\(\int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-\frac {c}{a x})^{7/2}} \, dx\) [488]

Optimal result

Integrand size = 24, antiderivative size = 267 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\frac {7 \left (1-\frac {1}{a x}\right )^{7/2}}{4 a \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {3 \left (1-\frac {1}{a x}\right )^{7/2}}{2 \left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {a \left (1-\frac {1}{a x}\right )^{7/2} x}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {\left (1-\frac {1}{a x}\right )^{7/2} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {11 \left (1-\frac {1}{a x}\right )^{7/2} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{4 \sqrt {2} a \left (c-\frac {c}{a x}\right )^{7/2}} \]

[Out]

(1-1/a/x)^(7/2)*arctanh((1+1/a/x)^(1/2))/a/(c-c/a/x)^(7/2)-11/8*(1-1/a/x)^(7/2)*arctanh(1/2*(1+1/a/x)^(1/2)*2^
(1/2))/a/(c-c/a/x)^(7/2)*2^(1/2)+7/4*(1-1/a/x)^(7/2)/a/(c-c/a/x)^(7/2)/(1+1/a/x)^(1/2)-3/2*(1-1/a/x)^(7/2)/(a-
1/x)/(c-c/a/x)^(7/2)/(1+1/a/x)^(1/2)+a*(1-1/a/x)^(7/2)*x/(a-1/x)/(c-c/a/x)^(7/2)/(1+1/a/x)^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6317, 6314, 105, 156, 157, 162, 65, 214, 212} \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\frac {\left (1-\frac {1}{a x}\right )^{7/2} \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right )}{a \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {11 \left (1-\frac {1}{a x}\right )^{7/2} \text {arctanh}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right )}{4 \sqrt {2} a \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {a x \left (1-\frac {1}{a x}\right )^{7/2}}{\left (a-\frac {1}{x}\right ) \sqrt {\frac {1}{a x}+1} \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {7 \left (1-\frac {1}{a x}\right )^{7/2}}{4 a \sqrt {\frac {1}{a x}+1} \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {3 \left (1-\frac {1}{a x}\right )^{7/2}}{2 \left (a-\frac {1}{x}\right ) \sqrt {\frac {1}{a x}+1} \left (c-\frac {c}{a x}\right )^{7/2}} \]

[In]

Int[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^(7/2)),x]

[Out]

(7*(1 - 1/(a*x))^(7/2))/(4*a*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(7/2)) - (3*(1 - 1/(a*x))^(7/2))/(2*(a - x^(-1))*
Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(7/2)) + (a*(1 - 1/(a*x))^(7/2)*x)/((a - x^(-1))*Sqrt[1 + 1/(a*x)]*(c - c/(a*x
))^(7/2)) + ((1 - 1/(a*x))^(7/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(c - c/(a*x))^(7/2)) - (11*(1 - 1/(a*x))^(7/2)
*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(4*Sqrt[2]*a*(c - c/(a*x))^(7/2))

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 156

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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]

Rule 157

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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 212

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

Rule 214

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

Rule 6314

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

Rule 6317

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (1-\frac {1}{a x}\right )^{7/2} \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^{7/2}} \, dx}{\left (c-\frac {c}{a x}\right )^{7/2}} \\ & = -\frac {\left (1-\frac {1}{a x}\right )^{7/2} \text {Subst}\left (\int \frac {1}{x^2 \left (1-\frac {x}{a}\right )^2 \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\left (c-\frac {c}{a x}\right )^{7/2}} \\ & = \frac {a \left (1-\frac {1}{a x}\right )^{7/2} x}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {\left (1-\frac {1}{a x}\right )^{7/2} \text {Subst}\left (\int \frac {-\frac {1}{2 a}-\frac {5 x}{2 a^2}}{x \left (1-\frac {x}{a}\right )^2 \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\left (c-\frac {c}{a x}\right )^{7/2}} \\ & = -\frac {3 \left (1-\frac {1}{a x}\right )^{7/2}}{2 \left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {a \left (1-\frac {1}{a x}\right )^{7/2} x}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {\left (a \left (1-\frac {1}{a x}\right )^{7/2}\right ) \text {Subst}\left (\int \frac {\frac {1}{a^2}+\frac {9 x}{2 a^3}}{x \left (1-\frac {x}{a}\right ) \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{2 \left (c-\frac {c}{a x}\right )^{7/2}} \\ & = \frac {7 \left (1-\frac {1}{a x}\right )^{7/2}}{4 a \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {3 \left (1-\frac {1}{a x}\right )^{7/2}}{2 \left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {a \left (1-\frac {1}{a x}\right )^{7/2} x}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {\left (a^2 \left (1-\frac {1}{a x}\right )^{7/2}\right ) \text {Subst}\left (\int \frac {\frac {1}{a^3}+\frac {7 x}{4 a^4}}{x \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 \left (c-\frac {c}{a x}\right )^{7/2}} \\ & = \frac {7 \left (1-\frac {1}{a x}\right )^{7/2}}{4 a \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {3 \left (1-\frac {1}{a x}\right )^{7/2}}{2 \left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {a \left (1-\frac {1}{a x}\right )^{7/2} x}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {\left (11 \left (1-\frac {1}{a x}\right )^{7/2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{8 a^2 \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {\left (1-\frac {1}{a x}\right )^{7/2} \text {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a \left (c-\frac {c}{a x}\right )^{7/2}} \\ & = \frac {7 \left (1-\frac {1}{a x}\right )^{7/2}}{4 a \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {3 \left (1-\frac {1}{a x}\right )^{7/2}}{2 \left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {a \left (1-\frac {1}{a x}\right )^{7/2} x}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {\left (1-\frac {1}{a x}\right )^{7/2} \text {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{\left (c-\frac {c}{a x}\right )^{7/2}}-\frac {\left (11 \left (1-\frac {1}{a x}\right )^{7/2}\right ) \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{4 a \left (c-\frac {c}{a x}\right )^{7/2}} \\ & = \frac {7 \left (1-\frac {1}{a x}\right )^{7/2}}{4 a \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {3 \left (1-\frac {1}{a x}\right )^{7/2}}{2 \left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {a \left (1-\frac {1}{a x}\right )^{7/2} x}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {\left (1-\frac {1}{a x}\right )^{7/2} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {11 \left (1-\frac {1}{a x}\right )^{7/2} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{4 \sqrt {2} a \left (c-\frac {c}{a x}\right )^{7/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.09 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.45 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\frac {\sqrt {1-\frac {1}{a x}} \left (2 a x (-3+2 a x)+11 (-1+a x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {a+\frac {1}{x}}{2 a}\right )+(4-4 a x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1+\frac {1}{a x}\right )\right )}{4 a c^3 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} (-1+a x)} \]

[In]

Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^(7/2)),x]

[Out]

(Sqrt[1 - 1/(a*x)]*(2*a*x*(-3 + 2*a*x) + 11*(-1 + a*x)*Hypergeometric2F1[-1/2, 1, 1/2, (a + x^(-1))/(2*a)] + (
4 - 4*a*x)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + 1/(a*x)]))/(4*a*c^3*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)]*(-1 + a
*x))

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.09

method	result	size
default	\(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (16 \sqrt {\left (a x +1\right ) x}\, a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, x^{2}-11 a^{\frac {5}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) x^{2}+4 \sqrt {\left (a x +1\right ) x}\, a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, x +8 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{3} \sqrt {\frac {1}{a}}\, x^{2}-28 \sqrt {\left (a x +1\right ) x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}-8 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}+11 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) \sqrt {a}\right )}{16 \left (a x -1\right )^{3} a^{\frac {3}{2}} c^{4} \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}}\)	\(290\)
risch	\(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}}+\frac {\left (\frac {\ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{2 a^{4} \sqrt {a^{2} c}}-\frac {\sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+3 \left (x -\frac {1}{a}\right ) a c +2 c}}{4 a^{6} c \left (x -\frac {1}{a}\right )}-\frac {11 \sqrt {2}\, \ln \left (\frac {4 c +3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+3 \left (x -\frac {1}{a}\right ) a c +2 c}}{x -\frac {1}{a}}\right )}{16 a^{5} \sqrt {c}}+\frac {\sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-\left (x +\frac {1}{a}\right ) a c}}{2 a^{6} c \left (x +\frac {1}{a}\right )}\right ) a^{3} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) a c x}}{c^{3} x \sqrt {\frac {c \left (a x -1\right )}{a x}}}\)	\(307\)

[In]

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

[Out]

1/16*((a*x-1)/(a*x+1))^(3/2)*(a*x+1)/(a*x-1)^3*(c*(a*x-1)/a/x)^(1/2)*x*(16*((a*x+1)*x)^(1/2)*a^(7/2)*(1/a)^(1/
2)*x^2-11*a^(5/2)*2^(1/2)*ln((2*2^(1/2)*(1/a)^(1/2)*((a*x+1)*x)^(1/2)*a+3*a*x+1)/(a*x-1))*x^2+4*((a*x+1)*x)^(1
/2)*a^(5/2)*(1/a)^(1/2)*x+8*ln(1/2*(2*((a*x+1)*x)^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2))*a^3*(1/a)^(1/2)*x^2-28*((a*x
+1)*x)^(1/2)*a^(3/2)*(1/a)^(1/2)-8*ln(1/2*(2*((a*x+1)*x)^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2))*a*(1/a)^(1/2)+11*2^(1
/2)*ln((2*2^(1/2)*(1/a)^(1/2)*((a*x+1)*x)^(1/2)*a+3*a*x+1)/(a*x-1))*a^(1/2))/a^(3/2)/c^4/(1/a)^(1/2)/((a*x+1)*
x)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 594, normalized size of antiderivative = 2.22 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\left [\frac {11 \, \sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt {2} {\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 8 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 8 \, {\left (4 \, a^{3} x^{3} + a^{2} x^{2} - 7 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{32 \, {\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}}, \frac {11 \, \sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {2} {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - 8 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 4 \, {\left (4 \, a^{3} x^{3} + a^{2} x^{2} - 7 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{16 \, {\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}}\right ] \]

[In]

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

[Out]

[1/32*(11*sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x - 4*sqrt(2)*(3*a^3
*x^3 + 4*a^2*x^2 + a*x)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 +
3*a*x - 1)) + 8*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^3*x^3 + 3*a^2*x^2 + a*x)*sq
rt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 8*(4*a^3*x^3 + a^2*x^2 - 7*a*x)*sqrt
((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4), 1/16*(11*sqrt(2)*(a^2*x^2
- 2*a*x + 1)*sqrt(-c)*arctan(2*sqrt(2)*(a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*
x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - 8*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*arctan(2*(a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*
x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 4*(4*a^3*x^3 + a^2*x^2 - 7*a*x)*sqrt((a
*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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