\(\int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-\frac {c}{a^2 x^2})^{7/2}} \, dx\) [879]

Optimal result

Integrand size = 24, antiderivative size = 357 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)^4}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{2 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)^3}+\frac {59 \sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)^2}-\frac {75 \sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}+\frac {9 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {201 \sqrt {1-\frac {1}{a^2 x^2}} \log (1+a x)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6332, 6328, 90} \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {75 \sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {59 \sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 (a x+1)^2 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{2 a c^3 (a x+1)^3 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 (a x+1)^4 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {9 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {201 \sqrt {1-\frac {1}{a^2 x^2}} \log (a x+1)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}} \]

[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 90

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]))

Rule 6328

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Dist[c^p/a^(2*p), Int[(u/x^(
2*p))*(-1 + a*x)^(p - n/2)*(1 + a*x)^(p + n/2), 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 6332

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])

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.24 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.30 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {\left (1-\frac {1}{a^2 x^2}\right )^{7/2} \left (64 x+\frac {208+478 a x+74 a^2 x^2-490 a^3 x^3-302 a^4 x^4}{a (-1+a x) (1+a x)^4}+\frac {9 \log (1-a x)}{a}-\frac {201 \log (1+a x)}{a}\right )}{64 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \]

[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)*(64*x + (208 + 478*a*x + 74*a^2*x^2 - 490*a^3*x^3 - 302*a^4*x^4)/(a*(-1 + a*x)*(1 + a
*x)^4) + (9*Log[1 - a*x])/a - (201*Log[1 + a*x])/a))/(64*(c - c/(a^2*x^2))^(7/2))

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.69

method	result	size
default	\(-\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \left (a x -1\right ) \left (-64 a^{6} x^{6}+201 \ln \left (a x +1\right ) x^{5} a^{5}-9 \ln \left (a x -1\right ) x^{5} a^{5}-192 a^{5} x^{5}+603 \ln \left (a x +1\right ) x^{4} a^{4}-27 \ln \left (a x -1\right ) x^{4} a^{4}+174 a^{4} x^{4}+402 a^{3} \ln \left (a x +1\right ) x^{3}-18 a^{3} \ln \left (a x -1\right ) x^{3}+618 a^{3} x^{3}-402 a^{2} \ln \left (a x +1\right ) x^{2}+18 a^{2} \ln \left (a x -1\right ) x^{2}+118 a^{2} x^{2}-603 a \ln \left (a x +1\right ) x +27 a \ln \left (a x -1\right ) x -414 a x -201 \ln \left (a x +1\right )+9 \ln \left (a x -1\right )-208\right )}{64 a^{8} x^{7} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {7}{2}}}\)	\(247\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.58 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {{\left (64 \, a^{6} x^{6} + 192 \, a^{5} x^{5} - 174 \, a^{4} x^{4} - 618 \, a^{3} x^{3} - 118 \, a^{2} x^{2} + 414 \, a x - 201 \, {\left (a^{5} x^{5} + 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} - 3 \, a x - 1\right )} \log \left (a x + 1\right ) + 9 \, {\left (a^{5} x^{5} + 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} - 3 \, a x - 1\right )} \log \left (a x - 1\right ) + 208\right )} \sqrt {a^{2} c}}{64 \, {\left (a^{7} c^{4} x^{5} + 3 \, a^{6} c^{4} x^{4} + 2 \, a^{5} c^{4} x^{3} - 2 \, a^{4} c^{4} x^{2} - 3 \, a^{3} c^{4} x - a^{2} c^{4}\right )}} \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}}} \,d x } \]

[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)

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\int \frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{7/2}} \,d x \]

[In]

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

[Out]

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