### 3.580 $$\int \frac{e^{3 \coth ^{-1}(a x)}}{(c-a^2 c x^2)^4} \, dx$$

Optimal. Leaf size=127 $-\frac{10 (3-4 a x) e^{3 \coth ^{-1}(a x)}}{63 a c^4 \left (1-a^2 x^2\right )^2}+\frac{8 (3-2 a x) e^{3 \coth ^{-1}(a x)}}{21 a c^4 \left (1-a^2 x^2\right )}-\frac{(1-2 a x) e^{3 \coth ^{-1}(a x)}}{9 a c^4 \left (1-a^2 x^2\right )^3}-\frac{16 e^{3 \coth ^{-1}(a x)}}{63 a c^4}$

[Out]

(-16*E^(3*ArcCoth[a*x]))/(63*a*c^4) - (E^(3*ArcCoth[a*x])*(1 - 2*a*x))/(9*a*c^4*(1 - a^2*x^2)^3) - (10*E^(3*Ar
cCoth[a*x])*(3 - 4*a*x))/(63*a*c^4*(1 - a^2*x^2)^2) + (8*E^(3*ArcCoth[a*x])*(3 - 2*a*x))/(21*a*c^4*(1 - a^2*x^
2))

________________________________________________________________________________________

Rubi [A]  time = 0.138726, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.091, Rules used = {6185, 6183} $-\frac{10 (3-4 a x) e^{3 \coth ^{-1}(a x)}}{63 a c^4 \left (1-a^2 x^2\right )^2}+\frac{8 (3-2 a x) e^{3 \coth ^{-1}(a x)}}{21 a c^4 \left (1-a^2 x^2\right )}-\frac{(1-2 a x) e^{3 \coth ^{-1}(a x)}}{9 a c^4 \left (1-a^2 x^2\right )^3}-\frac{16 e^{3 \coth ^{-1}(a x)}}{63 a c^4}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(3*ArcCoth[a*x])/(c - a^2*c*x^2)^4,x]

[Out]

(-16*E^(3*ArcCoth[a*x]))/(63*a*c^4) - (E^(3*ArcCoth[a*x])*(1 - 2*a*x))/(9*a*c^4*(1 - a^2*x^2)^3) - (10*E^(3*Ar
cCoth[a*x])*(3 - 4*a*x))/(63*a*c^4*(1 - a^2*x^2)^2) + (8*E^(3*ArcCoth[a*x])*(3 - 2*a*x))/(21*a*c^4*(1 - a^2*x^
2))

Rule 6185

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((n + 2*a*(p + 1)*x)*(c + d*x^
2)^(p + 1)*E^(n*ArcCoth[a*x]))/(a*c*(n^2 - 4*(p + 1)^2)), x] - Dist[(2*(p + 1)*(2*p + 3))/(c*(n^2 - 4*(p + 1)^
2)), Int[(c + d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !In
tegerQ[n/2] && LtQ[p, -1] && NeQ[p, -3/2] && NeQ[n^2 - 4*(p + 1)^2, 0] && (IntegerQ[p] ||  !IntegerQ[n])

Rule 6183

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[E^(n*ArcCoth[a*x])/(a*c*n), x] /; F
reeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !IntegerQ[n/2]

Rubi steps

\begin{align*} \int \frac{e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx &=-\frac{e^{3 \coth ^{-1}(a x)} (1-2 a x)}{9 a c^4 \left (1-a^2 x^2\right )^3}+\frac{10 \int \frac{e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx}{9 c}\\ &=-\frac{e^{3 \coth ^{-1}(a x)} (1-2 a x)}{9 a c^4 \left (1-a^2 x^2\right )^3}-\frac{10 e^{3 \coth ^{-1}(a x)} (3-4 a x)}{63 a c^4 \left (1-a^2 x^2\right )^2}+\frac{40 \int \frac{e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx}{21 c^2}\\ &=-\frac{e^{3 \coth ^{-1}(a x)} (1-2 a x)}{9 a c^4 \left (1-a^2 x^2\right )^3}-\frac{10 e^{3 \coth ^{-1}(a x)} (3-4 a x)}{63 a c^4 \left (1-a^2 x^2\right )^2}+\frac{8 e^{3 \coth ^{-1}(a x)} (3-2 a x)}{21 a c^4 \left (1-a^2 x^2\right )}-\frac{16 \int \frac{e^{3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{21 c^3}\\ &=-\frac{16 e^{3 \coth ^{-1}(a x)}}{63 a c^4}-\frac{e^{3 \coth ^{-1}(a x)} (1-2 a x)}{9 a c^4 \left (1-a^2 x^2\right )^3}-\frac{10 e^{3 \coth ^{-1}(a x)} (3-4 a x)}{63 a c^4 \left (1-a^2 x^2\right )^2}+\frac{8 e^{3 \coth ^{-1}(a x)} (3-2 a x)}{21 a c^4 \left (1-a^2 x^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.306899, size = 82, normalized size = 0.65 $-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (16 a^6 x^6-48 a^5 x^5+24 a^4 x^4+56 a^3 x^3-66 a^2 x^2+6 a x+19\right )}{63 c^4 (a x-1)^5 (a x+1)^2}$

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(3*ArcCoth[a*x])/(c - a^2*c*x^2)^4,x]

[Out]

-(Sqrt[1 - 1/(a^2*x^2)]*x*(19 + 6*a*x - 66*a^2*x^2 + 56*a^3*x^3 + 24*a^4*x^4 - 48*a^5*x^5 + 16*a^6*x^6))/(63*c
^4*(-1 + a*x)^5*(1 + a*x)^2)

________________________________________________________________________________________

Maple [A]  time = 0.135, size = 81, normalized size = 0.6 \begin{align*} -{\frac{16\,{x}^{6}{a}^{6}-48\,{x}^{5}{a}^{5}+24\,{x}^{4}{a}^{4}+56\,{x}^{3}{a}^{3}-66\,{a}^{2}{x}^{2}+6\,ax+19}{63\,{c}^{4} \left ({a}^{2}{x}^{2}-1 \right ) ^{3}a} \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x-1)/(a*x+1))^(3/2)/(-a^2*c*x^2+c)^4,x)

[Out]

-1/63*(16*a^6*x^6-48*a^5*x^5+24*a^4*x^4+56*a^3*x^3-66*a^2*x^2+6*a*x+19)/(a^2*x^2-1)^3/c^4/((a*x-1)/(a*x+1))^(3
/2)/a

________________________________________________________________________________________

Maxima [A]  time = 1.21075, size = 177, normalized size = 1.39 \begin{align*} \frac{1}{4032} \, a{\left (\frac{21 \,{\left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 18 \, \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{2} c^{4}} + \frac{\frac{54 \,{\left (a x - 1\right )}}{a x + 1} - \frac{189 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{420 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac{945 \,{\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} - 7}{a^{2} c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{2}}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4032*a*(21*(((a*x - 1)/(a*x + 1))^(3/2) - 18*sqrt((a*x - 1)/(a*x + 1)))/(a^2*c^4) + (54*(a*x - 1)/(a*x + 1)
- 189*(a*x - 1)^2/(a*x + 1)^2 + 420*(a*x - 1)^3/(a*x + 1)^3 - 945*(a*x - 1)^4/(a*x + 1)^4 - 7)/(a^2*c^4*((a*x
- 1)/(a*x + 1))^(9/2)))

________________________________________________________________________________________

Fricas [A]  time = 1.53683, size = 263, normalized size = 2.07 \begin{align*} -\frac{{\left (16 \, a^{6} x^{6} - 48 \, a^{5} x^{5} + 24 \, a^{4} x^{4} + 56 \, a^{3} x^{3} - 66 \, a^{2} x^{2} + 6 \, a x + 19\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{63 \,{\left (a^{7} c^{4} x^{6} - 4 \, a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, 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(1/((a*x-1)/(a*x+1))^(3/2)/(-a^2*c*x^2+c)^4,x, algorithm="fricas")

[Out]

-1/63*(16*a^6*x^6 - 48*a^5*x^5 + 24*a^4*x^4 + 56*a^3*x^3 - 66*a^2*x^2 + 6*a*x + 19)*sqrt((a*x - 1)/(a*x + 1))/
(a^7*c^4*x^6 - 4*a^6*c^4*x^5 + 5*a^5*c^4*x^4 - 5*a^3*c^4*x^2 + 4*a^2*c^4*x - a*c^4)

________________________________________________________________________________________

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))**(3/2)/(-a**2*c*x**2+c)**4,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.22066, size = 230, normalized size = 1.81 \begin{align*} \frac{1}{4032} \, a{\left (\frac{{\left (a x + 1\right )}^{4}{\left (\frac{54 \,{\left (a x - 1\right )}}{a x + 1} - \frac{189 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{420 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac{945 \,{\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} - 7\right )}}{{\left (a x - 1\right )}^{4} a^{2} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}} + \frac{21 \,{\left (\frac{{\left (a x - 1\right )} a^{4} c^{8} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} - 18 \, a^{4} c^{8} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{6} c^{12}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)/(-a^2*c*x^2+c)^4,x, algorithm="giac")

[Out]

1/4032*a*((a*x + 1)^4*(54*(a*x - 1)/(a*x + 1) - 189*(a*x - 1)^2/(a*x + 1)^2 + 420*(a*x - 1)^3/(a*x + 1)^3 - 94
5*(a*x - 1)^4/(a*x + 1)^4 - 7)/((a*x - 1)^4*a^2*c^4*sqrt((a*x - 1)/(a*x + 1))) + 21*((a*x - 1)*a^4*c^8*sqrt((a
*x - 1)/(a*x + 1))/(a*x + 1) - 18*a^4*c^8*sqrt((a*x - 1)/(a*x + 1)))/(a^6*c^12))