\(\int \frac {e^{n \coth ^{-1}(a x)}}{(c-a^2 c x^2)^4} \, dx\) [743]

Optimal result

Integrand size = 22, antiderivative size = 197 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {720 e^{n \coth ^{-1}(a x)}}{a c^4 n \left (36-n^2\right ) \left (64-20 n^2+n^4\right )}-\frac {e^{n \coth ^{-1}(a x)} (n-6 a x)}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}-\frac {30 e^{n \coth ^{-1}(a x)} (n-4 a x)}{a c^4 \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )^2}-\frac {360 e^{n \coth ^{-1}(a x)} (n-2 a x)}{a c^4 \left (4-n^2\right ) \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )} \]

[Out]

720*exp(n*arccoth(a*x))/a/c^4/n/(-n^2+36)/(n^4-20*n^2+64)-exp(n*arccoth(a*x))*(-6*a*x+n)/a/c^4/(-n^2+36)/(-a^2
*x^2+1)^3-30*exp(n*arccoth(a*x))*(-4*a*x+n)/a/c^4/(n^4-52*n^2+576)/(-a^2*x^2+1)^2-360*exp(n*arccoth(a*x))*(-2*
a*x+n)/a/c^4/(-n^2+36)/(n^4-20*n^2+64)/(-a^2*x^2+1)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6320, 6318} \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=-\frac {(n-6 a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}-\frac {360 (n-2 a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left (4-n^2\right ) \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )}-\frac {30 (n-4 a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )^2}+\frac {720 e^{n \coth ^{-1}(a x)}}{a c^4 n \left (36-n^2\right ) \left (n^4-20 n^2+64\right )} \]

[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 6318

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 6320

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*} \text {integral}& = -\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] (verified)

Time = 0.52 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.77 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=-\frac {e^{n \coth ^{-1}(a x)} \left (n^6-6 a n^5 x-120 a n^3 x \left (-2+a^2 x^2\right )+720 \left (-1+a^2 x^2\right )^3+10 n^4 \left (-5+3 a^2 x^2\right )-48 a n x \left (33-40 a^2 x^2+15 a^4 x^4\right )+8 n^2 \left (68-105 a^2 x^2+45 a^4 x^4\right )\right )}{a c^4 n \left (-36+n^2\right ) \left (-16+n^2\right ) \left (-4+n^2\right ) \left (-1+a^2 x^2\right )^3} \]

[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))

Maple [A] (verified)

Time = 59.38 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.85

method	result	size
gosper	\(-\frac {\left (720 a^{6} x^{6}-720 a^{5} x^{5} n +360 a^{4} n^{2} x^{4}-120 a^{3} n^{3} x^{3}-2160 a^{4} x^{4}+30 a^{2} n^{4} x^{2}+1920 a^{3} x^{3} n -6 a \,n^{5} x -840 a^{2} n^{2} x^{2}+n^{6}+240 a \,n^{3} x +2160 a^{2} x^{2}-50 n^{4}-1584 a n x +544 n^{2}-720\right ) {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{\left (a^{2} x^{2}-1\right )^{3} c^{4} a n \left (n^{6}-56 n^{4}+784 n^{2}-2304\right )}\)	\(167\)
risch	\(-\frac {\left (720 a^{6} x^{6}-720 a^{5} x^{5} n +360 a^{4} n^{2} x^{4}-120 a^{3} n^{3} x^{3}-2160 a^{4} x^{4}+30 a^{2} n^{4} x^{2}+1920 a^{3} x^{3} n -6 a \,n^{5} x -840 a^{2} n^{2} x^{2}+n^{6}+240 a \,n^{3} x +2160 a^{2} x^{2}-50 n^{4}-1584 a n x +544 n^{2}-720\right ) \left (a x -1\right )^{-\frac {n}{2}} \left (a x +1\right )^{\frac {n}{2}}}{c^{4} \left (n^{2}-36\right ) \left (n^{2}-16\right ) \left (n^{2}-4\right ) a n \left (a^{2} x^{2}-1\right )^{3}}\)	\(182\)
parallelrisch	\(\frac {-720 x^{6} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a^{6}+2160 x^{4} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a^{4}+720 a^{5} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} x^{5} n -360 x^{4} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a^{4} n^{2}+120 x^{3} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a^{3} n^{3}-30 x^{2} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a^{2} n^{4}+6 x \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a \,n^{5}+1584 x \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a n -2160 x^{2} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a^{2}-1920 a^{3} x^{3} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} n +840 x^{2} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a^{2} n^{2}-240 x \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a \,n^{3}+720 \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}-{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} n^{6}+50 \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} n^{4}-544 \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} n^{2}}{c^{4} \left (a^{2} x^{2}-1\right )^{3} a n \left (n^{6}-56 n^{4}+784 n^{2}-2304\right )}\)	\(274\)

[In]

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

[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)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.57 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {{\left (720 \, a^{6} x^{6} - 720 \, a^{5} n x^{5} + n^{6} + 360 \, {\left (a^{4} n^{2} - 6 \, a^{4}\right )} x^{4} - 50 \, n^{4} - 120 \, {\left (a^{3} n^{3} - 16 \, a^{3} n\right )} x^{3} + 30 \, {\left (a^{2} n^{4} - 28 \, a^{2} n^{2} + 72 \, a^{2}\right )} x^{2} + 544 \, n^{2} - 6 \, {\left (a n^{5} - 40 \, a n^{3} + 264 \, a n\right )} x - 720\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c^{4} n^{7} - 56 \, a c^{4} n^{5} + 784 \, a c^{4} n^{3} - {\left (a^{7} c^{4} n^{7} - 56 \, a^{7} c^{4} n^{5} + 784 \, a^{7} c^{4} n^{3} - 2304 \, a^{7} c^{4} n\right )} x^{6} - 2304 \, a c^{4} n + 3 \, {\left (a^{5} c^{4} n^{7} - 56 \, a^{5} c^{4} n^{5} + 784 \, a^{5} c^{4} n^{3} - 2304 \, a^{5} c^{4} n\right )} x^{4} - 3 \, {\left (a^{3} c^{4} n^{7} - 56 \, a^{3} c^{4} n^{5} + 784 \, a^{3} c^{4} n^{3} - 2304 \, a^{3} c^{4} n\right )} x^{2}} \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\text {Timed out} \]

[In]

integrate(exp(n*acoth(a*x))/(-a**2*c*x**2+c)**4,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{4}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{4}} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 4.83 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.59 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}\,\left (\frac {n^6-50\,n^4+544\,n^2-720}{a^7\,c^4\,n\,\left (n^6-56\,n^4+784\,n^2-2304\right )}-\frac {720\,x^5}{a^2\,c^4\,\left (n^6-56\,n^4+784\,n^2-2304\right )}-\frac {x^3\,\left (120\,n^2-1920\right )}{a^4\,c^4\,\left (n^6-56\,n^4+784\,n^2-2304\right )}+\frac {720\,x^6}{a\,c^4\,n\,\left (n^6-56\,n^4+784\,n^2-2304\right )}-\frac {6\,x\,\left (n^4-40\,n^2+264\right )}{a^6\,c^4\,\left (n^6-56\,n^4+784\,n^2-2304\right )}+\frac {x^2\,\left (30\,n^4-840\,n^2+2160\right )}{a^5\,c^4\,n\,\left (n^6-56\,n^4+784\,n^2-2304\right )}+\frac {x^4\,\left (360\,n^2-2160\right )}{a^3\,c^4\,n\,\left (n^6-56\,n^4+784\,n^2-2304\right )}\right )}{{\left (\frac {a\,x-1}{a\,x}\right )}^{n/2}\,\left (\frac {1}{a^6}-x^6+\frac {3\,x^4}{a^2}-\frac {3\,x^2}{a^4}\right )} \]

[In]

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

[Out]

(((a*x + 1)/(a*x))^(n/2)*((544*n^2 - 50*n^4 + n^6 - 720)/(a^7*c^4*n*(784*n^2 - 56*n^4 + n^6 - 2304)) - (720*x^
5)/(a^2*c^4*(784*n^2 - 56*n^4 + n^6 - 2304)) - (x^3*(120*n^2 - 1920))/(a^4*c^4*(784*n^2 - 56*n^4 + n^6 - 2304)
) + (720*x^6)/(a*c^4*n*(784*n^2 - 56*n^4 + n^6 - 2304)) - (6*x*(n^4 - 40*n^2 + 264))/(a^6*c^4*(784*n^2 - 56*n^
4 + n^6 - 2304)) + (x^2*(30*n^4 - 840*n^2 + 2160))/(a^5*c^4*n*(784*n^2 - 56*n^4 + n^6 - 2304)) + (x^4*(360*n^2
- 2160))/(a^3*c^4*n*(784*n^2 - 56*n^4 + n^6 - 2304))))/(((a*x - 1)/(a*x))^(n/2)*(1/a^6 - x^6 + (3*x^4)/a^2 -
(3*x^2)/a^4))