\(\int \frac {e^{-2 \coth ^{-1}(a x)}}{(c-\frac {c}{a^2 x^2})^{7/2}} \, dx\) [870]

Optimal result

Integrand size = 24, antiderivative size = 270 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=-\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}+\frac {10 (1-a x)^3}{3 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}+\frac {12 (1-a x)^4}{7 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}+\frac {82 (1-a x)^4 (1+a x)}{105 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}+\frac {2 (1-a x)^4 (1+a x)^2}{35 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}+\frac {2 (1-a x)^4 (1+a x)^3 (72+37 a x)}{35 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}+\frac {2 (1-a x)^{7/2} (1+a x)^{7/2} \arcsin (a x)}{a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7} \]

[Out]

-1/3*(-a*x+1)^2/a^2/(c-c/a^2/x^2)^(7/2)/x+10/3*(-a*x+1)^3/a^3/(c-c/a^2/x^2)^(7/2)/x^2+12/7*(-a*x+1)^4/a^4/(c-c
/a^2/x^2)^(7/2)/x^3+82/105*(-a*x+1)^4*(a*x+1)/a^5/(c-c/a^2/x^2)^(7/2)/x^4+2/35*(-a*x+1)^4*(a*x+1)^2/a^6/(c-c/a
^2/x^2)^(7/2)/x^5+2/35*(-a*x+1)^4*(a*x+1)^3*(37*a*x+72)/a^8/(c-c/a^2/x^2)^(7/2)/x^7+2*(-a*x+1)^(7/2)*(a*x+1)^(
7/2)*arcsin(a*x)/a^8/(c-c/a^2/x^2)^(7/2)/x^7

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6302, 6294, 6264, 100, 155, 148, 41, 222} \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=-\frac {(1-a x)^2}{3 a^2 x \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {2 (a x+1)^{7/2} (1-a x)^{7/2} \arcsin (a x)}{a^8 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {2 (a x+1)^3 (37 a x+72) (1-a x)^4}{35 a^8 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {2 (a x+1)^2 (1-a x)^4}{35 a^6 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {82 (a x+1) (1-a x)^4}{105 a^5 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {12 (1-a x)^4}{7 a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {10 (1-a x)^3}{3 a^3 x^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \]

[In]

Int[1/(E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^(7/2)),x]

[Out]

-1/3*(1 - a*x)^2/(a^2*(c - c/(a^2*x^2))^(7/2)*x) + (10*(1 - a*x)^3)/(3*a^3*(c - c/(a^2*x^2))^(7/2)*x^2) + (12*
(1 - a*x)^4)/(7*a^4*(c - c/(a^2*x^2))^(7/2)*x^3) + (82*(1 - a*x)^4*(1 + a*x))/(105*a^5*(c - c/(a^2*x^2))^(7/2)
*x^4) + (2*(1 - a*x)^4*(1 + a*x)^2)/(35*a^6*(c - c/(a^2*x^2))^(7/2)*x^5) + (2*(1 - a*x)^4*(1 + a*x)^3*(72 + 37
*a*x))/(35*a^8*(c - c/(a^2*x^2))^(7/2)*x^7) + (2*(1 - a*x)^(7/2)*(1 + a*x)^(7/2)*ArcSin[a*x])/(a^8*(c - c/(a^2
*x^2))^(7/2)*x^7)

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 155

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((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 - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]

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 )^{7/2}} \, dx \\ & = -\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {e^{-2 \text {arctanh}(a x)} x^7}{(1-a x)^{7/2} (1+a x)^{7/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7} \\ & = -\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {x^7}{(1-a x)^{5/2} (1+a x)^{9/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7} \\ & = -\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}+\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {x^5 (6+4 a x)}{(1-a x)^{3/2} (1+a x)^{9/2}} \, dx}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7} \\ & = -\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}+\frac {10 (1-a x)^3}{3 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}+\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {x^4 \left (-50 a-14 a^2 x\right )}{\sqrt {1-a x} (1+a x)^{9/2}} \, dx}{3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7} \\ & = -\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}+\frac {10 (1-a x)^3}{3 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}+\frac {12 (1-a x)^4}{7 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}+\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {x^3 \left (-144 a^2-62 a^3 x\right )}{\sqrt {1-a x} (1+a x)^{7/2}} \, dx}{21 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7} \\ & = -\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}+\frac {10 (1-a x)^3}{3 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}+\frac {12 (1-a x)^4}{7 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}+\frac {82 (1-a x)^4 (1+a x)}{105 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}+\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {x^2 \left (-246 a^3-228 a^4 x\right )}{\sqrt {1-a x} (1+a x)^{5/2}} \, dx}{105 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7} \\ & = -\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}+\frac {10 (1-a x)^3}{3 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}+\frac {12 (1-a x)^4}{7 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}+\frac {82 (1-a x)^4 (1+a x)}{105 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}+\frac {2 (1-a x)^4 (1+a x)^2}{35 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}+\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {x \left (-36 a^4-666 a^5 x\right )}{\sqrt {1-a x} (1+a x)^{3/2}} \, dx}{315 a^{10} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7} \\ & = -\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}+\frac {10 (1-a x)^3}{3 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}+\frac {12 (1-a x)^4}{7 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}+\frac {82 (1-a x)^4 (1+a x)}{105 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}+\frac {2 (1-a x)^4 (1+a x)^2}{35 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}+\frac {2 (1-a x)^4 (1+a x)^3 (72+37 a x)}{35 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}+\frac {\left (2 (1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7} \\ & = -\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}+\frac {10 (1-a x)^3}{3 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}+\frac {12 (1-a x)^4}{7 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}+\frac {82 (1-a x)^4 (1+a x)}{105 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}+\frac {2 (1-a x)^4 (1+a x)^2}{35 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}+\frac {2 (1-a x)^4 (1+a x)^3 (72+37 a x)}{35 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}+\frac {\left (2 (1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7} \\ & = -\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}+\frac {10 (1-a x)^3}{3 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}+\frac {12 (1-a x)^4}{7 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}+\frac {82 (1-a x)^4 (1+a x)}{105 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}+\frac {2 (1-a x)^4 (1+a x)^2}{35 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}+\frac {2 (1-a x)^4 (1+a x)^3 (72+37 a x)}{35 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}+\frac {2 (1-a x)^{7/2} (1+a x)^{7/2} \arcsin (a x)}{a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.49 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {432+654 a x-636 a^2 x^2-1226 a^3 x^3+74 a^4 x^4+562 a^5 x^5+105 a^6 x^6-210 (-1+a x) (1+a x)^3 \sqrt {-1+a^2 x^2} \log \left (a x+\sqrt {-1+a^2 x^2}\right )}{105 a^2 \sqrt {c-\frac {c}{a^2 x^2}} x (-1+a x) (c+a c x)^3} \]

[In]

Integrate[1/(E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^(7/2)),x]

[Out]

(432 + 654*a*x - 636*a^2*x^2 - 1226*a^3*x^3 + 74*a^4*x^4 + 562*a^5*x^5 + 105*a^6*x^6 - 210*(-1 + a*x)*(1 + a*x
)^3*Sqrt[-1 + a^2*x^2]*Log[a*x + Sqrt[-1 + a^2*x^2]])/(105*a^2*Sqrt[c - c/(a^2*x^2)]*x*(-1 + a*x)*(c + a*c*x)^
3)

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.37

method	result	size
risch	\(\frac {a^{2} x^{2}-1}{a^{2} c^{3} x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}+\frac {\left (-\frac {2 \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{a^{7} \sqrt {a^{2} c}}-\frac {\sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-2 \left (x +\frac {1}{a}\right ) a c}}{28 a^{12} c \left (x +\frac {1}{a}\right )^{4}}+\frac {39 \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-2 \left (x +\frac {1}{a}\right ) a c}}{140 a^{11} c \left (x +\frac {1}{a}\right )^{3}}-\frac {1753 \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-2 \left (x +\frac {1}{a}\right ) a c}}{1680 a^{10} c \left (x +\frac {1}{a}\right )^{2}}+\frac {3061 \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-2 \left (x +\frac {1}{a}\right ) a c}}{840 a^{9} c \left (x +\frac {1}{a}\right )}-\frac {\sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+2 \left (x -\frac {1}{a}\right ) a c}}{48 a^{10} c \left (x -\frac {1}{a}\right )^{2}}-\frac {7 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+2 \left (x -\frac {1}{a}\right ) a c}}{24 a^{9} c \left (x -\frac {1}{a}\right )}\right ) a^{6} \sqrt {c \left (a^{2} x^{2}-1\right )}}{c^{3} x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}\)	\(370\)
default	\(-\frac {\left (-105 c^{\frac {7}{2}} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} a^{7} x^{7}+96 c^{\frac {7}{2}} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} a^{6} x^{6}-553 x^{6} c^{\frac {7}{2}} a^{6} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}}+96 c^{\frac {7}{2}} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} a^{5} x^{5}+392 c^{\frac {7}{2}} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} a^{5} x^{5}-240 c^{\frac {7}{2}} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} a^{4} x^{4}+1540 c^{\frac {7}{2}} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} a^{4} x^{4}+210 \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} a^{6} c x -240 c^{\frac {7}{2}} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} a^{3} x^{3}-350 c^{\frac {7}{2}} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} a^{3} x^{3}+210 \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} a^{5} c +180 c^{\frac {7}{2}} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} a^{2} x^{2}-1470 c^{\frac {7}{2}} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} a^{2} x^{2}+180 c^{\frac {7}{2}} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} a x +42 c^{\frac {7}{2}} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} a x -30 c^{\frac {7}{2}} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}}+462 c^{\frac {7}{2}} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}}\right ) \left (a x -1\right )}{105 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} x^{7} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {7}{2}} a^{8} c^{\frac {7}{2}}}\)	\(572\)

[In]

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

[Out]

1/a^2*(a^2*x^2-1)/c^3/x/(c*(a^2*x^2-1)/a^2/x^2)^(1/2)+(-2/a^7*ln(a^2*c*x/(a^2*c)^(1/2)+(a^2*c*x^2-c)^(1/2))/(a
^2*c)^(1/2)-1/28/a^12/c/(x+1/a)^4*(a^2*c*(x+1/a)^2-2*(x+1/a)*a*c)^(1/2)+39/140/a^11/c/(x+1/a)^3*(a^2*c*(x+1/a)
^2-2*(x+1/a)*a*c)^(1/2)-1753/1680/a^10/c/(x+1/a)^2*(a^2*c*(x+1/a)^2-2*(x+1/a)*a*c)^(1/2)+3061/840/a^9/c/(x+1/a
)*(a^2*c*(x+1/a)^2-2*(x+1/a)*a*c)^(1/2)-1/48/a^10/c/(x-1/a)^2*(a^2*c*(x-1/a)^2+2*(x-1/a)*a*c)^(1/2)-7/24/a^9/c
/(x-1/a)*(a^2*c*(x-1/a)^2+2*(x-1/a)*a*c)^(1/2))*a^6/c^3/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 = 0.32 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.83 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\left [\frac {105 \, {\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt {c} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (105 \, a^{7} x^{7} + 562 \, a^{6} x^{6} + 74 \, a^{5} x^{5} - 1226 \, a^{4} x^{4} - 636 \, a^{3} x^{3} + 654 \, a^{2} x^{2} + 432 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{105 \, {\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}}, \frac {210 \, {\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + {\left (105 \, a^{7} x^{7} + 562 \, a^{6} x^{6} + 74 \, a^{5} x^{5} - 1226 \, a^{4} x^{4} - 636 \, a^{3} x^{3} + 654 \, a^{2} x^{2} + 432 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{105 \, {\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}}\right ] \]

[In]

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

[Out]

[1/105*(105*(a^6*x^6 + 2*a^5*x^5 - a^4*x^4 - 4*a^3*x^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) + (105*a^7*x^7 + 562*a^6*x^6 + 74*a^5*x^5 - 1226*a^4*x^4 - 63
6*a^3*x^3 + 654*a^2*x^2 + 432*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^7*c^4*x^6 + 2*a^6*c^4*x^5 - a^5*c^4*x^4
- 4*a^4*c^4*x^3 - a^3*c^4*x^2 + 2*a^2*c^4*x + a*c^4), 1/105*(210*(a^6*x^6 + 2*a^5*x^5 - a^4*x^4 - 4*a^3*x^3 -
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)) + (105
*a^7*x^7 + 562*a^6*x^6 + 74*a^5*x^5 - 1226*a^4*x^4 - 636*a^3*x^3 + 654*a^2*x^2 + 432*a*x)*sqrt((a^2*c*x^2 - c)
/(a^2*x^2)))/(a^7*c^4*x^6 + 2*a^6*c^4*x^5 - a^5*c^4*x^4 - 4*a^4*c^4*x^3 - a^3*c^4*x^2 + 2*a^2*c^4*x + a*c^4)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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