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]

16/35/((a*x-1)/(a*x+1))^(1/2)/a/c^4-1/35/((a*x-1)/(a*x+1))^(1/2)*(-6*a*x+1)/a/c^4/(-a^2*x^2+1)^3-2/35/((a*x-1)
/(a*x+1))^(1/2)*(-4*a*x+1)/a/c^4/(-a^2*x^2+1)^2-8/35/((a*x-1)/(a*x+1))^(1/2)*(-2*a*x+1)/a/c^4/(-a^2*x^2+1)

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Rubi [A]  time = 0.13, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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 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]

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])

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*}

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Mathematica [A]  time = 0.28, 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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fricas [A]  time = 0.55, size = 134, normalized size = 1.13 \[ \frac {{\left (16 \, a^{6} x^{6} - 16 \, a^{5} x^{5} - 40 \, a^{4} x^{4} + 40 \, a^{3} x^{3} + 30 \, a^{2} x^{2} - 30 \, a x - 5\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{35 \, {\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} \]

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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giac [A]  time = 0.16, size = 164, normalized size = 1.38 \[ \frac {\frac {{\left (a x + 1\right )}^{3} {\left (\frac {42 \, {\left (a x - 1\right )}}{a x + 1} - \frac {175 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {700 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 5\right )}}{{\left (a x - 1\right )}^{3} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {70 \, {\left (a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} + \frac {7 \, {\left (a x - 1\right )}^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + 525 \, \sqrt {\frac {a x - 1}{a x + 1}}}{2240 \, a c^{4}} \]

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*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*sqrt((a*x - 1)/(a*x + 1))) - 70*(a*x - 1)*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1) + 7*(a*x - 1)^2*sqrt
((a*x - 1)/(a*x + 1))/(a*x + 1)^2 + 525*sqrt((a*x - 1)/(a*x + 1)))/(a*c^4)

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maple [A]  time = 0.04, size = 81, normalized size = 0.68 \[ \frac {16 x^{6} a^{6}-16 x^{5} a^{5}-40 x^{4} a^{4}+40 x^{3} a^{3}+30 a^{2} x^{2}-30 a x -5}{35 \left (a^{2} x^{2}-1\right )^{3} c^{4} \sqrt {\frac {a x -1}{a x +1}}\, a} \]

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 = 0.31, size = 132, normalized size = 1.11 \[ \frac {1}{2240} \, a {\left (\frac {7 \, {\left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 10 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 75 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a^{2} c^{4}} + \frac {\frac {42 \, {\left (a x - 1\right )}}{a x + 1} - \frac {175 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {700 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 5}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}}}\right )} \]

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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mupad [B]  time = 0.06, size = 142, normalized size = 1.19 \[ \frac {15\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{64\,a\,c^4}-\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{32\,a\,c^4}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{320\,a\,c^4}-\frac {\frac {5\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {20\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {6\,\left (a\,x-1\right )}{5\,\left (a\,x+1\right )}+\frac {1}{7}}{64\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(15*((a*x - 1)/(a*x + 1))^(1/2))/(64*a*c^4) - ((a*x - 1)/(a*x + 1))^(3/2)/(32*a*c^4) + ((a*x - 1)/(a*x + 1))^(
5/2)/(320*a*c^4) - ((5*(a*x - 1)^2)/(a*x + 1)^2 - (20*(a*x - 1)^3)/(a*x + 1)^3 - (6*(a*x - 1))/(5*(a*x + 1)) +
1/7)/(64*a*c^4*((a*x - 1)/(a*x + 1))^(7/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{a^{8} x^{8} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 4 a^{6} x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + 6 a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 4 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx}{c^{4}} \]

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]

Integral(1/(a**8*x**8*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) - 4*a**6*x**6*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) + 6*a*
*4*x**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) - 4*a**2*x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) + sqrt(a*x/(a*x + 1)
- 1/(a*x + 1))), x)/c**4

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