\(\int \frac {e^{2 \coth ^{-1}(a x)}}{(c-\frac {c}{a^2 x^2})^{3/2}} \, dx\) [843]

Optimal result

Integrand size = 24, antiderivative size = 123 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=-\frac {(1+a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}+\frac {2 (5-2 a x) (1-a x) (1+a x)^2}{3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}-\frac {2 (1-a x)^{3/2} (1+a x)^{3/2} \arcsin (a x)}{a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3} \]

[Out]

-1/3*(a*x+1)^2/a^2/(c-c/a^2/x^2)^(3/2)/x+2/3*(-2*a*x+5)*(-a*x+1)*(a*x+1)^2/a^4/(c-c/a^2/x^2)^(3/2)/x^3-2*(-a*x
+1)^(3/2)*(a*x+1)^(3/2)*arcsin(a*x)/a^4/(c-c/a^2/x^2)^(3/2)/x^3

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6302, 6294, 6264, 100, 148, 41, 222} \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=-\frac {(a x+1)^2}{3 a^2 x \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}-\frac {2 (1-a x)^{3/2} (a x+1)^{3/2} \arcsin (a x)}{a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}+\frac {2 (5-2 a x) (1-a x) (a x+1)^2}{3 a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \]

[In]

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

[Out]

-1/3*(1 + a*x)^2/(a^2*(c - c/(a^2*x^2))^(3/2)*x) + (2*(5 - 2*a*x)*(1 - a*x)*(1 + a*x)^2)/(3*a^4*(c - c/(a^2*x^
2))^(3/2)*x^3) - (2*(1 - a*x)^(3/2)*(1 + a*x)^(3/2)*ArcSin[a*x])/(a^4*(c - c/(a^2*x^2))^(3/2)*x^3)

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 148

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[(b^2*d*e*g - a^2*d*f*h*m - a*b*(d*(f*g + e*h) - c*f*h*(m + 1)) + b*f*h*(b*c - a*d)*(m + 1)*x)*(a + b*x)
^(m + 1)*((c + d*x)^(n + 1)/(b^2*d*(b*c - a*d)*(m + 1))), x] + Dist[(a*d*f*h*m + b*(d*(f*g + e*h) - c*f*h*(m +
2)))/(b^2*d), Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[m
+ n + 2, 0] && NeQ[m, -1] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 6264

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

Rule 6294

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

Rule 6302

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx \\ & = -\frac {\left ((1-a x)^{3/2} (1+a x)^{3/2}\right ) \int \frac {e^{2 \text {arctanh}(a x)} x^3}{(1-a x)^{3/2} (1+a x)^{3/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3} \\ & = -\frac {\left ((1-a x)^{3/2} (1+a x)^{3/2}\right ) \int \frac {x^3}{(1-a x)^{5/2} \sqrt {1+a x}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3} \\ & = -\frac {(1+a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}+\frac {\left ((1-a x)^{3/2} (1+a x)^{3/2}\right ) \int \frac {x (2+4 a x)}{(1-a x)^{3/2} \sqrt {1+a x}} \, dx}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3} \\ & = -\frac {(1+a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}+\frac {2 (5-2 a x) (1-a x) (1+a x)^2}{3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}-\frac {\left (2 (1-a x)^{3/2} (1+a x)^{3/2}\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3} \\ & = -\frac {(1+a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}+\frac {2 (5-2 a x) (1-a x) (1+a x)^2}{3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}-\frac {\left (2 (1-a x)^{3/2} (1+a x)^{3/2}\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3} \\ & = -\frac {(1+a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}+\frac {2 (5-2 a x) (1-a x) (1+a x)^2}{3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}-\frac {2 (1-a x)^{3/2} (1+a x)^{3/2} \arcsin (a x)}{a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.77 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\frac {10-4 a x-11 a^2 x^2+3 a^3 x^3+6 (-1+a x) \sqrt {-1+a^2 x^2} \log \left (a x+\sqrt {-1+a^2 x^2}\right )}{3 a^2 c \sqrt {c-\frac {c}{a^2 x^2}} x (-1+a x)} \]

[In]

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

[Out]

(10 - 4*a*x - 11*a^2*x^2 + 3*a^3*x^3 + 6*(-1 + a*x)*Sqrt[-1 + a^2*x^2]*Log[a*x + Sqrt[-1 + a^2*x^2]])/(3*a^2*c
*Sqrt[c - c/(a^2*x^2)]*x*(-1 + a*x))

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.74

method	result	size
risch	\(\frac {a^{2} x^{2}-1}{a^{2} c x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}+\frac {\left (\frac {2 \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{a^{3} \sqrt {a^{2} c}}-\frac {\sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+2 \left (x -\frac {1}{a}\right ) a c}}{3 a^{6} c \left (x -\frac {1}{a}\right )^{2}}-\frac {8 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+2 \left (x -\frac {1}{a}\right ) a c}}{3 a^{5} c \left (x -\frac {1}{a}\right )}\right ) a^{2} \sqrt {c \left (a^{2} x^{2}-1\right )}}{c x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}\)	\(214\)
default	\(\frac {\left (3 c^{\frac {3}{2}} \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a^{3} x^{3}+4 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c^{\frac {3}{2}} a^{2} x^{2}-15 x^{2} a^{2} c^{\frac {3}{2}} \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+6 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a^{2} c x -4 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c^{\frac {3}{2}} a x -6 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a c -2 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c^{\frac {3}{2}}+12 c^{\frac {3}{2}} \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\right ) \left (a x +1\right )}{3 \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, x^{3} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {3}{2}} a^{4} c^{\frac {3}{2}}}\)	\(326\)

[In]

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

[Out]

1/a^2*(a^2*x^2-1)/c/x/(c*(a^2*x^2-1)/a^2/x^2)^(1/2)+(2/a^3*ln(a^2*c*x/(a^2*c)^(1/2)+(a^2*c*x^2-c)^(1/2))/(a^2*
c)^(1/2)-1/3/a^6/c/(x-1/a)^2*(a^2*c*(x-1/a)^2+2*(x-1/a)*a*c)^(1/2)-8/3/a^5/c/(x-1/a)*(a^2*c*(x-1/a)^2+2*(x-1/a
)*a*c)^(1/2))*a^2/c/x/(c*(a^2*x^2-1)/a^2/x^2)^(1/2)*(c*(a^2*x^2-1))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.28 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (3 \, a^{3} x^{3} - 14 \, a^{2} x^{2} + 10 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}, -\frac {6 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - {\left (3 \, a^{3} x^{3} - 14 \, a^{2} x^{2} + 10 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}\right ] \]

[In]

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

[Out]

[1/3*(3*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*log(2*a^2*c*x^2 + 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)
+ (3*a^3*x^3 - 14*a^2*x^2 + 10*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2), -1/
3*(6*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) -
(3*a^3*x^3 - 14*a^2*x^2 + 10*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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