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

Optimal. Leaf size=197 \[ -\frac{(n-6 a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}-\frac{360 (n-2 a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left (4-n^2\right ) \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )}-\frac{30 (n-4 a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )^2}+\frac{720 e^{n \coth ^{-1}(a x)}}{a c^4 n \left (36-n^2\right ) \left (n^4-20 n^2+64\right )} \]

[Out]

(720*E^(n*ArcCoth[a*x]))/(a*c^4*n*(36 - n^2)*(64 - 20*n^2 + n^4)) - (E^(n*ArcCoth[a*x])*(n - 6*a*x))/(a*c^4*(3
6 - n^2)*(1 - a^2*x^2)^3) - (30*E^(n*ArcCoth[a*x])*(n - 4*a*x))/(a*c^4*(16 - n^2)*(36 - n^2)*(1 - a^2*x^2)^2)
- (360*E^(n*ArcCoth[a*x])*(n - 2*a*x))/(a*c^4*(4 - n^2)*(16 - n^2)*(36 - n^2)*(1 - a^2*x^2))

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Rubi [A]  time = 0.180955, antiderivative size = 197, 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} \[ -\frac{(n-6 a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}-\frac{360 (n-2 a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left (4-n^2\right ) \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )}-\frac{30 (n-4 a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )^2}+\frac{720 e^{n \coth ^{-1}(a x)}}{a c^4 n \left (36-n^2\right ) \left (n^4-20 n^2+64\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(720*E^(n*ArcCoth[a*x]))/(a*c^4*n*(36 - n^2)*(64 - 20*n^2 + n^4)) - (E^(n*ArcCoth[a*x])*(n - 6*a*x))/(a*c^4*(3
6 - n^2)*(1 - a^2*x^2)^3) - (30*E^(n*ArcCoth[a*x])*(n - 4*a*x))/(a*c^4*(16 - n^2)*(36 - n^2)*(1 - a^2*x^2)^2)
- (360*E^(n*ArcCoth[a*x])*(n - 2*a*x))/(a*c^4*(4 - n^2)*(16 - n^2)*(36 - n^2)*(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^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx &=-\frac{e^{n \coth ^{-1}(a x)} (n-6 a x)}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}+\frac{30 \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx}{c \left (36-n^2\right )}\\ &=-\frac{e^{n \coth ^{-1}(a x)} (n-6 a x)}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}-\frac{30 e^{n \coth ^{-1}(a x)} (n-4 a x)}{a c^4 \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )^2}+\frac{360 \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx}{c^2 \left (576-52 n^2+n^4\right )}\\ &=-\frac{e^{n \coth ^{-1}(a x)} (n-6 a x)}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}-\frac{30 e^{n \coth ^{-1}(a x)} (n-4 a x)}{a c^4 \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )^2}-\frac{360 e^{n \coth ^{-1}(a x)} (n-2 a x)}{a c^4 \left (4-n^2\right ) \left (576-52 n^2+n^4\right ) \left (1-a^2 x^2\right )}+\frac{720 \int \frac{e^{n \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{c^3 \left (4-n^2\right ) \left (576-52 n^2+n^4\right )}\\ &=\frac{720 e^{n \coth ^{-1}(a x)}}{a c^4 n \left (4-n^2\right ) \left (576-52 n^2+n^4\right )}-\frac{e^{n \coth ^{-1}(a x)} (n-6 a x)}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}-\frac{30 e^{n \coth ^{-1}(a x)} (n-4 a x)}{a c^4 \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )^2}-\frac{360 e^{n \coth ^{-1}(a x)} (n-2 a x)}{a c^4 \left (4-n^2\right ) \left (576-52 n^2+n^4\right ) \left (1-a^2 x^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.300616, size = 152, normalized size = 0.77 \[ -\frac{\left (10 n^4 \left (3 a^2 x^2-5\right )-120 a n^3 x \left (a^2 x^2-2\right )+8 n^2 \left (45 a^4 x^4-105 a^2 x^2+68\right )-48 a n x \left (15 a^4 x^4-40 a^2 x^2+33\right )+720 \left (a^2 x^2-1\right )^3-6 a n^5 x+n^6\right ) e^{n \coth ^{-1}(a x)}}{a c^4 n \left (n^2-36\right ) \left (n^2-16\right ) \left (n^2-4\right ) \left (a^2 x^2-1\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((E^(n*ArcCoth[a*x])*(n^6 - 6*a*n^5*x - 120*a*n^3*x*(-2 + a^2*x^2) + 720*(-1 + a^2*x^2)^3 + 10*n^4*(-5 + 3*a^
2*x^2) - 48*a*n*x*(33 - 40*a^2*x^2 + 15*a^4*x^4) + 8*n^2*(68 - 105*a^2*x^2 + 45*a^4*x^4)))/(a*c^4*n*(-36 + n^2
)*(-16 + n^2)*(-4 + n^2)*(-1 + a^2*x^2)^3))

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Maple [A]  time = 0.046, size = 167, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 720\,{x}^{6}{a}^{6}-720\,{x}^{5}{a}^{5}n+360\,{a}^{4}{n}^{2}{x}^{4}-120\,{a}^{3}{n}^{3}{x}^{3}-2160\,{x}^{4}{a}^{4}+30\,{a}^{2}{n}^{4}{x}^{2}+1920\,{x}^{3}{a}^{3}n-6\,a{n}^{5}x-840\,{a}^{2}{n}^{2}{x}^{2}+{n}^{6}+240\,a{n}^{3}x+2160\,{a}^{2}{x}^{2}-50\,{n}^{4}-1584\,nax+544\,{n}^{2}-720 \right ){{\rm e}^{n{\rm arccoth} \left (ax\right )}}}{ \left ({a}^{2}{x}^{2}-1 \right ) ^{3}{c}^{4}an \left ({n}^{6}-56\,{n}^{4}+784\,{n}^{2}-2304 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arccoth(a*x))/(-a^2*c*x^2+c)^4,x)

[Out]

-(720*a^6*x^6-720*a^5*n*x^5+360*a^4*n^2*x^4-120*a^3*n^3*x^3-2160*a^4*x^4+30*a^2*n^4*x^2+1920*a^3*n*x^3-6*a*n^5
*x-840*a^2*n^2*x^2+n^6+240*a*n^3*x+2160*a^2*x^2-50*n^4-1584*a*n*x+544*n^2-720)*exp(n*arccoth(a*x))/(a^2*x^2-1)
^3/c^4/a/n/(n^6-56*n^4+784*n^2-2304)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{{\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(exp(n*arccoth(a*x))/(-a^2*c*x^2+c)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.78191, size = 683, normalized size = 3.47 \begin{align*} -\frac{{\left (720 \, a^{6} x^{6} + 720 \, a^{5} n x^{5} + n^{6} + 360 \,{\left (a^{4} n^{2} - 6 \, a^{4}\right )} x^{4} - 50 \, n^{4} + 120 \,{\left (a^{3} n^{3} - 16 \, a^{3} n\right )} x^{3} + 30 \,{\left (a^{2} n^{4} - 28 \, a^{2} n^{2} + 72 \, a^{2}\right )} x^{2} + 544 \, n^{2} + 6 \,{\left (a n^{5} - 40 \, a n^{3} + 264 \, a n\right )} x - 720\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a c^{4} n^{7} - 56 \, a c^{4} n^{5} + 784 \, a c^{4} n^{3} -{\left (a^{7} c^{4} n^{7} - 56 \, a^{7} c^{4} n^{5} + 784 \, a^{7} c^{4} n^{3} - 2304 \, a^{7} c^{4} n\right )} x^{6} - 2304 \, a c^{4} n + 3 \,{\left (a^{5} c^{4} n^{7} - 56 \, a^{5} c^{4} n^{5} + 784 \, a^{5} c^{4} n^{3} - 2304 \, a^{5} c^{4} n\right )} x^{4} - 3 \,{\left (a^{3} c^{4} n^{7} - 56 \, a^{3} c^{4} n^{5} + 784 \, a^{3} c^{4} n^{3} - 2304 \, a^{3} c^{4} n\right )} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arccoth(a*x))/(-a^2*c*x^2+c)^4,x, algorithm="fricas")

[Out]

-(720*a^6*x^6 + 720*a^5*n*x^5 + n^6 + 360*(a^4*n^2 - 6*a^4)*x^4 - 50*n^4 + 120*(a^3*n^3 - 16*a^3*n)*x^3 + 30*(
a^2*n^4 - 28*a^2*n^2 + 72*a^2)*x^2 + 544*n^2 + 6*(a*n^5 - 40*a*n^3 + 264*a*n)*x - 720)*((a*x - 1)/(a*x + 1))^(
1/2*n)/(a*c^4*n^7 - 56*a*c^4*n^5 + 784*a*c^4*n^3 - (a^7*c^4*n^7 - 56*a^7*c^4*n^5 + 784*a^7*c^4*n^3 - 2304*a^7*
c^4*n)*x^6 - 2304*a*c^4*n + 3*(a^5*c^4*n^7 - 56*a^5*c^4*n^5 + 784*a^5*c^4*n^3 - 2304*a^5*c^4*n)*x^4 - 3*(a^3*c
^4*n^7 - 56*a^3*c^4*n^5 + 784*a^3*c^4*n^3 - 2304*a^3*c^4*n)*x^2)

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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(exp(n*acoth(a*x))/(-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{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{{\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(exp(n*arccoth(a*x))/(-a^2*c*x^2+c)^4,x, algorithm="giac")

[Out]

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