3.1355 \(\int \frac{e^{n \tanh ^{-1}(a x)}}{(c-a^2 c x^2)^{7/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.186363, 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 = {6136, 6135} \[ -\frac{120 (n-a x) e^{n \tanh ^{-1}(a x)}}{a c^3 \left (1-n^2\right ) \left (9-n^2\right ) \left (25-n^2\right ) \sqrt{c-a^2 c x^2}}-\frac{20 (n-3 a x) e^{n \tanh ^{-1}(a x)}}{a c^2 \left (9-n^2\right ) \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac{(n-5 a x) e^{n \tanh ^{-1}(a x)}}{a c \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((E^(n*ArcTanh[a*x])*(n - 5*a*x))/(a*c*(25 - n^2)*(c - a^2*c*x^2)^(5/2))) - (20*E^(n*ArcTanh[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*ArcTanh[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 6136

Int[E^(ArcTanh[(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*ArcTanh[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*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && LtQ[p
, -1] &&  !IntegerQ[n] && NeQ[n^2 - 4*(p + 1)^2, 0] && IntegerQ[2*p]

Rule 6135

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[((n - a*x)*E^(n*ArcTanh[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 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx &=-\frac{e^{n \tanh ^{-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 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{c \left (25-n^2\right )}\\ &=-\frac{e^{n \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 = 0.221295, size = 182, normalized size = 1.1 \[ -\frac{\sqrt{1-a^2 x^2} (1-a x)^{\frac{1}{2} (-n-5)} (a x+1)^{\frac{n-5}{2}} \left (n^3 \left (30-20 a^2 x^2\right )+10 a n^2 x \left (6 a^2 x^2-11\right )+n \left (-120 a^4 x^4+260 a^2 x^2-149\right )+15 a x \left (8 a^4 x^4-20 a^2 x^2+15\right )+5 a n^4 x-n^5\right )}{a c^3 (n-5) (n-3) (n-1) (n+1) (n+3) (n+5) \sqrt{c-a^2 c x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(((1 - a*x)^((-5 - n)/2)*(1 + a*x)^((-5 + n)/2)*Sqrt[1 - a^2*x^2]*(-n^5 + 5*a*n^4*x + n^3*(30 - 20*a^2*x^2) +
10*a*n^2*x*(-11 + 6*a^2*x^2) + n*(-149 + 260*a^2*x^2 - 120*a^4*x^4) + 15*a*x*(15 - 20*a^2*x^2 + 8*a^4*x^4)))/
(a*c^3*(-5 + n)*(-3 + n)*(-1 + n)*(1 + n)*(3 + n)*(5 + n)*Sqrt[c - a^2*c*x^2]))

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Maple [A]  time = 0.03, 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{\it Artanh} \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*arctanh(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*arctanh(a*x))/a/(n^6-35*n^4+259*n^2-225)/(-a^2*c*x^2+c)^(7/2)

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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{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(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)

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Fricas [A]  time = 2.2288, 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*arctanh(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)

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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*atanh(a*x))/(-a**2*c*x**2+c)**(7/2),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{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(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)