### 3.750 $$\int \frac{e^{n \coth ^{-1}(a x)}}{(c-a^2 c x^2)^{9/2}} \, dx$$

Optimal. Leaf size=239 $-\frac{5040 (n-a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left (1-n^2\right ) \left (9-n^2\right ) \left (25-n^2\right ) \left (49-n^2\right ) \sqrt{c-a^2 c x^2}}-\frac{840 (n-3 a x) e^{n \coth ^{-1}(a x)}}{a c^3 \left (9-n^2\right ) \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac{42 (n-5 a x) e^{n \coth ^{-1}(a x)}}{a c^2 \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{(n-7 a x) e^{n \coth ^{-1}(a x)}}{a c \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{7/2}}$

[Out]

-((E^(n*ArcCoth[a*x])*(n - 7*a*x))/(a*c*(49 - n^2)*(c - a^2*c*x^2)^(7/2))) - (42*E^(n*ArcCoth[a*x])*(n - 5*a*x
))/(a*c^2*(25 - n^2)*(49 - n^2)*(c - a^2*c*x^2)^(5/2)) - (840*E^(n*ArcCoth[a*x])*(n - 3*a*x))/(a*c^3*(9 - n^2)
*(25 - n^2)*(49 - n^2)*(c - a^2*c*x^2)^(3/2)) - (5040*E^(n*ArcCoth[a*x])*(n - a*x))/(a*c^4*(1 - n^2)*(9 - n^2)
*(25 - n^2)*(49 - n^2)*Sqrt[c - a^2*c*x^2])

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Rubi [A]  time = 0.25439, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.083, Rules used = {6185, 6184} $-\frac{5040 (n-a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left (1-n^2\right ) \left (9-n^2\right ) \left (25-n^2\right ) \left (49-n^2\right ) \sqrt{c-a^2 c x^2}}-\frac{840 (n-3 a x) e^{n \coth ^{-1}(a x)}}{a c^3 \left (9-n^2\right ) \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac{42 (n-5 a x) e^{n \coth ^{-1}(a x)}}{a c^2 \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{(n-7 a x) e^{n \coth ^{-1}(a x)}}{a c \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{7/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(9/2),x]

[Out]

-((E^(n*ArcCoth[a*x])*(n - 7*a*x))/(a*c*(49 - n^2)*(c - a^2*c*x^2)^(7/2))) - (42*E^(n*ArcCoth[a*x])*(n - 5*a*x
))/(a*c^2*(25 - n^2)*(49 - n^2)*(c - a^2*c*x^2)^(5/2)) - (840*E^(n*ArcCoth[a*x])*(n - 3*a*x))/(a*c^3*(9 - n^2)
*(25 - n^2)*(49 - n^2)*(c - a^2*c*x^2)^(3/2)) - (5040*E^(n*ArcCoth[a*x])*(n - a*x))/(a*c^4*(1 - n^2)*(9 - n^2)
*(25 - n^2)*(49 - 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 )^{9/2}} \, dx &=-\frac{e^{n \coth ^{-1}(a x)} (n-7 a x)}{a c \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{7/2}}+\frac{42 \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx}{c \left (49-n^2\right )}\\ &=-\frac{e^{n \coth ^{-1}(a x)} (n-7 a x)}{a c \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{7/2}}-\frac{42 e^{n \coth ^{-1}(a x)} (n-5 a x)}{a c^2 \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{840 \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{c^2 \left (25-n^2\right ) \left (49-n^2\right )}\\ &=-\frac{e^{n \coth ^{-1}(a x)} (n-7 a x)}{a c \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{7/2}}-\frac{42 e^{n \coth ^{-1}(a x)} (n-5 a x)}{a c^2 \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{840 e^{n \coth ^{-1}(a x)} (n-3 a x)}{a c^3 \left (9-n^2\right ) \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}+\frac{5040 \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{c^3 \left (9-n^2\right ) \left (25-n^2\right ) \left (49-n^2\right )}\\ &=-\frac{e^{n \coth ^{-1}(a x)} (n-7 a x)}{a c \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{7/2}}-\frac{42 e^{n \coth ^{-1}(a x)} (n-5 a x)}{a c^2 \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{840 e^{n \coth ^{-1}(a x)} (n-3 a x)}{a c^3 \left (9-n^2\right ) \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac{5040 e^{n \coth ^{-1}(a x)} (n-a x)}{a c^4 \left (1-n^2\right ) \left (9-n^2\right ) \left (25-n^2\right ) \left (49-n^2\right ) \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 1.51471, size = 260, normalized size = 1.09 $\frac{a x^2 \left (1-\frac{1}{a^2 x^2}\right ) e^{n \coth ^{-1}(a x)} \left (-\frac{63 a x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left (3 \coth ^{-1}(a x)\right )}{n^2-9}+\frac{35 a x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left (5 \coth ^{-1}(a x)\right )}{n^2-25}-\frac{7 a x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left (7 \coth ^{-1}(a x)\right )}{n^2-49}+\frac{21 a n x \sqrt{1-\frac{1}{a^2 x^2}} \sinh \left (3 \coth ^{-1}(a x)\right )}{n^2-9}-\frac{7 a n x \sqrt{1-\frac{1}{a^2 x^2}} \sinh \left (5 \coth ^{-1}(a x)\right )}{n^2-25}+\frac{a n x \sqrt{1-\frac{1}{a^2 x^2}} \sinh \left (7 \coth ^{-1}(a x)\right )}{n^2-49}+\frac{35 a x}{n^2-1}-\frac{35 n}{n^2-1}\right )}{64 c^3 \left (c-a^2 c x^2\right )^{3/2}}$

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(9/2),x]

[Out]

(a*E^(n*ArcCoth[a*x])*(1 - 1/(a^2*x^2))*x^2*((-35*n)/(-1 + n^2) + (35*a*x)/(-1 + n^2) - (63*a*Sqrt[1 - 1/(a^2*
x^2)]*x*Cosh[3*ArcCoth[a*x]])/(-9 + n^2) + (35*a*Sqrt[1 - 1/(a^2*x^2)]*x*Cosh[5*ArcCoth[a*x]])/(-25 + n^2) - (
7*a*Sqrt[1 - 1/(a^2*x^2)]*x*Cosh[7*ArcCoth[a*x]])/(-49 + n^2) + (21*a*n*Sqrt[1 - 1/(a^2*x^2)]*x*Sinh[3*ArcCoth
[a*x]])/(-9 + n^2) - (7*a*n*Sqrt[1 - 1/(a^2*x^2)]*x*Sinh[5*ArcCoth[a*x]])/(-25 + n^2) + (a*n*Sqrt[1 - 1/(a^2*x
^2)]*x*Sinh[7*ArcCoth[a*x]])/(-49 + n^2)))/(64*c^3*(c - a^2*c*x^2)^(3/2))

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Maple [A]  time = 0.045, size = 218, normalized size = 0.9 \begin{align*}{\frac{ \left ( 5040\,{a}^{7}{x}^{7}-5040\,n{a}^{6}{x}^{6}+2520\,{a}^{5}{n}^{2}{x}^{5}-840\,{a}^{4}{n}^{3}{x}^{4}-17640\,{x}^{5}{a}^{5}+210\,{a}^{3}{n}^{4}{x}^{3}+15960\,{a}^{4}n{x}^{4}-42\,{a}^{2}{n}^{5}{x}^{2}-7140\,{a}^{3}{n}^{2}{x}^{3}+7\,a{n}^{6}x+2100\,{a}^{2}{n}^{3}{x}^{2}-{n}^{7}+22050\,{x}^{3}{a}^{3}-455\,a{n}^{4}x-17178\,{a}^{2}n{x}^{2}+77\,{n}^{5}+6433\,a{n}^{2}x-1519\,{n}^{3}-11025\,ax+6483\,n \right ) \left ( ax-1 \right ) \left ( ax+1 \right ){{\rm e}^{n{\rm arccoth} \left (ax\right )}}}{a \left ({n}^{8}-84\,{n}^{6}+1974\,{n}^{4}-12916\,{n}^{2}+11025 \right ) } \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{9}{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)^(9/2),x)

[Out]

(a*x-1)*(a*x+1)*(5040*a^7*x^7-5040*a^6*n*x^6+2520*a^5*n^2*x^5-840*a^4*n^3*x^4-17640*a^5*x^5+210*a^3*n^4*x^3+15
960*a^4*n*x^4-42*a^2*n^5*x^2-7140*a^3*n^2*x^3+7*a*n^6*x+2100*a^2*n^3*x^2-n^7+22050*a^3*x^3-455*a*n^4*x-17178*a
^2*n*x^2+77*n^5+6433*a*n^2*x-1519*n^3-11025*a*x+6483*n)*exp(n*arccoth(a*x))/a/(n^8-84*n^6+1974*n^4-12916*n^2+1
1025)/(-a^2*c*x^2+c)^(9/2)

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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 )}^{\frac{9}{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)^(9/2),x, algorithm="maxima")

[Out]

integrate(((a*x - 1)/(a*x + 1))^(1/2*n)/(-a^2*c*x^2 + c)^(9/2), x)

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Fricas [A]  time = 1.6778, size = 1030, normalized size = 4.31 \begin{align*} -\frac{{\left (5040 \, a^{7} x^{7} + 5040 \, a^{6} n x^{6} + n^{7} + 2520 \,{\left (a^{5} n^{2} - 7 \, a^{5}\right )} x^{5} - 77 \, n^{5} + 840 \,{\left (a^{4} n^{3} - 19 \, a^{4} n\right )} x^{4} + 210 \,{\left (a^{3} n^{4} - 34 \, a^{3} n^{2} + 105 \, a^{3}\right )} x^{3} + 1519 \, n^{3} + 42 \,{\left (a^{2} n^{5} - 50 \, a^{2} n^{3} + 409 \, a^{2} n\right )} x^{2} + 7 \,{\left (a n^{6} - 65 \, a n^{4} + 919 \, a n^{2} - 1575 \, a\right )} x - 6483 \, n\right )} \sqrt{-a^{2} c x^{2} + c} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a c^{5} n^{8} - 84 \, a c^{5} n^{6} + 1974 \, a c^{5} n^{4} +{\left (a^{9} c^{5} n^{8} - 84 \, a^{9} c^{5} n^{6} + 1974 \, a^{9} c^{5} n^{4} - 12916 \, a^{9} c^{5} n^{2} + 11025 \, a^{9} c^{5}\right )} x^{8} - 12916 \, a c^{5} n^{2} - 4 \,{\left (a^{7} c^{5} n^{8} - 84 \, a^{7} c^{5} n^{6} + 1974 \, a^{7} c^{5} n^{4} - 12916 \, a^{7} c^{5} n^{2} + 11025 \, a^{7} c^{5}\right )} x^{6} + 11025 \, a c^{5} + 6 \,{\left (a^{5} c^{5} n^{8} - 84 \, a^{5} c^{5} n^{6} + 1974 \, a^{5} c^{5} n^{4} - 12916 \, a^{5} c^{5} n^{2} + 11025 \, a^{5} c^{5}\right )} x^{4} - 4 \,{\left (a^{3} c^{5} n^{8} - 84 \, a^{3} c^{5} n^{6} + 1974 \, a^{3} c^{5} n^{4} - 12916 \, a^{3} c^{5} n^{2} + 11025 \, a^{3} c^{5}\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)^(9/2),x, algorithm="fricas")

[Out]

-(5040*a^7*x^7 + 5040*a^6*n*x^6 + n^7 + 2520*(a^5*n^2 - 7*a^5)*x^5 - 77*n^5 + 840*(a^4*n^3 - 19*a^4*n)*x^4 + 2
10*(a^3*n^4 - 34*a^3*n^2 + 105*a^3)*x^3 + 1519*n^3 + 42*(a^2*n^5 - 50*a^2*n^3 + 409*a^2*n)*x^2 + 7*(a*n^6 - 65
*a*n^4 + 919*a*n^2 - 1575*a)*x - 6483*n)*sqrt(-a^2*c*x^2 + c)*((a*x - 1)/(a*x + 1))^(1/2*n)/(a*c^5*n^8 - 84*a*
c^5*n^6 + 1974*a*c^5*n^4 + (a^9*c^5*n^8 - 84*a^9*c^5*n^6 + 1974*a^9*c^5*n^4 - 12916*a^9*c^5*n^2 + 11025*a^9*c^
5)*x^8 - 12916*a*c^5*n^2 - 4*(a^7*c^5*n^8 - 84*a^7*c^5*n^6 + 1974*a^7*c^5*n^4 - 12916*a^7*c^5*n^2 + 11025*a^7*
c^5)*x^6 + 11025*a*c^5 + 6*(a^5*c^5*n^8 - 84*a^5*c^5*n^6 + 1974*a^5*c^5*n^4 - 12916*a^5*c^5*n^2 + 11025*a^5*c^
5)*x^4 - 4*(a^3*c^5*n^8 - 84*a^3*c^5*n^6 + 1974*a^3*c^5*n^4 - 12916*a^3*c^5*n^2 + 11025*a^3*c^5)*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)**(9/2),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 )}^{\frac{9}{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)^(9/2),x, algorithm="giac")

[Out]

integrate(((a*x - 1)/(a*x + 1))^(1/2*n)/(-a^2*c*x^2 + c)^(9/2), x)