3.597 $$\int \frac{e^{-\coth ^{-1}(a x)}}{(c-a^2 c x^2)^4} \, dx$$

Optimal. Leaf size=127 $\frac{8 (2 a x+1) e^{-\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )}+\frac{2 (4 a x+1) e^{-\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )^2}+\frac{(6 a x+1) e^{-\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )^3}-\frac{16 e^{-\coth ^{-1}(a x)}}{35 a c^4}$

[Out]

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

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Rubi [A]  time = 0.135755, 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} $\frac{8 (2 a x+1) e^{-\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )}+\frac{2 (4 a x+1) e^{-\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )^2}+\frac{(6 a x+1) e^{-\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )^3}-\frac{16 e^{-\coth ^{-1}(a x)}}{35 a c^4}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(E^ArcCoth[a*x]*(c - a^2*c*x^2)^4),x]

[Out]

-16/(35*a*c^4*E^ArcCoth[a*x]) + (1 + 6*a*x)/(35*a*c^4*E^ArcCoth[a*x]*(1 - a^2*x^2)^3) + (2*(1 + 4*a*x))/(35*a*
c^4*E^ArcCoth[a*x]*(1 - a^2*x^2)^2) + (8*(1 + 2*a*x))/(35*a*c^4*E^ArcCoth[a*x]*(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 = 0.254699, size = 80, normalized size = 0.63 $-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (16 a^6 x^6+16 a^5 x^5-40 a^4 x^4-40 a^3 x^3+30 a^2 x^2+30 a x-5\right )}{35 (a x-1)^3 (a c x+c)^4}$

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(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*
(-1 + a*x)^3*(c + a*c*x)^4)

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Maple [A]  time = 0.046, size = 81, normalized size = 0.6 \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\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3}{c}^{4}a}\sqrt{{\frac{ax-1}{ax+1}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/35*((a*x-1)/(a*x+1))^(1/2)*(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

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Maxima [A]  time = 1.04753, size = 182, normalized size = 1.43 \begin{align*} \frac{1}{2240} \, a{\left (\frac{5 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - 42 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 175 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 700 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{4}} + \frac{7 \,{\left (\frac{10 \,{\left (a x - 1\right )}}{a x + 1} - \frac{75 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 1\right )}}{a^{2} c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.59774, size = 223, normalized size = 1.76 \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} - 3 \, a^{5} c^{4} x^{4} + 3 \, a^{3} c^{4} x^{2} - a c^{4}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a^{2} c x^{2} - c\right )}^{4}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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