\(\int \frac {e^{\coth ^{-1}(a x)}}{(c-\frac {c}{a x})^4} \, dx\) [386]

Optimal result

Integrand size = 20, antiderivative size = 171 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=-\frac {16 \left (a+\frac {1}{x}\right )}{7 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {4 \left (7 a+\frac {17}{x}\right )}{35 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {175 a+\frac {307}{x}}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {525 a+\frac {719}{x}}{105 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^4}+\frac {5 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^4} \]

[Out]

-16/7*(a+1/x)/a^2/c^4/(1-1/a^2/x^2)^(7/2)-4/35*(7*a+17/x)/a^2/c^4/(1-1/a^2/x^2)^(5/2)+1/105*(-175*a-307/x)/a^2
/c^4/(1-1/a^2/x^2)^(3/2)+5*arctanh((1-1/a^2/x^2)^(1/2))/a/c^4+1/105*(-525*a-719/x)/a^2/c^4/(1-1/a^2/x^2)^(1/2)
+x*(1-1/a^2/x^2)^(1/2)/c^4

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6312, 866, 1819, 821, 272, 65, 214} \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {5 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^4}-\frac {16 \left (a+\frac {1}{x}\right )}{7 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {4 \left (7 a+\frac {17}{x}\right )}{35 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {175 a+\frac {307}{x}}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {525 a+\frac {719}{x}}{105 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^4} \]

[In]

Int[E^ArcCoth[a*x]/(c - c/(a*x))^4,x]

[Out]

(-16*(a + x^(-1)))/(7*a^2*c^4*(1 - 1/(a^2*x^2))^(7/2)) - (4*(7*a + 17/x))/(35*a^2*c^4*(1 - 1/(a^2*x^2))^(5/2))
- (175*a + 307/x)/(105*a^2*c^4*(1 - 1/(a^2*x^2))^(3/2)) - (525*a + 719/x)/(105*a^2*c^4*Sqrt[1 - 1/(a^2*x^2)])
+ (Sqrt[1 - 1/(a^2*x^2)]*x)/c^4 + (5*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(a*c^4)

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 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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g
))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e
^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0
] && EqQ[Simplify[m + 2*p + 3], 0]

Rule 866

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^(2*m)/a
^m, Int[(f + g*x)^n*((a + c*x^2)^(m + p)/(d - e*x)^m), x], x] /; FreeQ[{a, c, d, e, f, g, n, p}, x] && NeQ[e*f
- d*g, 0] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[f, 0] && ILtQ[m, -1] &&  !(IGtQ[n, 0] && ILtQ[m +
n, 0] &&  !GtQ[p, 1])

Rule 1819

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rule 6312

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

Rubi steps \begin{align*} \text {integral}& = -\left (c \text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{x^2 \left (c-\frac {c x}{a}\right )^5} \, dx,x,\frac {1}{x}\right )\right ) \\ & = -\frac {\text {Subst}\left (\int \frac {\left (c+\frac {c x}{a}\right )^5}{x^2 \left (1-\frac {x^2}{a^2}\right )^{9/2}} \, dx,x,\frac {1}{x}\right )}{c^9} \\ & = -\frac {16 \left (a+\frac {1}{x}\right )}{7 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}+\frac {\text {Subst}\left (\int \frac {-7 c^5-\frac {35 c^5 x}{a}-\frac {61 c^5 x^2}{a^2}+\frac {7 c^5 x^3}{a^3}}{x^2 \left (1-\frac {x^2}{a^2}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{7 c^9} \\ & = -\frac {16 \left (a+\frac {1}{x}\right )}{7 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {4 \left (7 a+\frac {17}{x}\right )}{35 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {\text {Subst}\left (\int \frac {35 c^5+\frac {175 c^5 x}{a}+\frac {272 c^5 x^2}{a^2}}{x^2 \left (1-\frac {x^2}{a^2}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{35 c^9} \\ & = -\frac {16 \left (a+\frac {1}{x}\right )}{7 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {4 \left (7 a+\frac {17}{x}\right )}{35 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {175 a+\frac {307}{x}}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {-105 c^5-\frac {525 c^5 x}{a}-\frac {614 c^5 x^2}{a^2}}{x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{105 c^9} \\ & = -\frac {16 \left (a+\frac {1}{x}\right )}{7 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {4 \left (7 a+\frac {17}{x}\right )}{35 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {175 a+\frac {307}{x}}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {525 a+\frac {719}{x}}{105 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {\text {Subst}\left (\int \frac {105 c^5+\frac {525 c^5 x}{a}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{105 c^9} \\ & = -\frac {16 \left (a+\frac {1}{x}\right )}{7 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {4 \left (7 a+\frac {17}{x}\right )}{35 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {175 a+\frac {307}{x}}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {525 a+\frac {719}{x}}{105 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^4}-\frac {5 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a c^4} \\ & = -\frac {16 \left (a+\frac {1}{x}\right )}{7 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {4 \left (7 a+\frac {17}{x}\right )}{35 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {175 a+\frac {307}{x}}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {525 a+\frac {719}{x}}{105 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^4}-\frac {5 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 a c^4} \\ & = -\frac {16 \left (a+\frac {1}{x}\right )}{7 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {4 \left (7 a+\frac {17}{x}\right )}{35 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {175 a+\frac {307}{x}}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {525 a+\frac {719}{x}}{105 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^4}+\frac {(5 a) \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )}{c^4} \\ & = -\frac {16 \left (a+\frac {1}{x}\right )}{7 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {4 \left (7 a+\frac {17}{x}\right )}{35 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {175 a+\frac {307}{x}}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {525 a+\frac {719}{x}}{105 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^4}+\frac {5 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.65 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {824-1947 a x+485 a^2 x^2+1812 a^3 x^3-1339 a^4 x^4+105 a^5 x^5+525 a \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{105 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^3} \]

[In]

Integrate[E^ArcCoth[a*x]/(c - c/(a*x))^4,x]

[Out]

(824 - 1947*a*x + 485*a^2*x^2 + 1812*a^3*x^3 - 1339*a^4*x^4 + 105*a^5*x^5 + 525*a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1
+ a*x)^3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(105*a^2*c^4*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^3)

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.55

method	result	size
risch	\(\frac {a x -1}{a \,c^{4} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {5 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{4} \sqrt {a^{2}}}-\frac {57 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{35 a^{8} \left (x -\frac {1}{a}\right )^{3}}-\frac {446 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{105 a^{7} \left (x -\frac {1}{a}\right )^{2}}-\frac {1024 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{105 a^{6} \left (x -\frac {1}{a}\right )}-\frac {2 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{7 a^{9} \left (x -\frac {1}{a}\right )^{4}}\right ) a^{4} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{4} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(265\)
default	\(-\frac {-525 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}-525 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}+420 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}+2625 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{4} x^{4}+2625 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}-1076 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}-5250 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}-5250 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+970 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x +5250 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}+5250 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-299 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-2625 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x -2625 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x +525 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}+525 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )}{105 a \sqrt {a^{2}}\, \left (a x -1\right )^{4} c^{4} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {\frac {a x -1}{a x +1}}}\)	\(523\)

[In]

int(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^4,x,method=_RETURNVERBOSE)

[Out]

1/a*(a*x-1)/c^4/((a*x-1)/(a*x+1))^(1/2)+(5/a^4*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2))/(a^2)^(1/2)-57/35/a^8/(
x-1/a)^3*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2)-446/105/a^7/(x-1/a)^2*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2)-1024/105/a^
6/(x-1/a)*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2)-2/7/a^9/(x-1/a)^4*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2))*a^4/c^4/((a*x
-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^(1/2)/(a*x+1)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.19 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {525 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 525 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (105 \, a^{5} x^{5} - 1339 \, a^{4} x^{4} + 1812 \, a^{3} x^{3} + 485 \, a^{2} x^{2} - 1947 \, a x + 824\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{105 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} \]

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^4,x, algorithm="fricas")

[Out]

1/105*(525*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 525*(a^4*x^4 - 4
*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (105*a^5*x^5 - 1339*a^4*x^4 + 1812*a^3*
x^3 + 485*a^2*x^2 - 1947*a*x + 824)*sqrt((a*x - 1)/(a*x + 1)))/(a^5*c^4*x^4 - 4*a^4*c^4*x^3 + 6*a^3*c^4*x^2 -
4*a^2*c^4*x + a*c^4)

Sympy [F]

\[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {a^{4} \int \frac {x^{4}}{a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 4 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + 6 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 4 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx}{c^{4}} \]

[In]

integrate(1/((a*x-1)/(a*x+1))**(1/2)/(c-c/a/x)**4,x)

[Out]

a**4*Integral(x**4/(a**4*x**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) - 4*a**3*x**3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)
) + 6*a**2*x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) - 4*a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) + sqrt(a*x/(a*x +
1) - 1/(a*x + 1))), x)/c**4

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.99 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {1}{420} \, a {\left (\frac {\frac {111 \, {\left (a x - 1\right )}}{a x + 1} + \frac {469 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {2765 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac {4200 \, {\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 15}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} - a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}}} + \frac {2100 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac {2100 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{4}}\right )} \]

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^4,x, algorithm="maxima")

[Out]

1/420*a*((111*(a*x - 1)/(a*x + 1) + 469*(a*x - 1)^2/(a*x + 1)^2 + 2765*(a*x - 1)^3/(a*x + 1)^3 - 4200*(a*x - 1
)^4/(a*x + 1)^4 + 15)/(a^2*c^4*((a*x - 1)/(a*x + 1))^(9/2) - a^2*c^4*((a*x - 1)/(a*x + 1))^(7/2)) + 2100*log(s
qrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^4) - 2100*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^4))

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^4,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 [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.80 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {10\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^4}-\frac {\frac {67\,{\left (a\,x-1\right )}^2}{15\,{\left (a\,x+1\right )}^2}+\frac {79\,{\left (a\,x-1\right )}^3}{3\,{\left (a\,x+1\right )}^3}-\frac {40\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}+\frac {37\,\left (a\,x-1\right )}{35\,\left (a\,x+1\right )}+\frac {1}{7}}{4\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}-4\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}} \]

[In]

int(1/((c - c/(a*x))^4*((a*x - 1)/(a*x + 1))^(1/2)),x)

[Out]

(10*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(a*c^4) - ((67*(a*x - 1)^2)/(15*(a*x + 1)^2) + (79*(a*x - 1)^3)/(3*(a*
x + 1)^3) - (40*(a*x - 1)^4)/(a*x + 1)^4 + (37*(a*x - 1))/(35*(a*x + 1)) + 1/7)/(4*a*c^4*((a*x - 1)/(a*x + 1))
^(7/2) - 4*a*c^4*((a*x - 1)/(a*x + 1))^(9/2))