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

Optimal. Leaf size=322 \[ \frac{c^3 x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{5 a^6 x^5 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{6 a^7 x^6 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]

[Out]

-(c^3*Sqrt[c - c/(a^2*x^2)])/(6*a^7*Sqrt[1 - 1/(a^2*x^2)]*x^6) + (c^3*Sqrt[c - c/(a^2*x^2)])/(5*a^6*Sqrt[1 - 1
/(a^2*x^2)]*x^5) + (3*c^3*Sqrt[c - c/(a^2*x^2)])/(4*a^5*Sqrt[1 - 1/(a^2*x^2)]*x^4) - (c^3*Sqrt[c - c/(a^2*x^2)
])/(a^4*Sqrt[1 - 1/(a^2*x^2)]*x^3) - (3*c^3*Sqrt[c - c/(a^2*x^2)])/(2*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^2) + (3*c^3*
Sqrt[c - c/(a^2*x^2)])/(a^2*Sqrt[1 - 1/(a^2*x^2)]*x) + (c^3*Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2)] - (
c^3*Sqrt[c - c/(a^2*x^2)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)])

________________________________________________________________________________________

Rubi [A]  time = 0.172026, antiderivative size = 322, 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} \[ \frac{c^3 x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{5 a^6 x^5 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{6 a^7 x^6 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^3*Sqrt[c - c/(a^2*x^2)])/(6*a^7*Sqrt[1 - 1/(a^2*x^2)]*x^6) + (c^3*Sqrt[c - c/(a^2*x^2)])/(5*a^6*Sqrt[1 - 1
/(a^2*x^2)]*x^5) + (3*c^3*Sqrt[c - c/(a^2*x^2)])/(4*a^5*Sqrt[1 - 1/(a^2*x^2)]*x^4) - (c^3*Sqrt[c - c/(a^2*x^2)
])/(a^4*Sqrt[1 - 1/(a^2*x^2)]*x^3) - (3*c^3*Sqrt[c - c/(a^2*x^2)])/(2*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^2) + (3*c^3*
Sqrt[c - c/(a^2*x^2)])/(a^2*Sqrt[1 - 1/(a^2*x^2)]*x) + (c^3*Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2)] - (
c^3*Sqrt[c - c/(a^2*x^2)]*Log[x])/(a*Sqrt[1 - 1/(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 e^{-\coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^{7/2} \, dx &=\frac{\left (c^3 \sqrt{c-\frac{c}{a^2 x^2}}\right ) \int e^{-\coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^{7/2} \, dx}{\sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\left (c^3 \sqrt{c-\frac{c}{a^2 x^2}}\right ) \int \frac{(-1+a x)^4 (1+a x)^3}{x^7} \, dx}{a^7 \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\left (c^3 \sqrt{c-\frac{c}{a^2 x^2}}\right ) \int \left (a^7+\frac{1}{x^7}-\frac{a}{x^6}-\frac{3 a^2}{x^5}+\frac{3 a^3}{x^4}+\frac{3 a^4}{x^3}-\frac{3 a^5}{x^2}-\frac{a^6}{x}\right ) \, dx}{a^7 \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=-\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{6 a^7 \sqrt{1-\frac{1}{a^2 x^2}} x^6}+\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{5 a^6 \sqrt{1-\frac{1}{a^2 x^2}} x^5}+\frac{3 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 a^5 \sqrt{1-\frac{1}{a^2 x^2}} x^4}-\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{a^4 \sqrt{1-\frac{1}{a^2 x^2}} x^3}-\frac{3 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2}+\frac{3 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 \sqrt{1-\frac{1}{a^2 x^2}} x}+\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}} x}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}} \log (x)}{a \sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}

Mathematica [A]  time = 0.0698008, size = 97, normalized size = 0.3 \[ \frac{\left (c-\frac{c}{a^2 x^2}\right )^{7/2} \left (-\frac{3 a^4}{2 x^2}-\frac{a^3}{x^3}+\frac{3 a^2}{4 x^4}+a^7 x+\frac{3 a^5}{x}-a^6 \log (x)+\frac{a}{5 x^5}-\frac{1}{6 x^6}\right )}{a^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c - c/(a^2*x^2))^(7/2)*(-1/(6*x^6) + a/(5*x^5) + (3*a^2)/(4*x^4) - a^3/x^3 - (3*a^4)/(2*x^2) + (3*a^5)/x + a
^7*x - a^6*Log[x]))/(a^7*(1 - 1/(a^2*x^2))^(7/2))

________________________________________________________________________________________

Maple [A]  time = 0.236, size = 112, normalized size = 0.4 \begin{align*} -{\frac{x \left ( -60\,{a}^{7}{x}^{7}+60\,{a}^{6}\ln \left ( x \right ){x}^{6}-180\,{x}^{5}{a}^{5}+90\,{x}^{4}{a}^{4}+60\,{x}^{3}{a}^{3}-45\,{a}^{2}{x}^{2}-12\,ax+10 \right ) }{ \left ( 60\,ax-60 \right ) \left ({a}^{2}{x}^{2}-1 \right ) ^{3}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{{\frac{7}{2}}}\sqrt{{\frac{ax-1}{ax+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*((a*x-1)/(a*x+1))^(1/2)*(c*(a^2*x^2-1)/a^2/x^2)^(7/2)*x*(-60*a^7*x^7+60*a^6*ln(x)*x^6-180*x^5*a^5+90*x^4
*a^4+60*x^3*a^3-45*a^2*x^2-12*a*x+10)/(a*x-1)/(a^2*x^2-1)^3

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.53244, size = 216, normalized size = 0.67 \begin{align*} \frac{{\left (60 \, a^{7} c^{3} x^{7} - 60 \, a^{6} c^{3} x^{6} \log \left (x\right ) + 180 \, a^{5} c^{3} x^{5} - 90 \, a^{4} c^{3} x^{4} - 60 \, a^{3} c^{3} x^{3} + 45 \, a^{2} c^{3} x^{2} + 12 \, a c^{3} x - 10 \, c^{3}\right )} \sqrt{a^{2} c}}{60 \, a^{8} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(60*a^7*c^3*x^7 - 60*a^6*c^3*x^6*log(x) + 180*a^5*c^3*x^5 - 90*a^4*c^3*x^4 - 60*a^3*c^3*x^3 + 45*a^2*c^3*
x^2 + 12*a*c^3*x - 10*c^3)*sqrt(a^2*c)/(a^8*x^6)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

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