\(\int \frac {e^{-\coth ^{-1}(a x)}}{(c-\frac {c}{a x})^{7/2}} \, dx\) [470]

Optimal result

Integrand size = 24, antiderivative size = 277 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=-\frac {5 a \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}{4 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {35 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}{16 \left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {5 \left (1-\frac {1}{a x}\right )^{7/2} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {115 \left (1-\frac {1}{a x}\right )^{7/2} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{16 \sqrt {2} a \left (c-\frac {c}{a x}\right )^{7/2}} \]

[Out]

5*(1-1/a/x)^(7/2)*arctanh((1+1/a/x)^(1/2))/a/(c-c/a/x)^(7/2)-115/32*(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)-5/4*a*(1-1/a/x)^(7/2)*(1+1/a/x)^(1/2)/(a-1/x)^2/(c-c/a/x)^(7/2)-35/16*(1-
1/a/x)^(7/2)*(1+1/a/x)^(1/2)/(a-1/x)/(c-c/a/x)^(7/2)+a^2*(1-1/a/x)^(7/2)*x*(1+1/a/x)^(1/2)/(a-1/x)^2/(c-c/a/x)
^(7/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6317, 6314, 105, 21, 101, 156, 162, 65, 214, 212} \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\frac {a^2 x \sqrt {\frac {1}{a x}+1} \left (1-\frac {1}{a x}\right )^{7/2}}{\left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {5 \left (1-\frac {1}{a x}\right )^{7/2} \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right )}{a \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {115 \left (1-\frac {1}{a x}\right )^{7/2} \text {arctanh}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right )}{16 \sqrt {2} a \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {35 \sqrt {\frac {1}{a x}+1} \left (1-\frac {1}{a x}\right )^{7/2}}{16 \left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {5 a \sqrt {\frac {1}{a x}+1} \left (1-\frac {1}{a x}\right )^{7/2}}{4 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{7/2}} \]

[In]

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

[Out]

(-5*a*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)])/(4*(a - x^(-1))^2*(c - c/(a*x))^(7/2)) - (35*(1 - 1/(a*x))^(7/2)*
Sqrt[1 + 1/(a*x)])/(16*(a - x^(-1))*(c - c/(a*x))^(7/2)) + (a^2*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)]*x)/((a -
x^(-1))^2*(c - c/(a*x))^(7/2)) + (5*(1 - 1/(a*x))^(7/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(c - c/(a*x))^(7/2)) -
(115*(1 - 1/(a*x))^(7/2)*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(16*Sqrt[2]*a*(c - c/(a*x))^(7/2))

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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 101

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

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 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^{-\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 )^3 \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\left (c-\frac {c}{a x}\right )^{7/2}} \\ & = \frac {a^2 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right )^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 {-\frac {5}{2 a}-\frac {5 x}{2 a^2}}{x \left (1-\frac {x}{a}\right )^3 \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\left (c-\frac {c}{a x}\right )^{7/2}} \\ & = \frac {a^2 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {\left (5 \left (1-\frac {1}{a x}\right )^{7/2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x \left (1-\frac {x}{a}\right )^3} \, dx,x,\frac {1}{x}\right )}{2 a \left (c-\frac {c}{a x}\right )^{7/2}} \\ & = -\frac {5 a \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}{4 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {\left (5 \left (1-\frac {1}{a x}\right )^{7/2}\right ) \text {Subst}\left (\int \frac {-2-\frac {3 x}{2 a}}{x \left (1-\frac {x}{a}\right )^2 \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{4 a \left (c-\frac {c}{a x}\right )^{7/2}} \\ & = -\frac {5 a \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}{4 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {35 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}{16 \left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {\left (5 \left (1-\frac {1}{a x}\right )^{7/2}\right ) \text {Subst}\left (\int \frac {\frac {4}{a}+\frac {7 x}{4 a^2}}{x \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{8 \left (c-\frac {c}{a x}\right )^{7/2}} \\ & = -\frac {5 a \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}{4 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {35 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}{16 \left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {\left (115 \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 )}{32 a^2 \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {\left (5 \left (1-\frac {1}{a x}\right )^{7/2}\right ) \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 {5 a \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}{4 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {35 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}{16 \left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {\left (5 \left (1-\frac {1}{a x}\right )^{7/2}\right ) \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 (115 \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 )}{16 a \left (c-\frac {c}{a x}\right )^{7/2}} \\ & = -\frac {5 a \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}{4 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {35 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}{16 \left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {5 \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 {115 \left (1-\frac {1}{a x}\right )^{7/2} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{16 \sqrt {2} a \left (c-\frac {c}{a x}\right )^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.49 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\frac {\sqrt {1-\frac {1}{a x}} \left (2 a \sqrt {1+\frac {1}{a x}} x \left (35-55 a x+16 a^2 x^2\right )+160 (-1+a x)^2 \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )-115 \sqrt {2} (-1+a x)^2 \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )\right )}{32 a c^3 \sqrt {c-\frac {c}{a x}} (-1+a x)^2} \]

[In]

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

[Out]

(Sqrt[1 - 1/(a*x)]*(2*a*Sqrt[1 + 1/(a*x)]*x*(35 - 55*a*x + 16*a^2*x^2) + 160*(-1 + a*x)^2*ArcTanh[Sqrt[1 + 1/(
a*x)]] - 115*Sqrt[2]*(-1 + a*x)^2*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]]))/(32*a*c^3*Sqrt[c - c/(a*x)]*(-1 + a*x)^
2)

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.14

method	result	size
risch	\(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}}+\frac {\left (\frac {5 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{2 a^{4} \sqrt {a^{2} c}}-\frac {\sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+3 \left (x -\frac {1}{a}\right ) a c +2 c}}{4 a^{7} c \left (x -\frac {1}{a}\right )^{2}}-\frac {23 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+3 \left (x -\frac {1}{a}\right ) a c +2 c}}{16 a^{6} c \left (x -\frac {1}{a}\right )}-\frac {115 \sqrt {2}\, \ln \left (\frac {4 c +3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+3 \left (x -\frac {1}{a}\right ) a c +2 c}}{x -\frac {1}{a}}\right )}{64 a^{5} \sqrt {c}}\right ) a^{3} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) a c x}}{c^{3} x \sqrt {\frac {c \left (a x -1\right )}{a x}}}\)	\(316\)
default	\(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (64 \sqrt {\left (a x +1\right ) x}\, a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, x^{2}-115 a^{\frac {5}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) x^{2}-220 \sqrt {\left (a x +1\right ) x}\, a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, x +160 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{3} \sqrt {\frac {1}{a}}\, x^{2}+230 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) x +140 \sqrt {\left (a x +1\right ) x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}-320 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{2} \sqrt {\frac {1}{a}}\, x +160 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}-115 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) \sqrt {a}\right )}{64 a^{\frac {3}{2}} c^{4} \left (a x -1\right )^{3} \sqrt {\left (a x +1\right ) x}\, \sqrt {\frac {1}{a}}}\)	\(371\)

[In]

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

[Out]

1/a*(a*x+1)/c^3*((a*x-1)/(a*x+1))^(1/2)/(c*(a*x-1)/a/x)^(1/2)+(5/2/a^4*ln((1/2*a*c+a^2*c*x)/(a^2*c)^(1/2)+(a^2
*c*x^2+a*c*x)^(1/2))/(a^2*c)^(1/2)-1/4/a^7/c/(x-1/a)^2*(a^2*c*(x-1/a)^2+3*(x-1/a)*a*c+2*c)^(1/2)-23/16/a^6/c/(
x-1/a)*(a^2*c*(x-1/a)^2+3*(x-1/a)*a*c+2*c)^(1/2)-115/64/a^5/c^(1/2)*2^(1/2)*ln((4*c+3*(x-1/a)*a*c+2*2^(1/2)*c^
(1/2)*(a^2*c*(x-1/a)^2+3*(x-1/a)*a*c+2*c)^(1/2))/(x-1/a)))/c^3*a^3*((a*x-1)/(a*x+1))^(1/2)/x/(c*(a*x-1)/a/x)^(
1/2)*((a*x+1)*a*c*x)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 668, normalized size of antiderivative = 2.41 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\left [\frac {115 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt {2} {\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 160 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 8 \, {\left (16 \, a^{4} x^{4} - 39 \, a^{3} x^{3} - 20 \, a^{2} x^{2} + 35 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{128 \, {\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}}, \frac {115 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {2} {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - 160 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 4 \, {\left (16 \, a^{4} x^{4} - 39 \, a^{3} x^{3} - 20 \, a^{2} x^{2} + 35 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{64 \, {\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}}\right ] \]

[In]

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

[Out]

[1/128*(115*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*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)) + 160*(a^3*x^3 - 3*a^2*x^2 + 3*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)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 8*(16*a^
4*x^4 - 39*a^3*x^3 - 20*a^2*x^2 + 35*a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^4*c^4*x^3 - 3*
a^3*c^4*x^2 + 3*a^2*c^4*x - a*c^4), 1/64*(115*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*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)) -
160*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*arctan(2*(a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqr
t((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 4*(16*a^4*x^4 - 39*a^3*x^3 - 20*a^2*x^2 + 35*a*x)*sqrt((a*x
- 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^4*c^4*x^3 - 3*a^3*c^4*x^2 + 3*a^2*c^4*x - a*c^4)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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