∫ e− tanh−1(ax) ____________________(c− c ____

Optimal. Leaf size=265 \[ \frac {\left (1-a^2 x^2\right )^{5/2}}{a^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}+\frac {\left (1-a^2 x^2\right )^{5/2}}{8 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{5/2}}{8 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 (1+a x)^2}-\frac {\left (1-a^2 x^2\right )^{5/2}}{a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 (1+a x)}+\frac {7 \left (1-a^2 x^2\right )^{5/2} \log (1-a x)}{16 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}-\frac {23 \left (1-a^2 x^2\right )^{5/2} \log (1+a x)}{16 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5} \]

(-a^2*x^2+1)^(5/2)/a^5/(c-c/a^2/x^2)^(5/2)/x^4+1/8*(-a^2*x^2+1)^(5/2)/a^6/(c-c/a^2/x^2)^(5/2)/x^5/(-a*x+1)+1/8

*(-a^2*x^2+1)^(5/2)/a^6/(c-c/a^2/x^2)^(5/2)/x^5/(a*x+1)^2-(-a^2*x^2+1)^(5/2)/a^6/(c-c/a^2/x^2)^(5/2)/x^5/(a*x+

1)+7/16*(-a^2*x^2+1)^(5/2)*ln(-a*x+1)/a^6/(c-c/a^2/x^2)^(5/2)/x^5-23/16*(-a^2*x^2+1)^(5/2)*ln(a*x+1)/a^6/(c-c/

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Rubi [A]
time = 0.15, antiderivative size = 265, normalized size of antiderivative = 1.00, 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 = {6295, 6285, 90} \begin {gather*} \frac {\left (1-a^2 x^2\right )^{5/2}}{8 a^6 x^5 (1-a x) \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}-\frac {\left (1-a^2 x^2\right )^{5/2}}{a^6 x^5 (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}+\frac {\left (1-a^2 x^2\right )^{5/2}}{8 a^6 x^5 (a x+1)^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}+\frac {7 \left (1-a^2 x^2\right )^{5/2} \log (1-a x)}{16 a^6 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}-\frac {23 \left (1-a^2 x^2\right )^{5/2} \log (a x+1)}{16 a^6 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}+\frac {\left (1-a^2 x^2\right )^{5/2}}{a^5 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \end {gather*}

(1 - a^2*x^2)^(5/2)/(a^5*(c - c/(a^2*x^2))^(5/2)*x^4) + (1 - a^2*x^2)^(5/2)/(8*a^6*(c - c/(a^2*x^2))^(5/2)*x^5

*(1 - a*x)) + (1 - a^2*x^2)^(5/2)/(8*a^6*(c - c/(a^2*x^2))^(5/2)*x^5*(1 + a*x)^2) - (1 - a^2*x^2)^(5/2)/(a^6*(

c - c/(a^2*x^2))^(5/2)*x^5*(1 + a*x)) + (7*(1 - a^2*x^2)^(5/2)*Log[1 - a*x])/(16*a^6*(c - c/(a^2*x^2))^(5/2)*x

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] &&

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

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/x^(2*p))*(1 + c*(x^2/d))^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] &

\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx &=\frac {\left (1-a^2 x^2\right )^{5/2} \int \frac {e^{-\tanh ^{-1}(a x)} x^5}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}\\ &=\frac {\left (1-a^2 x^2\right )^{5/2} \int \frac {x^5}{(1-a x)^2 (1+a x)^3} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}\\ &=\frac {\left (1-a^2 x^2\right )^{5/2} \int \left (\frac {1}{a^5}+\frac {1}{8 a^5 (-1+a x)^2}+\frac {7}{16 a^5 (-1+a x)}-\frac {1}{4 a^5 (1+a x)^3}+\frac {1}{a^5 (1+a x)^2}-\frac {23}{16 a^5 (1+a x)}\right ) \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}\\ &=\frac {\left (1-a^2 x^2\right )^{5/2}}{a^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}+\frac {\left (1-a^2 x^2\right )^{5/2}}{8 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{5/2}}{8 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 (1+a x)^2}-\frac {\left (1-a^2 x^2\right )^{5/2}}{a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 (1+a x)}+\frac {7 \left (1-a^2 x^2\right )^{5/2} \log (1-a x)}{16 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}-\frac {23 \left (1-a^2 x^2\right )^{5/2} \log (1+a x)}{16 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 88, normalized size = 0.33 \begin {gather*} \frac {\left (1-a^2 x^2\right )^{5/2} \left (2 \left (8 a x+\frac {1}{1-a x}+\frac {1}{(1+a x)^2}-\frac {8}{1+a x}\right )+7 \log (1-a x)-23 \log (1+a x)\right )}{16 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5} \end {gather*}

((1 - a^2*x^2)^(5/2)*(2*(8*a*x + (1 - a*x)^(-1) + (1 + a*x)^(-2) - 8/(1 + a*x)) + 7*Log[1 - a*x] - 23*Log[1 +

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method	result	size

default	\(-\frac {\left (a x -1\right ) \sqrt {-a^{2} x^{2}+1}\, \left (-16 a^{4} x^{4}+23 \ln \left (a x +1\right ) a^{3} x^{3}-7 \ln \left (a x -1\right ) a^{3} x^{3}-16 a^{3} x^{3}+23 a^{2} x^{2} \ln \left (a x +1\right )-7 \ln \left (a x -1\right ) a^{2} x^{2}+34 a^{2} x^{2}-23 a x \ln \left (a x +1\right )+7 \ln \left (a x -1\right ) a x +18 a x -23 \ln \left (a x +1\right )+7 \ln \left (a x -1\right )-12\right )}{16 a^{6} x^{5} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {5}{2}}}\)	\(167\)

-1/16*(a*x-1)*(-a^2*x^2+1)^(1/2)*(-16*a^4*x^4+23*ln(a*x+1)*a^3*x^3-7*ln(a*x-1)*a^3*x^3-16*a^3*x^3+23*a^2*x^2*l

n(a*x+1)-7*ln(a*x-1)*a^2*x^2+34*a^2*x^2-23*a*x*ln(a*x+1)+7*ln(a*x-1)*a*x+18*a*x-23*ln(a*x+1)+7*ln(a*x-1)-12)/a

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

integral(sqrt(-a^2*x^2 + 1)*a^6*x^6*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^7*c^3*x^7 + a^6*c^3*x^6 - 3*a^5*c^3*x^5

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{\left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{\frac {5}{2}} \left (a x + 1\right )}\, dx \end {gather*}

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP

UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {1-a^2\,x^2}}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{5/2}\,\left (a\,x+1\right )} \,d x \end {gather*}

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3.8.21 \(\int \frac {e^{-\tanh ^{-1}(a x)}}{(c-\frac {c}{a^2 x^2})^{5/2}} \, dx\) [721]