3.847 \(\int e^{3 \coth ^{-1}(a x)} (c-\frac{c}{a^2 x^2})^{7/2} \, dx\)

Optimal. Leaf size=324 \[ \frac{c^3 x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{5 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{5 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{3 a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{5 a^6 x^5 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{6 a^7 x^6 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 c^3 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]

[Out]

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

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Rubi [A]  time = 0.15883, antiderivative size = 324, 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} \[ \frac{c^3 x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{5 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{5 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{3 a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{5 a^6 x^5 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{6 a^7 x^6 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 c^3 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^(7/2),x]

[Out]

-(c^3*Sqrt[c - c/(a^2*x^2)])/(6*a^7*Sqrt[1 - 1/(a^2*x^2)]*x^6) - (3*c^3*Sqrt[c - c/(a^2*x^2)])/(5*a^6*Sqrt[1 -
1/(a^2*x^2)]*x^5) - (c^3*Sqrt[c - c/(a^2*x^2)])/(4*a^5*Sqrt[1 - 1/(a^2*x^2)]*x^4) + (5*c^3*Sqrt[c - c/(a^2*x^
2)])/(3*a^4*Sqrt[1 - 1/(a^2*x^2)]*x^3) + (5*c^3*Sqrt[c - c/(a^2*x^2)])/(2*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^2) - (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)] +
(3*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^{3 \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^{3 \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)^2 (1+a x)^5}{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{3 a}{x^6}+\frac{a^2}{x^5}-\frac{5 a^3}{x^4}-\frac{5 a^4}{x^3}+\frac{a^5}{x^2}+\frac{3 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{3 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{5 a^6 \sqrt{1-\frac{1}{a^2 x^2}} x^5}-\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 a^5 \sqrt{1-\frac{1}{a^2 x^2}} x^4}+\frac{5 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{3 a^4 \sqrt{1-\frac{1}{a^2 x^2}} x^3}+\frac{5 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2}-\frac{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{3 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 = 0.0833184, size = 94, normalized size = 0.29 \[ \frac{\left (c-\frac{c}{a^2 x^2}\right )^{7/2} \left (3 a^6 \log (x)-\frac{-60 a^7 x^7+60 a^5 x^5-150 a^4 x^4-100 a^3 x^3+15 a^2 x^2+36 a x+10}{60 x^6}\right )}{a^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^(7/2),x]

[Out]

((c - c/(a^2*x^2))^(7/2)*(-(10 + 36*a*x + 15*a^2*x^2 - 100*a^3*x^3 - 150*a^4*x^4 + 60*a^5*x^5 - 60*a^7*x^7)/(6
0*x^6) + 3*a^6*Log[x]))/(a^7*(1 - 1/(a^2*x^2))^(7/2))

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Maple [A]  time = 0.247, size = 112, normalized size = 0.4 \begin{align*}{\frac{ \left ( 60\,{a}^{7}{x}^{7}+180\,{a}^{6}\ln \left ( x \right ){x}^{6}-60\,{x}^{5}{a}^{5}+150\,{x}^{4}{a}^{4}+100\,{x}^{3}{a}^{3}-15\,{a}^{2}{x}^{2}-36\,ax-10 \right ) x}{60\, \left ( ax+1 \right ) ^{3} \left ({a}^{2}{x}^{2}-1 \right ) ^{2}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{{\frac{7}{2}}} \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{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)^(7/2),x)

[Out]

1/60*(60*a^7*x^7+180*a^6*ln(x)*x^6-60*x^5*a^5+150*x^4*a^4+100*x^3*a^3-15*a^2*x^2-36*a*x-10)*(c*(a^2*x^2-1)/a^2
/x^2)^(7/2)*x/(a*x+1)^3/(a^2*x^2-1)^2/((a*x-1)/(a*x+1))^(3/2)

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

[Out]

integrate((c - c/(a^2*x^2))^(7/2)/((a*x - 1)/(a*x + 1))^(3/2), x)

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Fricas [A]  time = 1.65373, size = 219, normalized size = 0.68 \begin{align*} \frac{{\left (60 \, a^{7} c^{3} x^{7} + 180 \, a^{6} c^{3} x^{6} \log \left (x\right ) - 60 \, a^{5} c^{3} x^{5} + 150 \, a^{4} c^{3} x^{4} + 100 \, a^{3} c^{3} x^{3} - 15 \, a^{2} c^{3} x^{2} - 36 \, 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(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a^2/x^2)^(7/2),x, algorithm="fricas")

[Out]

1/60*(60*a^7*c^3*x^7 + 180*a^6*c^3*x^6*log(x) - 60*a^5*c^3*x^5 + 150*a^4*c^3*x^4 + 100*a^3*c^3*x^3 - 15*a^2*c^
3*x^2 - 36*a*c^3*x - 10*c^3)*sqrt(a^2*c)/(a^8*x^6)

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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(1/((a*x-1)/(a*x+1))**(3/2)*(c-c/a**2/x**2)**(7/2),x)

[Out]

Timed out

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

[Out]

integrate((c - c/(a^2*x^2))^(7/2)/((a*x - 1)/(a*x + 1))^(3/2), x)