3.563 \(\int \frac{e^{\coth ^{-1}(a x)}}{(c-a^2 c x^2)^4} \, dx\)

Optimal. Leaf size=119 \[ -\frac{(1-6 a x) e^{\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )^3}-\frac{8 (1-2 a x) e^{\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )}-\frac{2 (1-4 a x) e^{\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )^2}+\frac{16 e^{\coth ^{-1}(a x)}}{35 a c^4} \]

[Out]

(16*E^ArcCoth[a*x])/(35*a*c^4) - (E^ArcCoth[a*x]*(1 - 6*a*x))/(35*a*c^4*(1 - a^2*x^2)^3) - (2*E^ArcCoth[a*x]*(
1 - 4*a*x))/(35*a*c^4*(1 - a^2*x^2)^2) - (8*E^ArcCoth[a*x]*(1 - 2*a*x))/(35*a*c^4*(1 - a^2*x^2))

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Rubi [A]  time = 0.127988, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {6185, 6183} \[ -\frac{(1-6 a x) e^{\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )^3}-\frac{8 (1-2 a x) e^{\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )}-\frac{2 (1-4 a x) e^{\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )^2}+\frac{16 e^{\coth ^{-1}(a x)}}{35 a c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*E^ArcCoth[a*x])/(35*a*c^4) - (E^ArcCoth[a*x]*(1 - 6*a*x))/(35*a*c^4*(1 - a^2*x^2)^3) - (2*E^ArcCoth[a*x]*(
1 - 4*a*x))/(35*a*c^4*(1 - a^2*x^2)^2) - (8*E^ArcCoth[a*x]*(1 - 2*a*x))/(35*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^{\coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx &=-\frac{e^{\coth ^{-1}(a x)} (1-6 a x)}{35 a c^4 \left (1-a^2 x^2\right )^3}+\frac{6 \int \frac{e^{\coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx}{7 c}\\ &=-\frac{e^{\coth ^{-1}(a x)} (1-6 a x)}{35 a c^4 \left (1-a^2 x^2\right )^3}-\frac{2 e^{\coth ^{-1}(a x)} (1-4 a x)}{35 a c^4 \left (1-a^2 x^2\right )^2}+\frac{24 \int \frac{e^{\coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx}{35 c^2}\\ &=-\frac{e^{\coth ^{-1}(a x)} (1-6 a x)}{35 a c^4 \left (1-a^2 x^2\right )^3}-\frac{2 e^{\coth ^{-1}(a x)} (1-4 a x)}{35 a c^4 \left (1-a^2 x^2\right )^2}-\frac{8 e^{\coth ^{-1}(a x)} (1-2 a x)}{35 a c^4 \left (1-a^2 x^2\right )}+\frac{16 \int \frac{e^{\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{35 c^3}\\ &=\frac{16 e^{\coth ^{-1}(a x)}}{35 a c^4}-\frac{e^{\coth ^{-1}(a x)} (1-6 a x)}{35 a c^4 \left (1-a^2 x^2\right )^3}-\frac{2 e^{\coth ^{-1}(a x)} (1-4 a x)}{35 a c^4 \left (1-a^2 x^2\right )^2}-\frac{8 e^{\coth ^{-1}(a x)} (1-2 a x)}{35 a c^4 \left (1-a^2 x^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.247455, size = 82, normalized size = 0.69 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (16 a^6 x^6-16 a^5 x^5-40 a^4 x^4+40 a^3 x^3+30 a^2 x^2-30 a x-5\right )}{35 c^4 (a x-1)^4 (a x+1)^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[1 - 1/(a^2*x^2)]*x*(-5 - 30*a*x + 30*a^2*x^2 + 40*a^3*x^3 - 40*a^4*x^4 - 16*a^5*x^5 + 16*a^6*x^6))/(35*c
^4*(-1 + a*x)^4*(1 + a*x)^3)

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Maple [A]  time = 0.044, size = 81, normalized size = 0.7 \begin{align*}{\frac{16\,{x}^{6}{a}^{6}-16\,{x}^{5}{a}^{5}-40\,{x}^{4}{a}^{4}+40\,{x}^{3}{a}^{3}+30\,{a}^{2}{x}^{2}-30\,ax-5}{35\,{c}^{4} \left ({a}^{2}{x}^{2}-1 \right ) ^{3}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^2*c*x^2+c)^4,x)

[Out]

1/35*(16*a^6*x^6-16*a^5*x^5-40*a^4*x^4+40*a^3*x^3+30*a^2*x^2-30*a*x-5)/(a^2*x^2-1)^3/c^4/((a*x-1)/(a*x+1))^(1/
2)/a

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Maxima [A]  time = 1.10038, size = 178, normalized size = 1.5 \begin{align*} \frac{1}{2240} \, a{\left (\frac{7 \,{\left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - 10 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 75 \, \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{2} c^{4}} + \frac{\frac{42 \,{\left (a x - 1\right )}}{a x + 1} - \frac{175 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{700 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 5}{a^{2} c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}}}\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^2*c*x^2+c)^4,x, algorithm="maxima")

[Out]

1/2240*a*(7*(((a*x - 1)/(a*x + 1))^(5/2) - 10*((a*x - 1)/(a*x + 1))^(3/2) + 75*sqrt((a*x - 1)/(a*x + 1)))/(a^2
*c^4) + (42*(a*x - 1)/(a*x + 1) - 175*(a*x - 1)^2/(a*x + 1)^2 + 700*(a*x - 1)^3/(a*x + 1)^3 - 5)/(a^2*c^4*((a*
x - 1)/(a*x + 1))^(7/2)))

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Fricas [A]  time = 1.55682, size = 278, normalized size = 2.34 \begin{align*} \frac{{\left (16 \, a^{6} x^{6} - 16 \, a^{5} x^{5} - 40 \, a^{4} x^{4} + 40 \, a^{3} x^{3} + 30 \, a^{2} x^{2} - 30 \, a x - 5\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{35 \,{\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, 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))^(1/2)/(-a^2*c*x^2+c)^4,x, algorithm="fricas")

[Out]

1/35*(16*a^6*x^6 - 16*a^5*x^5 - 40*a^4*x^4 + 40*a^3*x^3 + 30*a^2*x^2 - 30*a*x - 5)*sqrt((a*x - 1)/(a*x + 1))/(
a^7*c^4*x^6 - 2*a^6*c^4*x^5 - a^5*c^4*x^4 + 4*a^4*c^4*x^3 - a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4)

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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**2*c*x**2+c)**4,x)

[Out]

Timed out

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Giac [A]  time = 1.12541, size = 259, normalized size = 2.18 \begin{align*} \frac{1}{2240} \, a{\left (\frac{{\left (a x + 1\right )}^{3}{\left (\frac{42 \,{\left (a x - 1\right )}}{a x + 1} - \frac{175 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{700 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 5\right )}}{{\left (a x - 1\right )}^{3} a^{2} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}} - \frac{7 \,{\left (\frac{10 \,{\left (a x - 1\right )} a^{8} c^{16} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} - \frac{{\left (a x - 1\right )}^{2} a^{8} c^{16} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} - 75 \, a^{8} c^{16} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{10} c^{20}}\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^2*c*x^2+c)^4,x, algorithm="giac")

[Out]

1/2240*a*((a*x + 1)^3*(42*(a*x - 1)/(a*x + 1) - 175*(a*x - 1)^2/(a*x + 1)^2 + 700*(a*x - 1)^3/(a*x + 1)^3 - 5)
/((a*x - 1)^3*a^2*c^4*sqrt((a*x - 1)/(a*x + 1))) - 7*(10*(a*x - 1)*a^8*c^16*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1
) - (a*x - 1)^2*a^8*c^16*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1)^2 - 75*a^8*c^16*sqrt((a*x - 1)/(a*x + 1)))/(a^10*
c^20))