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

Optimal. Leaf size=299 \[ \frac{a^7 x^8 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}}+\frac{3 a^5 x^6 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}}-\frac{3 a^4 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{2 \left (1-a^2 x^2\right )^{7/2}}-\frac{a^3 x^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}}+\frac{3 a^2 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{4 \left (1-a^2 x^2\right )^{7/2}}+\frac{a x^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{5 \left (1-a^2 x^2\right )^{7/2}}-\frac{x \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{6 \left (1-a^2 x^2\right )^{7/2}}-\frac{a^6 x^7 \log (x) \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}} \]

[Out]

-((c - c/(a^2*x^2))^(7/2)*x)/(6*(1 - a^2*x^2)^(7/2)) + (a*(c - c/(a^2*x^2))^(7/2)*x^2)/(5*(1 - a^2*x^2)^(7/2))
+ (3*a^2*(c - c/(a^2*x^2))^(7/2)*x^3)/(4*(1 - a^2*x^2)^(7/2)) - (a^3*(c - c/(a^2*x^2))^(7/2)*x^4)/(1 - a^2*x^
2)^(7/2) - (3*a^4*(c - c/(a^2*x^2))^(7/2)*x^5)/(2*(1 - a^2*x^2)^(7/2)) + (3*a^5*(c - c/(a^2*x^2))^(7/2)*x^6)/(
1 - a^2*x^2)^(7/2) + (a^7*(c - c/(a^2*x^2))^(7/2)*x^8)/(1 - a^2*x^2)^(7/2) - (a^6*(c - c/(a^2*x^2))^(7/2)*x^7*
Log[x])/(1 - a^2*x^2)^(7/2)

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Rubi [A]  time = 0.192087, antiderivative size = 299, 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 = {6160, 6150, 88} \[ \frac{a^7 x^8 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}}+\frac{3 a^5 x^6 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}}-\frac{3 a^4 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{2 \left (1-a^2 x^2\right )^{7/2}}-\frac{a^3 x^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}}+\frac{3 a^2 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{4 \left (1-a^2 x^2\right )^{7/2}}+\frac{a x^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{5 \left (1-a^2 x^2\right )^{7/2}}-\frac{x \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{6 \left (1-a^2 x^2\right )^{7/2}}-\frac{a^6 x^7 \log (x) \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((c - c/(a^2*x^2))^(7/2)*x)/(6*(1 - a^2*x^2)^(7/2)) + (a*(c - c/(a^2*x^2))^(7/2)*x^2)/(5*(1 - a^2*x^2)^(7/2))
+ (3*a^2*(c - c/(a^2*x^2))^(7/2)*x^3)/(4*(1 - a^2*x^2)^(7/2)) - (a^3*(c - c/(a^2*x^2))^(7/2)*x^4)/(1 - a^2*x^
2)^(7/2) - (3*a^4*(c - c/(a^2*x^2))^(7/2)*x^5)/(2*(1 - a^2*x^2)^(7/2)) + (3*a^5*(c - c/(a^2*x^2))^(7/2)*x^6)/(
1 - a^2*x^2)^(7/2) + (a^7*(c - c/(a^2*x^2))^(7/2)*x^8)/(1 - a^2*x^2)^(7/2) - (a^6*(c - c/(a^2*x^2))^(7/2)*x^7*
Log[x])/(1 - a^2*x^2)^(7/2)

Rule 6160

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbol] :> Dist[(x^(2*p)*(c + d/x^2)^p)/
(1 + (c*x^2)/d)^p, Int[(u*(1 + (c*x^2)/d)^p*E^(n*ArcTanh[a*x]))/x^(2*p), x], x] /; FreeQ[{a, c, d, n, p}, x] &
& EqQ[c + a^2*d, 0] &&  !IntegerQ[p] &&  !IntegerQ[n/2]

Rule 6150

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[x^m*(1 -
a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p
] || GtQ[c, 0])

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^{-\tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^{7/2} \, dx &=\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac{e^{-\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{7/2}}{x^7} \, dx}{\left (1-a^2 x^2\right )^{7/2}}\\ &=\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac{(1-a x)^4 (1+a x)^3}{x^7} \, dx}{\left (1-a^2 x^2\right )^{7/2}}\\ &=\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7\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}{\left (1-a^2 x^2\right )^{7/2}}\\ &=-\frac{\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x}{6 \left (1-a^2 x^2\right )^{7/2}}+\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2}{5 \left (1-a^2 x^2\right )^{7/2}}+\frac{3 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3}{4 \left (1-a^2 x^2\right )^{7/2}}-\frac{a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^4}{\left (1-a^2 x^2\right )^{7/2}}-\frac{3 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^5}{2 \left (1-a^2 x^2\right )^{7/2}}+\frac{3 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^6}{\left (1-a^2 x^2\right )^{7/2}}+\frac{a^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^8}{\left (1-a^2 x^2\right )^{7/2}}-\frac{a^6 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7 \log (x)}{\left (1-a^2 x^2\right )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.0565689, size = 98, normalized size = 0.33 \[ -\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}} \left (60 a^7 x^7+180 a^5 x^5-90 a^4 x^4-60 a^3 x^3+45 a^2 x^2-60 a^6 x^6 \log (x)+12 a x-10\right )}{60 a^6 x^5 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.155, size = 102, normalized size = 0.3 \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 ) }{60\, \left ({a}^{2}{x}^{2}-1 \right ) ^{4}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{{\frac{7}{2}}}\sqrt{-{a}^{2}{x}^{2}+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^2*x^2+1)^(1/2),x)

[Out]

-1/60*(c*(a^2*x^2-1)/a^2/x^2)^(7/2)*x/(a^2*x^2-1)^4*(-a^2*x^2+1)^(1/2)*(-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)

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

[Out]

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

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Fricas [A]  time = 2.27702, size = 1142, normalized size = 3.82 \begin{align*} \left [\frac{30 \,{\left (a^{7} c^{3} x^{7} - a^{5} c^{3} x^{5}\right )} \sqrt{-c} \log \left (\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} +{\left (a x^{5} - a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right ) +{\left (60 \, a^{7} c^{3} x^{7} + 180 \, a^{5} c^{3} x^{5} - 90 \, a^{4} c^{3} x^{4} -{\left (60 \, a^{7} + 180 \, a^{5} - 90 \, a^{4} - 60 \, a^{3} + 45 \, a^{2} + 12 \, a - 10\right )} c^{3} x^{6} - 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} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{60 \,{\left (a^{8} x^{7} - a^{6} x^{5}\right )}}, -\frac{60 \,{\left (a^{7} c^{3} x^{7} - a^{5} c^{3} x^{5}\right )} \sqrt{c} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left (a x^{3} + a x\right )} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} -{\left (a^{2} + 1\right )} c x^{2} + c}\right ) -{\left (60 \, a^{7} c^{3} x^{7} + 180 \, a^{5} c^{3} x^{5} - 90 \, a^{4} c^{3} x^{4} -{\left (60 \, a^{7} + 180 \, a^{5} - 90 \, a^{4} - 60 \, a^{3} + 45 \, a^{2} + 12 \, a - 10\right )} c^{3} x^{6} - 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} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{60 \,{\left (a^{8} x^{7} - a^{6} x^{5}\right )}}\right ] \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^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

[1/60*(30*(a^7*c^3*x^7 - a^5*c^3*x^5)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 + (a*x^5 - a*x)*sqrt(-a^2*x^
2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) + (60*a^7*c^3*x^7 + 180*a^5*c^3*x^5 - 90
*a^4*c^3*x^4 - (60*a^7 + 180*a^5 - 90*a^4 - 60*a^3 + 45*a^2 + 12*a - 10)*c^3*x^6 - 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*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^8*x^7 - a^6*x^5), -1/60*(60
*(a^7*c^3*x^7 - a^5*c^3*x^5)*sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3 + a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2
*x^2))/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) - (60*a^7*c^3*x^7 + 180*a^5*c^3*x^5 - 90*a^4*c^3*x^4 - (60*a^7 + 180
*a^5 - 90*a^4 - 60*a^3 + 45*a^2 + 12*a - 10)*c^3*x^6 - 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*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^8*x^7 - a^6*x^5)]

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

[Out]

Timed out

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

[Out]

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