### 3.650 $$\int \frac{e^{-\coth ^{-1}(a x)}}{(c-a^2 c x^2)^{7/2}} \, dx$$

Optimal. Leaf size=276 $-\frac{a^6 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}{8 (1-a x) \left (c-a^2 c x^2\right )^{7/2}}+\frac{3 a^6 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}{16 (a x+1) \left (c-a^2 c x^2\right )^{7/2}}-\frac{a^6 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}{32 (1-a x)^2 \left (c-a^2 c x^2\right )^{7/2}}+\frac{3 a^6 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}{32 (a x+1)^2 \left (c-a^2 c x^2\right )^{7/2}}+\frac{a^6 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}{24 (a x+1)^3 \left (c-a^2 c x^2\right )^{7/2}}-\frac{5 a^6 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} \tanh ^{-1}(a x)}{16 \left (c-a^2 c x^2\right )^{7/2}}$

[Out]

-(a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(32*(1 - a*x)^2*(c - a^2*c*x^2)^(7/2)) - (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/
(8*(1 - a*x)*(c - a^2*c*x^2)^(7/2)) + (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(24*(1 + a*x)^3*(c - a^2*c*x^2)^(7/2))
+ (3*a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(32*(1 + a*x)^2*(c - a^2*c*x^2)^(7/2)) + (3*a^6*(1 - 1/(a^2*x^2))^(7/2)
*x^7)/(16*(1 + a*x)*(c - a^2*c*x^2)^(7/2)) - (5*a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7*ArcTanh[a*x])/(16*(c - a^2*c*x
^2)^(7/2))

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Rubi [A]  time = 0.220785, antiderivative size = 276, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.167, Rules used = {6192, 6193, 44, 207} $-\frac{a^6 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}{8 (1-a x) \left (c-a^2 c x^2\right )^{7/2}}+\frac{3 a^6 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}{16 (a x+1) \left (c-a^2 c x^2\right )^{7/2}}-\frac{a^6 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}{32 (1-a x)^2 \left (c-a^2 c x^2\right )^{7/2}}+\frac{3 a^6 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}{32 (a x+1)^2 \left (c-a^2 c x^2\right )^{7/2}}+\frac{a^6 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}{24 (a x+1)^3 \left (c-a^2 c x^2\right )^{7/2}}-\frac{5 a^6 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} \tanh ^{-1}(a x)}{16 \left (c-a^2 c x^2\right )^{7/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(E^ArcCoth[a*x]*(c - a^2*c*x^2)^(7/2)),x]

[Out]

-(a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(32*(1 - a*x)^2*(c - a^2*c*x^2)^(7/2)) - (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/
(8*(1 - a*x)*(c - a^2*c*x^2)^(7/2)) + (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(24*(1 + a*x)^3*(c - a^2*c*x^2)^(7/2))
+ (3*a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(32*(1 + a*x)^2*(c - a^2*c*x^2)^(7/2)) + (3*a^6*(1 - 1/(a^2*x^2))^(7/2)
*x^7)/(16*(1 + a*x)*(c - a^2*c*x^2)^(7/2)) - (5*a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7*ArcTanh[a*x])/(16*(c - a^2*c*x
^2)^(7/2))

Rule 6192

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

Rule 6193

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Dist[c^p/a^(2*p), Int[(u*(-1
+ a*x)^(p - n/2)*(1 + a*x)^(p + n/2))/x^(2*p), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] &&  !
IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && IntegersQ[2*p, p + n/2]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{e^{-\coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx &=\frac{\left (\left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac{e^{-\coth ^{-1}(a x)}}{\left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7} \, dx}{\left (c-a^2 c x^2\right )^{7/2}}\\ &=\frac{\left (a^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac{1}{(-1+a x)^3 (1+a x)^4} \, dx}{\left (c-a^2 c x^2\right )^{7/2}}\\ &=\frac{\left (a^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7\right ) \int \left (\frac{1}{16 (-1+a x)^3}-\frac{1}{8 (-1+a x)^2}-\frac{1}{8 (1+a x)^4}-\frac{3}{16 (1+a x)^3}-\frac{3}{16 (1+a x)^2}+\frac{5}{16 \left (-1+a^2 x^2\right )}\right ) \, dx}{\left (c-a^2 c x^2\right )^{7/2}}\\ &=-\frac{a^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}{32 (1-a x)^2 \left (c-a^2 c x^2\right )^{7/2}}-\frac{a^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}{8 (1-a x) \left (c-a^2 c x^2\right )^{7/2}}+\frac{a^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}{24 (1+a x)^3 \left (c-a^2 c x^2\right )^{7/2}}+\frac{3 a^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}{32 (1+a x)^2 \left (c-a^2 c x^2\right )^{7/2}}+\frac{3 a^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}{16 (1+a x) \left (c-a^2 c x^2\right )^{7/2}}+\frac{\left (5 a^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac{1}{-1+a^2 x^2} \, dx}{16 \left (c-a^2 c x^2\right )^{7/2}}\\ &=-\frac{a^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}{32 (1-a x)^2 \left (c-a^2 c x^2\right )^{7/2}}-\frac{a^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}{8 (1-a x) \left (c-a^2 c x^2\right )^{7/2}}+\frac{a^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}{24 (1+a x)^3 \left (c-a^2 c x^2\right )^{7/2}}+\frac{3 a^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}{32 (1+a x)^2 \left (c-a^2 c x^2\right )^{7/2}}+\frac{3 a^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}{16 (1+a x) \left (c-a^2 c x^2\right )^{7/2}}-\frac{5 a^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7 \tanh ^{-1}(a x)}{16 \left (c-a^2 c x^2\right )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.0861285, size = 99, normalized size = 0.36 $\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (-15 a^4 x^4-15 a^3 x^3+25 a^2 x^2+25 a x+15 (a x-1)^2 (a x+1)^3 \tanh ^{-1}(a x)-8\right )}{48 (a x-1)^2 (a c x+c)^3 \sqrt{c-a^2 c x^2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/(E^ArcCoth[a*x]*(c - a^2*c*x^2)^(7/2)),x]

[Out]

(Sqrt[1 - 1/(a^2*x^2)]*x*(-8 + 25*a*x + 25*a^2*x^2 - 15*a^3*x^3 - 15*a^4*x^4 + 15*(-1 + a*x)^2*(1 + a*x)^3*Arc
Tanh[a*x]))/(48*(-1 + a*x)^2*(c + a*c*x)^3*Sqrt[c - a^2*c*x^2])

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Maple [A]  time = 0.151, size = 241, normalized size = 0.9 \begin{align*} -{\frac{15\,\ln \left ( ax+1 \right ){x}^{5}{a}^{5}-15\,\ln \left ( ax-1 \right ){x}^{5}{a}^{5}+15\,\ln \left ( ax+1 \right ){a}^{4}{x}^{4}-15\,\ln \left ( ax-1 \right ){a}^{4}{x}^{4}-30\,{x}^{4}{a}^{4}-30\,{a}^{3}{x}^{3}\ln \left ( ax+1 \right ) +30\,\ln \left ( ax-1 \right ){x}^{3}{a}^{3}-30\,{x}^{3}{a}^{3}-30\,\ln \left ( ax+1 \right ){a}^{2}{x}^{2}+30\,\ln \left ( ax-1 \right ){a}^{2}{x}^{2}+50\,{a}^{2}{x}^{2}+15\,ax\ln \left ( ax+1 \right ) -15\,\ln \left ( ax-1 \right ) xa+50\,ax+15\,\ln \left ( ax+1 \right ) -15\,\ln \left ( ax-1 \right ) -16}{96\, \left ( ax+1 \right ) ^{2} \left ({a}^{2}{x}^{2}-1 \right ){c}^{4}a \left ( ax-1 \right ) ^{2}}\sqrt{{\frac{ax-1}{ax+1}}}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96*((a*x-1)/(a*x+1))^(1/2)/(a*x+1)^2*(-c*(a^2*x^2-1))^(1/2)*(15*ln(a*x+1)*x^5*a^5-15*ln(a*x-1)*x^5*a^5+15*l
n(a*x+1)*a^4*x^4-15*ln(a*x-1)*a^4*x^4-30*x^4*a^4-30*a^3*x^3*ln(a*x+1)+30*ln(a*x-1)*x^3*a^3-30*x^3*a^3-30*ln(a*
x+1)*a^2*x^2+30*ln(a*x-1)*a^2*x^2+50*a^2*x^2+15*a*x*ln(a*x+1)-15*ln(a*x-1)*x*a+50*a*x+15*ln(a*x+1)-15*ln(a*x-1
)-16)/(a^2*x^2-1)/c^4/a/(a*x-1)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a x - 1}{a x + 1}}}{{\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(((a*x-1)/(a*x+1))^(1/2)/(-a^2*c*x^2+c)^(7/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.92782, size = 393, normalized size = 1.42 \begin{align*} -\frac{15 \,{\left (a^{6} x^{5} + a^{5} x^{4} - 2 \, a^{4} x^{3} - 2 \, a^{3} x^{2} + a^{2} x + a\right )} \sqrt{-c} \log \left (\frac{a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c} \sqrt{-c} x + c}{a^{2} x^{2} - 1}\right ) - 2 \,{\left (15 \, a^{4} x^{4} + 15 \, a^{3} x^{3} - 25 \, a^{2} x^{2} - 25 \, a x + 8\right )} \sqrt{-a^{2} c}}{96 \,{\left (a^{7} c^{4} x^{5} + a^{6} c^{4} x^{4} - 2 \, a^{5} c^{4} x^{3} - 2 \, a^{4} c^{4} x^{2} + a^{3} c^{4} x + a^{2} c^{4}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(1/2)/(-a^2*c*x^2+c)^(7/2),x, algorithm="fricas")

[Out]

-1/96*(15*(a^6*x^5 + a^5*x^4 - 2*a^4*x^3 - 2*a^3*x^2 + a^2*x + a)*sqrt(-c)*log((a^2*c*x^2 - 2*sqrt(-a^2*c)*sqr
t(-c)*x + c)/(a^2*x^2 - 1)) - 2*(15*a^4*x^4 + 15*a^3*x^3 - 25*a^2*x^2 - 25*a*x + 8)*sqrt(-a^2*c))/(a^7*c^4*x^5
+ a^6*c^4*x^4 - 2*a^5*c^4*x^3 - 2*a^4*c^4*x^2 + a^3*c^4*x + a^2*c^4)

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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(((a*x-1)/(a*x+1))**(1/2)/(-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{\sqrt{\frac{a x - 1}{a x + 1}}}{{\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(((a*x-1)/(a*x+1))^(1/2)/(-a^2*c*x^2+c)^(7/2),x, algorithm="giac")

[Out]

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