\(\int \frac {e^{\coth ^{-1}(a x)}}{(c-\frac {c}{a^2 x^2})^{5/2}} \, dx\) [836]

Optimal result

Integrand size = 22, antiderivative size = 263 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^2 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{8 a c^2 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^2}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{a c^2 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{8 a c^2 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}+\frac {23 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{16 a c^2 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {7 \sqrt {1-\frac {1}{a^2 x^2}} \log (1+a x)}{16 a c^2 \sqrt {c-\frac {c}{a^2 x^2}}} \]

[Out]

x*(1-1/a^2/x^2)^(1/2)/c^2/(c-c/a^2/x^2)^(1/2)-1/8*(1-1/a^2/x^2)^(1/2)/a/c^2/(-a*x+1)^2/(c-c/a^2/x^2)^(1/2)+(1-
1/a^2/x^2)^(1/2)/a/c^2/(-a*x+1)/(c-c/a^2/x^2)^(1/2)-1/8*(1-1/a^2/x^2)^(1/2)/a/c^2/(a*x+1)/(c-c/a^2/x^2)^(1/2)+
23/16*ln(-a*x+1)*(1-1/a^2/x^2)^(1/2)/a/c^2/(c-c/a^2/x^2)^(1/2)-7/16*ln(a*x+1)*(1-1/a^2/x^2)^(1/2)/a/c^2/(c-c/a
^2/x^2)^(1/2)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6332, 6328, 90} \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^2 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{a c^2 (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{8 a c^2 (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{8 a c^2 (1-a x)^2 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {23 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{16 a c^2 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {7 \sqrt {1-\frac {1}{a^2 x^2}} \log (a x+1)}{16 a c^2 \sqrt {c-\frac {c}{a^2 x^2}}} \]

[In]

Int[E^ArcCoth[a*x]/(c - c/(a^2*x^2))^(5/2),x]

[Out]

(Sqrt[1 - 1/(a^2*x^2)]*x)/(c^2*Sqrt[c - c/(a^2*x^2)]) - Sqrt[1 - 1/(a^2*x^2)]/(8*a*c^2*Sqrt[c - c/(a^2*x^2)]*(
1 - a*x)^2) + Sqrt[1 - 1/(a^2*x^2)]/(a*c^2*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)) - Sqrt[1 - 1/(a^2*x^2)]/(8*a*c^2*S
qrt[c - c/(a^2*x^2)]*(1 + a*x)) + (23*Sqrt[1 - 1/(a^2*x^2)]*Log[1 - a*x])/(16*a*c^2*Sqrt[c - c/(a^2*x^2)]) - (
7*Sqrt[1 - 1/(a^2*x^2)]*Log[1 + a*x])/(16*a*c^2*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^{\coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2}} \, dx}{c^2 \sqrt {c-\frac {c}{a^2 x^2}}} \\ & = \frac {\left (a^5 \sqrt {1-\frac {1}{a^2 x^2}}\right ) \int \frac {x^5}{(-1+a x)^3 (1+a x)^2} \, dx}{c^2 \sqrt {c-\frac {c}{a^2 x^2}}} \\ & = \frac {\left (a^5 \sqrt {1-\frac {1}{a^2 x^2}}\right ) \int \left (\frac {1}{a^5}+\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)}+\frac {1}{8 a^5 (1+a x)^2}-\frac {7}{16 a^5 (1+a x)}\right ) \, dx}{c^2 \sqrt {c-\frac {c}{a^2 x^2}}} \\ & = \frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^2 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{8 a c^2 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^2}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{a c^2 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{8 a c^2 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}+\frac {23 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{16 a c^2 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {7 \sqrt {1-\frac {1}{a^2 x^2}} \log (1+a x)}{16 a c^2 \sqrt {c-\frac {c}{a^2 x^2}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.37 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\frac {\left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (x-\frac {1}{8 a (-1+a x)^2}+\frac {1}{a-a^2 x}-\frac {1}{8 a+8 a^2 x}+\frac {23 \log (1-a x)}{16 a}-\frac {7 \log (1+a x)}{16 a}\right )}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \]

[In]

Integrate[E^ArcCoth[a*x]/(c - c/(a^2*x^2))^(5/2),x]

[Out]

((1 - 1/(a^2*x^2))^(5/2)*(x - 1/(8*a*(-1 + a*x)^2) + (a - a^2*x)^(-1) - (8*a + 8*a^2*x)^(-1) + (23*Log[1 - a*x
])/(16*a) - (7*Log[1 + a*x])/(16*a)))/(c - c/(a^2*x^2))^(5/2)

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.67

method	result	size
default	\(-\frac {\left (a x -1\right ) \left (a x +1\right ) \left (-16 a^{4} x^{4}+7 a^{3} \ln \left (a x +1\right ) x^{3}-23 a^{3} \ln \left (a x -1\right ) x^{3}+16 a^{3} x^{3}-7 a^{2} \ln \left (a x +1\right ) x^{2}+23 a^{2} \ln \left (a x -1\right ) x^{2}+34 a^{2} x^{2}-7 a \ln \left (a x +1\right ) x +23 a \ln \left (a x -1\right ) x -18 a x +7 \ln \left (a x +1\right )-23 \ln \left (a x -1\right )-12\right )}{16 \sqrt {\frac {a x -1}{a x +1}}\, a^{6} x^{5} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {5}{2}}}\)	\(175\)

[In]

int(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a^2/x^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/16/((a*x-1)/(a*x+1))^(1/2)*(a*x-1)*(a*x+1)*(-16*a^4*x^4+7*a^3*ln(a*x+1)*x^3-23*a^3*ln(a*x-1)*x^3+16*a^3*x^3
-7*a^2*ln(a*x+1)*x^2+23*a^2*ln(a*x-1)*x^2+34*a^2*x^2-7*a*ln(a*x+1)*x+23*a*ln(a*x-1)*x-18*a*x+7*ln(a*x+1)-23*ln
(a*x-1)-12)/a^6/x^5/(c*(a^2*x^2-1)/a^2/x^2)^(5/2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.52 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\frac {{\left (16 \, a^{4} x^{4} - 16 \, a^{3} x^{3} - 34 \, a^{2} x^{2} + 18 \, a x - 7 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x + 1\right ) + 23 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x - 1\right ) + 12\right )} \sqrt {a^{2} c}}{16 \, {\left (a^{5} c^{3} x^{3} - a^{4} c^{3} x^{2} - a^{3} c^{3} x + a^{2} c^{3}\right )}} \]

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a^2/x^2)^(5/2),x, algorithm="fricas")

[Out]

1/16*(16*a^4*x^4 - 16*a^3*x^3 - 34*a^2*x^2 + 18*a*x - 7*(a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*x + 1) + 23*(a^3*x
^3 - a^2*x^2 - a*x + 1)*log(a*x - 1) + 12)*sqrt(a^2*c)/(a^5*c^3*x^3 - a^4*c^3*x^2 - a^3*c^3*x + a^2*c^3)

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate(1/((a*x-1)/(a*x+1))**(1/2)/(c-c/a**2/x**2)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {5}{2}} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a^2/x^2)^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((c - c/(a^2*x^2))^(5/2)*sqrt((a*x - 1)/(a*x + 1))), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a^2/x^2)^(5/2),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. \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\int \frac {1}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{5/2}\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]

[In]

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

[Out]

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