### 3.749 $$\int \frac{e^{n \coth ^{-1}(a x)}}{(c-a^2 c x^2)^{7/2}} \, dx$$

Optimal. Leaf size=166 $-\frac{120 (n-a x) e^{n \coth ^{-1}(a x)}}{a c^3 \left (1-n^2\right ) \left (9-n^2\right ) \left (25-n^2\right ) \sqrt{c-a^2 c x^2}}-\frac{20 (n-3 a x) e^{n \coth ^{-1}(a x)}}{a c^2 \left (9-n^2\right ) \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac{(n-5 a x) e^{n \coth ^{-1}(a x)}}{a c \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}$

[Out]

-((E^(n*ArcCoth[a*x])*(n - 5*a*x))/(a*c*(25 - n^2)*(c - a^2*c*x^2)^(5/2))) - (20*E^(n*ArcCoth[a*x])*(n - 3*a*x
))/(a*c^2*(9 - n^2)*(25 - n^2)*(c - a^2*c*x^2)^(3/2)) - (120*E^(n*ArcCoth[a*x])*(n - a*x))/(a*c^3*(1 - n^2)*(9
- n^2)*(25 - n^2)*Sqrt[c - a^2*c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.176648, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.083, Rules used = {6185, 6184} $-\frac{120 (n-a x) e^{n \coth ^{-1}(a x)}}{a c^3 \left (1-n^2\right ) \left (9-n^2\right ) \left (25-n^2\right ) \sqrt{c-a^2 c x^2}}-\frac{20 (n-3 a x) e^{n \coth ^{-1}(a x)}}{a c^2 \left (9-n^2\right ) \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac{(n-5 a x) e^{n \coth ^{-1}(a x)}}{a c \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(7/2),x]

[Out]

-((E^(n*ArcCoth[a*x])*(n - 5*a*x))/(a*c*(25 - n^2)*(c - a^2*c*x^2)^(5/2))) - (20*E^(n*ArcCoth[a*x])*(n - 3*a*x
))/(a*c^2*(9 - n^2)*(25 - n^2)*(c - a^2*c*x^2)^(3/2)) - (120*E^(n*ArcCoth[a*x])*(n - a*x))/(a*c^3*(1 - n^2)*(9
- n^2)*(25 - n^2)*Sqrt[c - a^2*c*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 6184

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[((n - a*x)*E^(n*ArcCoth[a*x]))
/(a*c*(n^2 - 1)*Sqrt[c + d*x^2]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !IntegerQ[n]

Rubi steps

\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx &=-\frac{e^{n \coth ^{-1}(a x)} (n-5 a x)}{a c \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{20 \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{c \left (25-n^2\right )}\\ &=-\frac{e^{n \coth ^{-1}(a x)} (n-5 a x)}{a c \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{20 e^{n \coth ^{-1}(a x)} (n-3 a x)}{a c^2 \left (9-n^2\right ) \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}+\frac{120 \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{c^2 \left (9-n^2\right ) \left (25-n^2\right )}\\ &=-\frac{e^{n \coth ^{-1}(a x)} (n-5 a x)}{a c \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{20 e^{n \coth ^{-1}(a x)} (n-3 a x)}{a c^2 \left (9-n^2\right ) \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac{120 e^{n \coth ^{-1}(a x)} (n-a x)}{a c^3 \left (1-n^2\right ) \left (9-n^2\right ) \left (25-n^2\right ) \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 1.46125, size = 299, normalized size = 1.8 $-\frac{a^2 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} e^{n \coth ^{-1}(a x)} \left (\frac{10 n^5}{a x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{340 n^3}{a x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{10 \left (n^4-34 n^2+225\right )}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2250 n}{a x \sqrt{1-\frac{1}{a^2 x^2}}}-5 n^4 \cosh \left (5 \coth ^{-1}(a x)\right )+50 n^2 \cosh \left (5 \coth ^{-1}(a x)\right )+15 \left (n^4-26 n^2+25\right ) \cosh \left (3 \coth ^{-1}(a x)\right )-5 n^5 \sinh \left (3 \coth ^{-1}(a x)\right )+n^5 \sinh \left (5 \coth ^{-1}(a x)\right )+130 n^3 \sinh \left (3 \coth ^{-1}(a x)\right )-10 n^3 \sinh \left (5 \coth ^{-1}(a x)\right )-125 n \sinh \left (3 \coth ^{-1}(a x)\right )+9 n \sinh \left (5 \coth ^{-1}(a x)\right )-45 \cosh \left (5 \coth ^{-1}(a x)\right )\right )}{16 c^2 (n-5) (n-3) (n-1) (n+1) (n+3) (n+5) \left (c-a^2 c x^2\right )^{3/2}}$

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(7/2),x]

[Out]

-(a^2*E^(n*ArcCoth[a*x])*(1 - 1/(a^2*x^2))^(3/2)*x^3*((-10*(225 - 34*n^2 + n^4))/Sqrt[1 - 1/(a^2*x^2)] + (2250
*n)/(a*Sqrt[1 - 1/(a^2*x^2)]*x) - (340*n^3)/(a*Sqrt[1 - 1/(a^2*x^2)]*x) + (10*n^5)/(a*Sqrt[1 - 1/(a^2*x^2)]*x)
+ 15*(25 - 26*n^2 + n^4)*Cosh[3*ArcCoth[a*x]] - 45*Cosh[5*ArcCoth[a*x]] + 50*n^2*Cosh[5*ArcCoth[a*x]] - 5*n^4
*Cosh[5*ArcCoth[a*x]] - 125*n*Sinh[3*ArcCoth[a*x]] + 130*n^3*Sinh[3*ArcCoth[a*x]] - 5*n^5*Sinh[3*ArcCoth[a*x]]
+ 9*n*Sinh[5*ArcCoth[a*x]] - 10*n^3*Sinh[5*ArcCoth[a*x]] + n^5*Sinh[5*ArcCoth[a*x]]))/(16*c^2*(-5 + n)*(-3 +
n)*(-1 + n)*(1 + n)*(3 + n)*(5 + n)*(c - a^2*c*x^2)^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.042, size = 140, normalized size = 0.8 \begin{align*}{\frac{ \left ( 120\,{x}^{5}{a}^{5}-120\,n{a}^{4}{x}^{4}+60\,{a}^{3}{n}^{2}{x}^{3}-20\,{a}^{2}{n}^{3}{x}^{2}-300\,{x}^{3}{a}^{3}+5\,a{n}^{4}x+260\,n{a}^{2}{x}^{2}-{n}^{5}-110\,a{n}^{2}x+30\,{n}^{3}+225\,ax-149\,n \right ) \left ( ax-1 \right ) \left ( ax+1 \right ){{\rm e}^{n{\rm arccoth} \left (ax\right )}}}{a \left ({n}^{6}-35\,{n}^{4}+259\,{n}^{2}-225 \right ) } \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{7}{2}}}} \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)^(7/2),x)

[Out]

(a*x-1)*(a*x+1)*(120*a^5*x^5-120*a^4*n*x^4+60*a^3*n^2*x^3-20*a^2*n^3*x^2-300*a^3*x^3+5*a*n^4*x+260*a^2*n*x^2-n
^5-110*a*n^2*x+30*n^3+225*a*x-149*n)*exp(n*arccoth(a*x))/a/(n^6-35*n^4+259*n^2-225)/(-a^2*c*x^2+c)^(7/2)

________________________________________________________________________________________

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 )}^{\frac{7}{2}}}\,{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)^(7/2),x, algorithm="maxima")

[Out]

integrate(((a*x - 1)/(a*x + 1))^(1/2*n)/(-a^2*c*x^2 + c)^(7/2), x)

________________________________________________________________________________________

Fricas [A]  time = 1.69961, size = 622, normalized size = 3.75 \begin{align*} -\frac{{\left (120 \, a^{5} x^{5} + 120 \, a^{4} n x^{4} + n^{5} + 60 \,{\left (a^{3} n^{2} - 5 \, a^{3}\right )} x^{3} - 30 \, n^{3} + 20 \,{\left (a^{2} n^{3} - 13 \, a^{2} n\right )} x^{2} + 5 \,{\left (a n^{4} - 22 \, a n^{2} + 45 \, a\right )} x + 149 \, n\right )} \sqrt{-a^{2} c x^{2} + c} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a c^{4} n^{6} - 35 \, a c^{4} n^{4} + 259 \, a c^{4} n^{2} -{\left (a^{7} c^{4} n^{6} - 35 \, a^{7} c^{4} n^{4} + 259 \, a^{7} c^{4} n^{2} - 225 \, a^{7} c^{4}\right )} x^{6} - 225 \, a c^{4} + 3 \,{\left (a^{5} c^{4} n^{6} - 35 \, a^{5} c^{4} n^{4} + 259 \, a^{5} c^{4} n^{2} - 225 \, a^{5} c^{4}\right )} x^{4} - 3 \,{\left (a^{3} c^{4} n^{6} - 35 \, a^{3} c^{4} n^{4} + 259 \, a^{3} c^{4} n^{2} - 225 \, a^{3} c^{4}\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)^(7/2),x, algorithm="fricas")

[Out]

-(120*a^5*x^5 + 120*a^4*n*x^4 + n^5 + 60*(a^3*n^2 - 5*a^3)*x^3 - 30*n^3 + 20*(a^2*n^3 - 13*a^2*n)*x^2 + 5*(a*n
^4 - 22*a*n^2 + 45*a)*x + 149*n)*sqrt(-a^2*c*x^2 + c)*((a*x - 1)/(a*x + 1))^(1/2*n)/(a*c^4*n^6 - 35*a*c^4*n^4
+ 259*a*c^4*n^2 - (a^7*c^4*n^6 - 35*a^7*c^4*n^4 + 259*a^7*c^4*n^2 - 225*a^7*c^4)*x^6 - 225*a*c^4 + 3*(a^5*c^4*
n^6 - 35*a^5*c^4*n^4 + 259*a^5*c^4*n^2 - 225*a^5*c^4)*x^4 - 3*(a^3*c^4*n^6 - 35*a^3*c^4*n^4 + 259*a^3*c^4*n^2
- 225*a^3*c^4)*x^2)

________________________________________________________________________________________

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)**(7/2),x)

[Out]

Timed out

________________________________________________________________________________________

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 )}^{\frac{7}{2}}}\,{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)^(7/2),x, algorithm="giac")

[Out]

integrate(((a*x - 1)/(a*x + 1))^(1/2*n)/(-a^2*c*x^2 + c)^(7/2), x)