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

Optimal. Leaf size=359 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a c^3 (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{5 \sqrt{1-\frac{1}{a^2 x^2}}}{16 a c^3 (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{11 \sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 (1-a x)^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{24 a c^3 (1-a x)^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{51 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{32 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{19 \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{32 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}} \]

[Out]

(Sqrt[1 - 1/(a^2*x^2)]*x)/(c^3*Sqrt[c - c/(a^2*x^2)]) + Sqrt[1 - 1/(a^2*x^2)]/(24*a*c^3*Sqrt[c - c/(a^2*x^2)]*
(1 - a*x)^3) - (11*Sqrt[1 - 1/(a^2*x^2)])/(32*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)^2) + (3*Sqrt[1 - 1/(a^2*x^
2)])/(2*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)) + Sqrt[1 - 1/(a^2*x^2)]/(32*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 + a*
x)^2) - (5*Sqrt[1 - 1/(a^2*x^2)])/(16*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)) + (51*Sqrt[1 - 1/(a^2*x^2)]*Log[1
- a*x])/(32*a*c^3*Sqrt[c - c/(a^2*x^2)]) - (19*Sqrt[1 - 1/(a^2*x^2)]*Log[1 + a*x])/(32*a*c^3*Sqrt[c - c/(a^2*
x^2)])

________________________________________________________________________________________

Rubi [A]  time = 0.194875, antiderivative size = 359, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6197, 6193, 88} \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a c^3 (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{5 \sqrt{1-\frac{1}{a^2 x^2}}}{16 a c^3 (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{11 \sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 (1-a x)^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{24 a c^3 (1-a x)^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{51 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{32 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{19 \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{32 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 1/(a^2*x^2)]*x)/(c^3*Sqrt[c - c/(a^2*x^2)]) + Sqrt[1 - 1/(a^2*x^2)]/(24*a*c^3*Sqrt[c - c/(a^2*x^2)]*
(1 - a*x)^3) - (11*Sqrt[1 - 1/(a^2*x^2)])/(32*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)^2) + (3*Sqrt[1 - 1/(a^2*x^
2)])/(2*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)) + Sqrt[1 - 1/(a^2*x^2)]/(32*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 + a*
x)^2) - (5*Sqrt[1 - 1/(a^2*x^2)])/(16*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)) + (51*Sqrt[1 - 1/(a^2*x^2)]*Log[1
- a*x])/(32*a*c^3*Sqrt[c - c/(a^2*x^2)]) - (19*Sqrt[1 - 1/(a^2*x^2)]*Log[1 + a*x])/(32*a*c^3*Sqrt[c - c/(a^2*
x^2)])

Rule 6197

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

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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{e^{\coth ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^{7/2}} \, dx &=\frac{\sqrt{1-\frac{1}{a^2 x^2}} \int \frac{e^{\coth ^{-1}(a x)}}{\left (1-\frac{1}{a^2 x^2}\right )^{7/2}} \, dx}{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}\\ &=\frac{\left (a^7 \sqrt{1-\frac{1}{a^2 x^2}}\right ) \int \frac{x^7}{(-1+a x)^4 (1+a x)^3} \, dx}{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}\\ &=\frac{\left (a^7 \sqrt{1-\frac{1}{a^2 x^2}}\right ) \int \left (\frac{1}{a^7}+\frac{1}{8 a^7 (-1+a x)^4}+\frac{11}{16 a^7 (-1+a x)^3}+\frac{3}{2 a^7 (-1+a x)^2}+\frac{51}{32 a^7 (-1+a x)}-\frac{1}{16 a^7 (1+a x)^3}+\frac{5}{16 a^7 (1+a x)^2}-\frac{19}{32 a^7 (1+a x)}\right ) \, dx}{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}\\ &=\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{24 a c^3 \sqrt{c-\frac{c}{a^2 x^2}} (1-a x)^3}-\frac{11 \sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 \sqrt{c-\frac{c}{a^2 x^2}} (1-a x)^2}+\frac{3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a c^3 \sqrt{c-\frac{c}{a^2 x^2}} (1-a x)}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 \sqrt{c-\frac{c}{a^2 x^2}} (1+a x)^2}-\frac{5 \sqrt{1-\frac{1}{a^2 x^2}}}{16 a c^3 \sqrt{c-\frac{c}{a^2 x^2}} (1+a x)}+\frac{51 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{32 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{19 \sqrt{1-\frac{1}{a^2 x^2}} \log (1+a x)}{32 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}}\\ \end{align*}

Mathematica [A]  time = 0.147996, size = 104, normalized size = 0.29 \[ \frac{\left (1-\frac{1}{a^2 x^2}\right )^{7/2} \left (96 a x+\frac{144}{1-a x}-\frac{30}{a x+1}-\frac{33}{(a x-1)^2}+\frac{3}{(a x+1)^2}-\frac{4}{(a x-1)^3}+153 \log (1-a x)-57 \log (a x+1)\right )}{96 a \left (c-\frac{c}{a^2 x^2}\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1 - 1/(a^2*x^2))^(7/2)*(96*a*x + 144/(1 - a*x) - 4/(-1 + a*x)^3 - 33/(-1 + a*x)^2 + 3/(1 + a*x)^2 - 30/(1 +
a*x) + 153*Log[1 - a*x] - 57*Log[1 + a*x]))/(96*a*(c - c/(a^2*x^2))^(7/2))

________________________________________________________________________________________

Maple [A]  time = 0.258, size = 247, normalized size = 0.7 \begin{align*} -{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) \left ( -96\,{x}^{6}{a}^{6}+57\,\ln \left ( ax+1 \right ){x}^{5}{a}^{5}-153\,\ln \left ( ax-1 \right ){x}^{5}{a}^{5}+96\,{x}^{5}{a}^{5}-57\,\ln \left ( ax+1 \right ){a}^{4}{x}^{4}+153\,\ln \left ( ax-1 \right ){a}^{4}{x}^{4}+366\,{x}^{4}{a}^{4}-114\,{a}^{3}{x}^{3}\ln \left ( ax+1 \right ) +306\,\ln \left ( ax-1 \right ){x}^{3}{a}^{3}-222\,{x}^{3}{a}^{3}+114\,\ln \left ( ax+1 \right ){a}^{2}{x}^{2}-306\,\ln \left ( ax-1 \right ){a}^{2}{x}^{2}-338\,{a}^{2}{x}^{2}+57\,ax\ln \left ( ax+1 \right ) -153\,\ln \left ( ax-1 \right ) xa+122\,ax-57\,\ln \left ( ax+1 \right ) +153\,\ln \left ( ax-1 \right ) +88 \right ) }{96\,{a}^{8}{x}^{7}}{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96/((a*x-1)/(a*x+1))^(1/2)*(a*x-1)*(a*x+1)*(-96*x^6*a^6+57*ln(a*x+1)*x^5*a^5-153*ln(a*x-1)*x^5*a^5+96*x^5*a
^5-57*ln(a*x+1)*a^4*x^4+153*ln(a*x-1)*a^4*x^4+366*x^4*a^4-114*a^3*x^3*ln(a*x+1)+306*ln(a*x-1)*x^3*a^3-222*x^3*
a^3+114*ln(a*x+1)*a^2*x^2-306*ln(a*x-1)*a^2*x^2-338*a^2*x^2+57*a*x*ln(a*x+1)-153*ln(a*x-1)*x*a+122*a*x-57*ln(a
*x+1)+153*ln(a*x-1)+88)/a^8/x^7/(c*(a^2*x^2-1)/a^2/x^2)^(7/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{7}{2}} \sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.62203, size = 443, normalized size = 1.23 \begin{align*} \frac{{\left (96 \, a^{6} x^{6} - 96 \, a^{5} x^{5} - 366 \, a^{4} x^{4} + 222 \, a^{3} x^{3} + 338 \, a^{2} x^{2} - 122 \, a x - 57 \,{\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \log \left (a x + 1\right ) + 153 \,{\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \log \left (a x - 1\right ) - 88\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(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a^2/x^2)^(7/2),x, algorithm="fricas")

[Out]

1/96*(96*a^6*x^6 - 96*a^5*x^5 - 366*a^4*x^4 + 222*a^3*x^3 + 338*a^2*x^2 - 122*a*x - 57*(a^5*x^5 - a^4*x^4 - 2*
a^3*x^3 + 2*a^2*x^2 + a*x - 1)*log(a*x + 1) + 153*(a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1)*log(a*
x - 1) - 88)*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)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{7}{2}} \sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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