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]

-exp(n*arccoth(a*x))*(-7*a*x+n)/a/c/(-n^2+49)/(-a^2*c*x^2+c)^(7/2)-42*exp(n*arccoth(a*x))*(-5*a*x+n)/a/c^2/(n^
4-74*n^2+1225)/(-a^2*c*x^2+c)^(5/2)-840*exp(n*arccoth(a*x))*(-3*a*x+n)/a/c^3/(-n^2+49)/(n^4-34*n^2+225)/(-a^2*
c*x^2+c)^(3/2)-5040*exp(n*arccoth(a*x))*(-a*x+n)/a/c^4/(n^4-74*n^2+1225)/(n^4-10*n^2+9)/(-a^2*c*x^2+c)^(1/2)

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Rubi [A]  time = 0.25, antiderivative size = 239, normalized size of antiderivative = 1.00, 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 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]

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])

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*}

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Mathematica [A]  time = 1.68, 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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fricas [A]  time = 0.58, size = 451, normalized size = 1.89 \[ -\frac {{\left (5040 \, a^{7} x^{7} + 5040 \, a^{6} n x^{6} + n^{7} + 2520 \, {\left (a^{5} n^{2} - 7 \, a^{5}\right )} x^{5} - 77 \, n^{5} + 840 \, {\left (a^{4} n^{3} - 19 \, a^{4} n\right )} x^{4} + 210 \, {\left (a^{3} n^{4} - 34 \, a^{3} n^{2} + 105 \, a^{3}\right )} x^{3} + 1519 \, n^{3} + 42 \, {\left (a^{2} n^{5} - 50 \, a^{2} n^{3} + 409 \, a^{2} n\right )} x^{2} + 7 \, {\left (a n^{6} - 65 \, a n^{4} + 919 \, a n^{2} - 1575 \, a\right )} x - 6483 \, n\right )} \sqrt {-a^{2} c x^{2} + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{a c^{5} n^{8} - 84 \, a c^{5} n^{6} + 1974 \, a c^{5} n^{4} + {\left (a^{9} c^{5} n^{8} - 84 \, a^{9} c^{5} n^{6} + 1974 \, a^{9} c^{5} n^{4} - 12916 \, a^{9} c^{5} n^{2} + 11025 \, a^{9} c^{5}\right )} x^{8} - 12916 \, a c^{5} n^{2} - 4 \, {\left (a^{7} c^{5} n^{8} - 84 \, a^{7} c^{5} n^{6} + 1974 \, a^{7} c^{5} n^{4} - 12916 \, a^{7} c^{5} n^{2} + 11025 \, a^{7} c^{5}\right )} x^{6} + 11025 \, a c^{5} + 6 \, {\left (a^{5} c^{5} n^{8} - 84 \, a^{5} c^{5} n^{6} + 1974 \, a^{5} c^{5} n^{4} - 12916 \, a^{5} c^{5} n^{2} + 11025 \, a^{5} c^{5}\right )} x^{4} - 4 \, {\left (a^{3} c^{5} n^{8} - 84 \, a^{3} c^{5} n^{6} + 1974 \, a^{3} c^{5} n^{4} - 12916 \, a^{3} c^{5} n^{2} + 11025 \, a^{3} c^{5}\right )} x^{2}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}}}\,{d x} \]

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)

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maple [A]  time = 0.04, size = 218, normalized size = 0.91 \[ \frac {\left (a x -1\right ) \left (a x +1\right ) \left (5040 a^{7} x^{7}-5040 n \,a^{6} x^{6}+2520 a^{5} n^{2} x^{5}-840 a^{4} n^{3} x^{4}-17640 x^{5} a^{5}+210 a^{3} n^{4} x^{3}+15960 n \,a^{4} x^{4}-42 a^{2} n^{5} x^{2}-7140 a^{3} n^{2} x^{3}+7 a \,n^{6} x +2100 a^{2} n^{3} x^{2}-n^{7}+22050 x^{3} a^{3}-455 a \,n^{4} x -17178 a^{2} n \,x^{2}+77 n^{5}+6433 a \,n^{2} x -1519 n^{3}-11025 a x +6483 n \right ) {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )}}{a \left (n^{8}-84 n^{6}+1974 n^{4}-12916 n^{2}+11025\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}}} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}}}\,{d x} \]

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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mupad [B]  time = 1.98, size = 441, normalized size = 1.85 \[ -\frac {{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}\,\left (\frac {5040\,x^7}{c^4\,\left (n^8-84\,n^6+1974\,n^4-12916\,n^2+11025\right )}+\frac {-n^7+77\,n^5-1519\,n^3+6483\,n}{a^7\,c^4\,\left (n^8-84\,n^6+1974\,n^4-12916\,n^2+11025\right )}-\frac {5040\,n\,x^6}{a\,c^4\,\left (n^8-84\,n^6+1974\,n^4-12916\,n^2+11025\right )}+\frac {x^5\,\left (2520\,n^2-17640\right )}{a^2\,c^4\,\left (n^8-84\,n^6+1974\,n^4-12916\,n^2+11025\right )}+\frac {x^3\,\left (210\,n^4-7140\,n^2+22050\right )}{a^4\,c^4\,\left (n^8-84\,n^6+1974\,n^4-12916\,n^2+11025\right )}+\frac {7\,x\,\left (n^6-65\,n^4+919\,n^2-1575\right )}{a^6\,c^4\,\left (n^8-84\,n^6+1974\,n^4-12916\,n^2+11025\right )}-\frac {840\,n\,x^4\,\left (n^2-19\right )}{a^3\,c^4\,\left (n^8-84\,n^6+1974\,n^4-12916\,n^2+11025\right )}-\frac {42\,n\,x^2\,\left (n^4-50\,n^2+409\right )}{a^5\,c^4\,\left (n^8-84\,n^6+1974\,n^4-12916\,n^2+11025\right )}\right )}{{\left (\frac {a\,x-1}{a\,x}\right )}^{n/2}\,\left (\frac {\sqrt {c-a^2\,c\,x^2}}{a^6}-x^6\,\sqrt {c-a^2\,c\,x^2}+\frac {3\,x^4\,\sqrt {c-a^2\,c\,x^2}}{a^2}-\frac {3\,x^2\,\sqrt {c-a^2\,c\,x^2}}{a^4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*acoth(a*x))/(c - a^2*c*x^2)^(9/2),x)

[Out]

-(((a*x + 1)/(a*x))^(n/2)*((5040*x^7)/(c^4*(1974*n^4 - 12916*n^2 - 84*n^6 + n^8 + 11025)) + (6483*n - 1519*n^3
+ 77*n^5 - n^7)/(a^7*c^4*(1974*n^4 - 12916*n^2 - 84*n^6 + n^8 + 11025)) - (5040*n*x^6)/(a*c^4*(1974*n^4 - 129
16*n^2 - 84*n^6 + n^8 + 11025)) + (x^5*(2520*n^2 - 17640))/(a^2*c^4*(1974*n^4 - 12916*n^2 - 84*n^6 + n^8 + 110
25)) + (x^3*(210*n^4 - 7140*n^2 + 22050))/(a^4*c^4*(1974*n^4 - 12916*n^2 - 84*n^6 + n^8 + 11025)) + (7*x*(919*
n^2 - 65*n^4 + n^6 - 1575))/(a^6*c^4*(1974*n^4 - 12916*n^2 - 84*n^6 + n^8 + 11025)) - (840*n*x^4*(n^2 - 19))/(
a^3*c^4*(1974*n^4 - 12916*n^2 - 84*n^6 + n^8 + 11025)) - (42*n*x^2*(n^4 - 50*n^2 + 409))/(a^5*c^4*(1974*n^4 -
12916*n^2 - 84*n^6 + n^8 + 11025))))/(((a*x - 1)/(a*x))^(n/2)*((c - a^2*c*x^2)^(1/2)/a^6 - x^6*(c - a^2*c*x^2)
^(1/2) + (3*x^4*(c - a^2*c*x^2)^(1/2))/a^2 - (3*x^2*(c - a^2*c*x^2)^(1/2))/a^4))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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