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

Optimal. Leaf size=360 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{75 \sqrt{1-\frac{1}{a^2 x^2}}}{16 a c^3 (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{59 \sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 (1-a x)^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{2 a c^3 (1-a x)^3 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{16 a c^3 (1-a x)^4 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{201 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{64 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{9 \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{64 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}} \]

[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)]/(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)) - Sqrt[1 - 1/(a^2*x^2)]/(32*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)) + (201*Sqrt[1 - 1/(a^2*x^2)]*Log[1 -
a*x])/(64*a*c^3*Sqrt[c - c/(a^2*x^2)]) - (9*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 = 0.206108, antiderivative size = 360, 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} \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{75 \sqrt{1-\frac{1}{a^2 x^2}}}{16 a c^3 (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{59 \sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 (1-a x)^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{2 a c^3 (1-a x)^3 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{16 a c^3 (1-a x)^4 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{201 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{64 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{9 \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{64 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[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)]/(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)) - Sqrt[1 - 1/(a^2*x^2)]/(32*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)) + (201*Sqrt[1 - 1/(a^2*x^2)]*Log[1 -
a*x])/(64*a*c^3*Sqrt[c - c/(a^2*x^2)]) - (9*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)^5 (1+a x)^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 \left (\frac{1}{a^7}+\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)}+\frac{1}{32 a^7 (1+a x)^2}-\frac{9}{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}}}{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{\sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 \sqrt{c-\frac{c}{a^2 x^2}} (1+a x)}+\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}}}-\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}}}\\ \end{align*}

Mathematica [A]  time = 0.126351, size = 140, normalized size = 0.39 \[ \frac{a^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} \left (\frac{x}{a^7}+\frac{75}{16 a^8 (1-a x)}-\frac{1}{32 a^8 (a x+1)}-\frac{59}{32 a^8 (1-a x)^2}+\frac{1}{2 a^8 (1-a x)^3}-\frac{1}{16 a^8 (1-a x)^4}+\frac{201 \log (1-a x)}{64 a^8}-\frac{9 \log (a x+1)}{64 a^8}\right )}{\left (c-\frac{c}{a^2 x^2}\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^7*(1 - 1/(a^2*x^2))^(7/2)*(x/a^7 - 1/(16*a^8*(1 - a*x)^4) + 1/(2*a^8*(1 - a*x)^3) - 59/(32*a^8*(1 - a*x)^2)
+ 75/(16*a^8*(1 - a*x)) - 1/(32*a^8*(1 + a*x)) + (201*Log[1 - a*x])/(64*a^8) - (9*Log[1 + a*x])/(64*a^8)))/(c
- c/(a^2*x^2))^(7/2)

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Maple [A]  time = 0.249, size = 247, normalized size = 0.7 \begin{align*} -{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) \left ( -64\,{x}^{6}{a}^{6}+9\,\ln \left ( ax+1 \right ){x}^{5}{a}^{5}-201\,\ln \left ( ax-1 \right ){x}^{5}{a}^{5}+192\,{x}^{5}{a}^{5}-27\,\ln \left ( ax+1 \right ){a}^{4}{x}^{4}+603\,\ln \left ( ax-1 \right ){a}^{4}{x}^{4}+174\,{x}^{4}{a}^{4}+18\,{a}^{3}{x}^{3}\ln \left ( ax+1 \right ) -402\,\ln \left ( ax-1 \right ){x}^{3}{a}^{3}-618\,{x}^{3}{a}^{3}+18\,\ln \left ( ax+1 \right ){a}^{2}{x}^{2}-402\,\ln \left ( ax-1 \right ){a}^{2}{x}^{2}+118\,{a}^{2}{x}^{2}-27\,ax\ln \left ( ax+1 \right ) +603\,\ln \left ( ax-1 \right ) xa+414\,ax+9\,\ln \left ( ax+1 \right ) -201\,\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(1/((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+9*ln(a*x+1)*x^5*a^5-201*ln(a*x-1)*x^5*a^5+192*x^5*a
^5-27*ln(a*x+1)*a^4*x^4+603*ln(a*x-1)*a^4*x^4+174*x^4*a^4+18*a^3*x^3*ln(a*x+1)-402*ln(a*x-1)*x^3*a^3-618*x^3*a
^3+18*ln(a*x+1)*a^2*x^2-402*ln(a*x-1)*a^2*x^2+118*a^2*x^2-27*a*x*ln(a*x+1)+603*ln(a*x-1)*x*a+414*a*x+9*ln(a*x+
1)-201*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{1}{{\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(1/((c - c/(a^2*x^2))^(7/2)*((a*x - 1)/(a*x + 1))^(3/2)), x)

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Fricas [A]  time = 1.82671, size = 460, normalized size = 1.28 \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 - 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 ) + 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 ) + 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(1/((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 - 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) + 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) + 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(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{1}{{\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(1/((c - c/(a^2*x^2))^(7/2)*((a*x - 1)/(a*x + 1))^(3/2)), x)