3.853 \(\int \frac{e^{3 \coth ^{-1}(a x)}}{(c-\frac{c}{a^2 x^2})^{5/2}} \, dx\)

Optimal. Leaf size=267 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{31 \sqrt{1-\frac{1}{a^2 x^2}}}{8 a c^2 (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{9 \sqrt{1-\frac{1}{a^2 x^2}}}{8 a c^2 (1-a x)^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{6 a c^2 (1-a x)^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{49 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{16 a c^2 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{16 a c^2 \sqrt{c-\frac{c}{a^2 x^2}}} \]

[Out]

(Sqrt[1 - 1/(a^2*x^2)]*x)/(c^2*Sqrt[c - c/(a^2*x^2)]) + Sqrt[1 - 1/(a^2*x^2)]/(6*a*c^2*Sqrt[c - c/(a^2*x^2)]*(
1 - a*x)^3) - (9*Sqrt[1 - 1/(a^2*x^2)])/(8*a*c^2*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)^2) + (31*Sqrt[1 - 1/(a^2*x^2)
])/(8*a*c^2*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)) + (49*Sqrt[1 - 1/(a^2*x^2)]*Log[1 - a*x])/(16*a*c^2*Sqrt[c - c/(a
^2*x^2)]) - (Sqrt[1 - 1/(a^2*x^2)]*Log[1 + a*x])/(16*a*c^2*Sqrt[c - c/(a^2*x^2)])

________________________________________________________________________________________

Rubi [A]  time = 0.176567, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6197, 6193, 88} \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{31 \sqrt{1-\frac{1}{a^2 x^2}}}{8 a c^2 (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{9 \sqrt{1-\frac{1}{a^2 x^2}}}{8 a c^2 (1-a x)^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{6 a c^2 (1-a x)^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{49 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{16 a c^2 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{16 a c^2 \sqrt{c-\frac{c}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcCoth[a*x])/(c - c/(a^2*x^2))^(5/2),x]

[Out]

(Sqrt[1 - 1/(a^2*x^2)]*x)/(c^2*Sqrt[c - c/(a^2*x^2)]) + Sqrt[1 - 1/(a^2*x^2)]/(6*a*c^2*Sqrt[c - c/(a^2*x^2)]*(
1 - a*x)^3) - (9*Sqrt[1 - 1/(a^2*x^2)])/(8*a*c^2*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)^2) + (31*Sqrt[1 - 1/(a^2*x^2)
])/(8*a*c^2*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)) + (49*Sqrt[1 - 1/(a^2*x^2)]*Log[1 - a*x])/(16*a*c^2*Sqrt[c - c/(a
^2*x^2)]) - (Sqrt[1 - 1/(a^2*x^2)]*Log[1 + a*x])/(16*a*c^2*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^{3 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^{5/2}} \, dx &=\frac{\sqrt{1-\frac{1}{a^2 x^2}} \int \frac{e^{3 \coth ^{-1}(a x)}}{\left (1-\frac{1}{a^2 x^2}\right )^{5/2}} \, dx}{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}\\ &=\frac{\left (a^5 \sqrt{1-\frac{1}{a^2 x^2}}\right ) \int \frac{x^5}{(-1+a x)^4 (1+a x)} \, dx}{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}\\ &=\frac{\left (a^5 \sqrt{1-\frac{1}{a^2 x^2}}\right ) \int \left (\frac{1}{a^5}+\frac{1}{2 a^5 (-1+a x)^4}+\frac{9}{4 a^5 (-1+a x)^3}+\frac{31}{8 a^5 (-1+a x)^2}+\frac{49}{16 a^5 (-1+a x)}-\frac{1}{16 a^5 (1+a x)}\right ) \, dx}{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}\\ &=\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{6 a c^2 \sqrt{c-\frac{c}{a^2 x^2}} (1-a x)^3}-\frac{9 \sqrt{1-\frac{1}{a^2 x^2}}}{8 a c^2 \sqrt{c-\frac{c}{a^2 x^2}} (1-a x)^2}+\frac{31 \sqrt{1-\frac{1}{a^2 x^2}}}{8 a c^2 \sqrt{c-\frac{c}{a^2 x^2}} (1-a x)}+\frac{49 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{16 a c^2 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}} \log (1+a x)}{16 a c^2 \sqrt{c-\frac{c}{a^2 x^2}}}\\ \end{align*}

Mathematica [A]  time = 0.105571, size = 86, normalized size = 0.32 \[ \frac{\left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (48 a x+\frac{186}{1-a x}-\frac{54}{(a x-1)^2}-\frac{8}{(a x-1)^3}+147 \log (1-a x)-3 \log (a x+1)\right )}{48 a \left (c-\frac{c}{a^2 x^2}\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(3*ArcCoth[a*x])/(c - c/(a^2*x^2))^(5/2),x]

[Out]

((1 - 1/(a^2*x^2))^(5/2)*(48*a*x + 186/(1 - a*x) - 8/(-1 + a*x)^3 - 54/(-1 + a*x)^2 + 147*Log[1 - a*x] - 3*Log
[1 + a*x]))/(48*a*(c - c/(a^2*x^2))^(5/2))

________________________________________________________________________________________

Maple [A]  time = 0.234, size = 175, normalized size = 0.7 \begin{align*} -{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) \left ( -48\,{x}^{4}{a}^{4}+3\,{a}^{3}{x}^{3}\ln \left ( ax+1 \right ) -147\,\ln \left ( ax-1 \right ){x}^{3}{a}^{3}+144\,{x}^{3}{a}^{3}-9\,\ln \left ( ax+1 \right ){a}^{2}{x}^{2}+441\,\ln \left ( ax-1 \right ){a}^{2}{x}^{2}+42\,{a}^{2}{x}^{2}+9\,ax\ln \left ( ax+1 \right ) -441\,\ln \left ( ax-1 \right ) xa-270\,ax-3\,\ln \left ( ax+1 \right ) +147\,\ln \left ( ax-1 \right ) +140 \right ) }{48\,{a}^{6}{x}^{5}} \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^(5/2),x)

[Out]

-1/48/((a*x-1)/(a*x+1))^(3/2)*(a*x-1)*(a*x+1)*(-48*x^4*a^4+3*a^3*x^3*ln(a*x+1)-147*ln(a*x-1)*x^3*a^3+144*x^3*a
^3-9*ln(a*x+1)*a^2*x^2+441*ln(a*x-1)*a^2*x^2+42*a^2*x^2+9*a*x*ln(a*x+1)-441*ln(a*x-1)*x*a-270*a*x-3*ln(a*x+1)+
147*ln(a*x-1)+140)/a^6/x^5/(c*(a^2*x^2-1)/a^2/x^2)^(5/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{5}{2}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((c - c/(a^2*x^2))^(5/2)*((a*x - 1)/(a*x + 1))^(3/2)), x)

________________________________________________________________________________________

Fricas [A]  time = 1.60547, size = 313, normalized size = 1.17 \begin{align*} \frac{{\left (48 \, a^{4} x^{4} - 144 \, a^{3} x^{3} - 42 \, a^{2} x^{2} + 270 \, a x - 3 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (a x + 1\right ) + 147 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (a x - 1\right ) - 140\right )} \sqrt{a^{2} c}}{48 \,{\left (a^{5} c^{3} x^{3} - 3 \, a^{4} c^{3} x^{2} + 3 \, a^{3} c^{3} x - a^{2} c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^(5/2),x, algorithm="fricas")

[Out]

1/48*(48*a^4*x^4 - 144*a^3*x^3 - 42*a^2*x^2 + 270*a*x - 3*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log(a*x + 1) + 147
*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log(a*x - 1) - 140)*sqrt(a^2*c)/(a^5*c^3*x^3 - 3*a^4*c^3*x^2 + 3*a^3*c^3*x
- a^2*c^3)

________________________________________________________________________________________

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))**(3/2)/(c-c/a**2/x**2)**(5/2),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{5}{2}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^(5/2),x, algorithm="giac")

[Out]

integrate(1/((c - c/(a^2*x^2))^(5/2)*((a*x - 1)/(a*x + 1))^(3/2)), x)