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

Optimal. Leaf size=357 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{75 \sqrt{1-\frac{1}{a^2 x^2}}}{16 a c^3 (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{59 \sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{2 a c^3 (a x+1)^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{16 a c^3 (a x+1)^4 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{9 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{64 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{201 \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{64 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}} \]

[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)]/(32*a*c^3*Sqrt[c - c/(a^2*x^2)]*
(1 - a*x)) + Sqrt[1 - 1/(a^2*x^2)]/(16*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)^4) - Sqrt[1 - 1/(a^2*x^2)]/(2*a*c
^3*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)^3) + (59*Sqrt[1 - 1/(a^2*x^2)])/(32*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)^2
) - (75*Sqrt[1 - 1/(a^2*x^2)])/(16*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)) + (9*Sqrt[1 - 1/(a^2*x^2)]*Log[1 - a
*x])/(64*a*c^3*Sqrt[c - c/(a^2*x^2)]) - (201*Sqrt[1 - 1/(a^2*x^2)]*Log[1 + a*x])/(64*a*c^3*Sqrt[c - c/(a^2*x^2
)])

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Rubi [A]  time = 0.198547, antiderivative size = 357, 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} \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{75 \sqrt{1-\frac{1}{a^2 x^2}}}{16 a c^3 (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{59 \sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{2 a c^3 (a x+1)^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{16 a c^3 (a x+1)^4 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{9 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{64 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{201 \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{64 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(3*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)]/(32*a*c^3*Sqrt[c - c/(a^2*x^2)]*
(1 - a*x)) + Sqrt[1 - 1/(a^2*x^2)]/(16*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)^4) - Sqrt[1 - 1/(a^2*x^2)]/(2*a*c
^3*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)^3) + (59*Sqrt[1 - 1/(a^2*x^2)])/(32*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)^2
) - (75*Sqrt[1 - 1/(a^2*x^2)])/(16*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)) + (9*Sqrt[1 - 1/(a^2*x^2)]*Log[1 - a
*x])/(64*a*c^3*Sqrt[c - c/(a^2*x^2)]) - (201*Sqrt[1 - 1/(a^2*x^2)]*Log[1 + a*x])/(64*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^{-3 \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^{-3 \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)^2 (1+a x)^5} \, 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}{32 a^7 (-1+a x)^2}+\frac{9}{64 a^7 (-1+a x)}-\frac{1}{4 a^7 (1+a x)^5}+\frac{3}{2 a^7 (1+a x)^4}-\frac{59}{16 a^7 (1+a x)^3}+\frac{75}{16 a^7 (1+a x)^2}-\frac{201}{64 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}}}{32 a c^3 \sqrt{c-\frac{c}{a^2 x^2}} (1-a x)}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{16 a c^3 \sqrt{c-\frac{c}{a^2 x^2}} (1+a x)^4}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{2 a c^3 \sqrt{c-\frac{c}{a^2 x^2}} (1+a x)^3}+\frac{59 \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{75 \sqrt{1-\frac{1}{a^2 x^2}}}{16 a c^3 \sqrt{c-\frac{c}{a^2 x^2}} (1+a x)}+\frac{9 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{64 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{201 \sqrt{1-\frac{1}{a^2 x^2}} \log (1+a x)}{64 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}}\\ \end{align*}

Mathematica [A]  time = 0.179346, size = 105, normalized size = 0.29 \[ \frac{\left (1-\frac{1}{a^2 x^2}\right )^{7/2} \left (2 \left (32 a x+\frac{1}{1-a x}-\frac{150}{a x+1}+\frac{59}{(a x+1)^2}-\frac{16}{(a x+1)^3}+\frac{2}{(a x+1)^4}\right )+9 \log (1-a x)-201 \log (a x+1)\right )}{64 a \left (c-\frac{c}{a^2 x^2}\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^(7/2)),x]

[Out]

((1 - 1/(a^2*x^2))^(7/2)*(2*(32*a*x + (1 - a*x)^(-1) + 2/(1 + a*x)^4 - 16/(1 + a*x)^3 + 59/(1 + a*x)^2 - 150/(
1 + a*x)) + 9*Log[1 - a*x] - 201*Log[1 + a*x]))/(64*a*(c - c/(a^2*x^2))^(7/2))

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Maple [A]  time = 0.263, size = 247, normalized size = 0.7 \begin{align*} -{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) \left ( -64\,{x}^{6}{a}^{6}+201\,\ln \left ( ax+1 \right ){x}^{5}{a}^{5}-9\,\ln \left ( ax-1 \right ){x}^{5}{a}^{5}-192\,{x}^{5}{a}^{5}+603\,\ln \left ( ax+1 \right ){a}^{4}{x}^{4}-27\,\ln \left ( ax-1 \right ){a}^{4}{x}^{4}+174\,{x}^{4}{a}^{4}+402\,{a}^{3}{x}^{3}\ln \left ( ax+1 \right ) -18\,\ln \left ( ax-1 \right ){x}^{3}{a}^{3}+618\,{x}^{3}{a}^{3}-402\,\ln \left ( ax+1 \right ){a}^{2}{x}^{2}+18\,\ln \left ( ax-1 \right ){a}^{2}{x}^{2}+118\,{a}^{2}{x}^{2}-603\,ax\ln \left ( ax+1 \right ) +27\,\ln \left ( ax-1 \right ) xa-414\,ax-201\,\ln \left ( ax+1 \right ) +9\,\ln \left ( ax-1 \right ) -208 \right ) }{64\,{a}^{8}{x}^{7}} \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{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64*((a*x-1)/(a*x+1))^(3/2)*(a*x+1)*(a*x-1)*(-64*x^6*a^6+201*ln(a*x+1)*x^5*a^5-9*ln(a*x-1)*x^5*a^5-192*x^5*a
^5+603*ln(a*x+1)*a^4*x^4-27*ln(a*x-1)*a^4*x^4+174*x^4*a^4+402*a^3*x^3*ln(a*x+1)-18*ln(a*x-1)*x^3*a^3+618*x^3*a
^3-402*ln(a*x+1)*a^2*x^2+18*ln(a*x-1)*a^2*x^2+118*a^2*x^2-603*a*x*ln(a*x+1)+27*ln(a*x-1)*x*a-414*a*x-201*ln(a*
x+1)+9*ln(a*x-1)-208)/a^8/x^7/(c*(a^2*x^2-1)/a^2/x^2)^(7/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{{\left (c - \frac{c}{a^{2} x^{2}}\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))^(3/2)/(c-c/a^2/x^2)^(7/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.65509, size = 460, normalized size = 1.29 \begin{align*} \frac{{\left (64 \, a^{6} x^{6} + 192 \, a^{5} x^{5} - 174 \, a^{4} x^{4} - 618 \, a^{3} x^{3} - 118 \, a^{2} x^{2} + 414 \, a x - 201 \,{\left (a^{5} x^{5} + 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} - 3 \, a x - 1\right )} \log \left (a x + 1\right ) + 9 \,{\left (a^{5} x^{5} + 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} - 3 \, a x - 1\right )} \log \left (a x - 1\right ) + 208\right )} \sqrt{a^{2} c}}{64 \,{\left (a^{7} c^{4} x^{5} + 3 \, a^{6} c^{4} x^{4} + 2 \, a^{5} c^{4} x^{3} - 2 \, a^{4} c^{4} x^{2} - 3 \, 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))^(3/2)/(c-c/a^2/x^2)^(7/2),x, algorithm="fricas")

[Out]

1/64*(64*a^6*x^6 + 192*a^5*x^5 - 174*a^4*x^4 - 618*a^3*x^3 - 118*a^2*x^2 + 414*a*x - 201*(a^5*x^5 + 3*a^4*x^4
+ 2*a^3*x^3 - 2*a^2*x^2 - 3*a*x - 1)*log(a*x + 1) + 9*(a^5*x^5 + 3*a^4*x^4 + 2*a^3*x^3 - 2*a^2*x^2 - 3*a*x - 1
)*log(a*x - 1) + 208)*sqrt(a^2*c)/(a^7*c^4*x^5 + 3*a^6*c^4*x^4 + 2*a^5*c^4*x^3 - 2*a^4*c^4*x^2 - 3*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))**(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 (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{{\left (c - \frac{c}{a^{2} x^{2}}\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))^(3/2)/(c-c/a^2/x^2)^(7/2),x, algorithm="giac")

[Out]

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