4}{a^2}-\frac{7 x}{a^3}+\frac{2 x^2}{a^4}\right ) \sqrt{1-\frac{x^2}{a^2}}}{\left (1-\frac{x}{a}\right )^6} \, dx,x,\frac{1}{x}\right )}{c^5}\\ &=\frac{a^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{c^5 \left (a-\frac{1}{x}\right )^5}+\frac{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{c^5 \left (a-\frac{1}{x}\right )^4}-\frac{a^4 \operatorname{Subst}\left (\int \frac{\left (\frac{18}{a^6}-\frac{20 x}{a^7}\right ) \sqrt{1-\frac{x^2}{a^2}}}{\left (1-\frac{x}{a}\right )^6} \, dx,x,\frac{1}{x}\right )}{2 c^5}\\ &=\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{9 c^5 \left (a-\frac{1}{x}\right )^6}+\frac{a^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{c^5 \left (a-\frac{1}{x}\right )^5}+\frac{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{c^5 \left (a-\frac{1}{x}\right )^4}-\frac{29 \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x^2}{a^2}}}{\left (1-\frac{x}{a}\right )^5} \, dx,x,\frac{1}{x}\right )}{3 a^2 c^5}\\ &=\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{9 c^5 \left (a-\frac{1}{x}\right )^6}-\frac{8 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{21 c^5 \left (a-\frac{1}{x}\right )^5}+\frac{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{c^5 \left (a-\frac{1}{x}\right )^4}-\frac{58 \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x^2}{a^2}}}{\left (1-\frac{x}{a}\right )^4} \, dx,x,\frac{1}{x}\right )}{21 a^2 c^5}\\ &=\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{9 c^5 \left (a-\frac{1}{x}\right )^6}-\frac{8 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{21 c^5 \left (a-\frac{1}{x}\right )^5}+\frac{47 a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{105 c^5 \left (a-\frac{1}{x}\right )^4}-\frac{58 \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x^2}{a^2}}}{\left (1-\frac{x}{a}\right )^3} \, dx,x,\frac{1}{x}\right )}{105 a^2 c^5}\\ &=\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{9 c^5 \left (a-\frac{1}{x}\right )^6}-\frac{8 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{21 c^5 \left (a-\frac{1}{x}\right )^5}+\frac{47 a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{105 c^5 \left (a-\frac{1}{x}\right )^4}-\frac{58 a^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{315 c^5 \left (a-\frac{1}{x}\right )^3}\\ \end{align*}

Mathematica [A]  time = 0.0616556, size = 59, normalized size = 0.44 $-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a^4 x^4-10 a^3 x^3+21 a^2 x^2-25 a x-58\right )}{315 c^5 (a x-1)^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(Sqrt[1 - 1/(a^2*x^2)]*x*(-58 - 25*a*x + 21*a^2*x^2 - 10*a^3*x^3 + 2*a^4*x^4))/(315*c^5*(-1 + a*x)^5)

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Maple [A]  time = 0.046, size = 58, 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 ) }{315\,{c}^{5} \left ( ax-1 \right ) ^{4}a}{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [A]  time = 1.02514, size = 96, normalized size = 0.72 \begin{align*} -\frac{\frac{135 \,{\left (a x - 1\right )}}{a x + 1} - \frac{189 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{105 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 35}{2520 \, a c^{5} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^5,x, algorithm="maxima")

[Out]

-1/2520*(135*(a*x - 1)/(a*x + 1) - 189*(a*x - 1)^2/(a*x + 1)^2 + 105*(a*x - 1)^3/(a*x + 1)^3 - 35)/(a*c^5*((a*
x - 1)/(a*x + 1))^(9/2))

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Fricas [A]  time = 1.59009, size = 247, normalized size = 1.86 \begin{align*} -\frac{{\left (2 \, a^{5} x^{5} - 8 \, a^{4} x^{4} + 11 \, a^{3} x^{3} - 4 \, a^{2} x^{2} - 83 \, a x - 58\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{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(1/((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^5,x, algorithm="fricas")

[Out]

-1/315*(2*a^5*x^5 - 8*a^4*x^4 + 11*a^3*x^3 - 4*a^2*x^2 - 83*a*x - 58)*sqrt((a*x - 1)/(a*x + 1))/(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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.14683, size = 115, normalized size = 0.86 \begin{align*} -\frac{{\left (a x + 1\right )}^{4}{\left (\frac{135 \,{\left (a x - 1\right )}}{a x + 1} - \frac{189 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{105 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 35\right )}}{2520 \,{\left (a x - 1\right )}^{4} a c^{5} \sqrt{\frac{a x - 1}{a x + 1}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^5,x, algorithm="giac")

[Out]

-1/2520*(a*x + 1)^4*(135*(a*x - 1)/(a*x + 1) - 189*(a*x - 1)^2/(a*x + 1)^2 + 105*(a*x - 1)^3/(a*x + 1)^3 - 35)
/((a*x - 1)^4*a*c^5*sqrt((a*x - 1)/(a*x + 1)))