\(\int \frac {e^{3 \coth ^{-1}(a x)}}{(c-\frac {c}{a x})^4} \, dx\) [403]

Optimal result

Integrand size = 22, antiderivative size = 204 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {16 \left (9 a-\frac {5}{x}\right )}{63 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {64 \left (a+\frac {1}{x}\right )}{9 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}-\frac {8 \left (21 a+\frac {41}{x}\right )}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {735 a+\frac {1417}{x}}{315 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {2205 a+\frac {3149}{x}}{315 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^4}+\frac {7 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^4} \]

[Out]

16/63*(9*a-5/x)/a^2/c^4/(1-1/a^2/x^2)^(7/2)-64/9*(a+1/x)/a^2/c^4/(1-1/a^2/x^2)^(9/2)-8/105*(21*a+41/x)/a^2/c^4
/(1-1/a^2/x^2)^(5/2)+1/315*(-735*a-1417/x)/a^2/c^4/(1-1/a^2/x^2)^(3/2)+7*arctanh((1-1/a^2/x^2)^(1/2))/a/c^4+1/
315*(-2205*a-3149/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 = 0.47 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6312, 866, 1819, 821, 272, 65, 214} \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {7 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^4}+\frac {16 \left (9 a-\frac {5}{x}\right )}{63 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {64 \left (a+\frac {1}{x}\right )}{9 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}-\frac {8 \left (21 a+\frac {41}{x}\right )}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {735 a+\frac {1417}{x}}{315 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {2205 a+\frac {3149}{x}}{315 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^4} \]

[In]

Int[E^(3*ArcCoth[a*x])/(c - c/(a*x))^4,x]

[Out]

(16*(9*a - 5/x))/(63*a^2*c^4*(1 - 1/(a^2*x^2))^(7/2)) - (64*(a + x^(-1)))/(9*a^2*c^4*(1 - 1/(a^2*x^2))^(9/2))
- (8*(21*a + 41/x))/(105*a^2*c^4*(1 - 1/(a^2*x^2))^(5/2)) - (735*a + 1417/x)/(315*a^2*c^4*(1 - 1/(a^2*x^2))^(3
/2)) - (2205*a + 3149/x)/(315*a^2*c^4*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[1 - 1/(a^2*x^2)]*x)/c^4 + (7*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^3 \text {Subst}\left (\int \frac {\left (1-\frac {x^2}{a^2}\right )^{3/2}}{x^2 \left (c-\frac {c x}{a}\right )^7} \, dx,x,\frac {1}{x}\right )\right ) \\ & = -\frac {\text {Subst}\left (\int \frac {\left (c+\frac {c x}{a}\right )^7}{x^2 \left (1-\frac {x^2}{a^2}\right )^{11/2}} \, dx,x,\frac {1}{x}\right )}{c^{11}} \\ & = -\frac {64 \left (a+\frac {1}{x}\right )}{9 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}+\frac {\text {Subst}\left (\int \frac {-9 c^7-\frac {63 c^7 x}{a}-\frac {134 c^7 x^2}{a^2}+\frac {198 c^7 x^3}{a^3}+\frac {63 c^7 x^4}{a^4}+\frac {9 c^7 x^5}{a^5}}{x^2 \left (1-\frac {x^2}{a^2}\right )^{9/2}} \, dx,x,\frac {1}{x}\right )}{9 c^{11}} \\ & = \frac {16 \left (9 a-\frac {5}{x}\right )}{63 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {64 \left (a+\frac {1}{x}\right )}{9 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}-\frac {\text {Subst}\left (\int \frac {63 c^7+\frac {441 c^7 x}{a}+\frac {921 c^7 x^2}{a^2}+\frac {63 c^7 x^3}{a^3}}{x^2 \left (1-\frac {x^2}{a^2}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{63 c^{11}} \\ & = \frac {16 \left (9 a-\frac {5}{x}\right )}{63 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {64 \left (a+\frac {1}{x}\right )}{9 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}-\frac {8 \left (21 a+\frac {41}{x}\right )}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}+\frac {\text {Subst}\left (\int \frac {-315 c^7-\frac {2205 c^7 x}{a}-\frac {3936 c^7 x^2}{a^2}}{x^2 \left (1-\frac {x^2}{a^2}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{315 c^{11}} \\ & = \frac {16 \left (9 a-\frac {5}{x}\right )}{63 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {64 \left (a+\frac {1}{x}\right )}{9 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}-\frac {8 \left (21 a+\frac {41}{x}\right )}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {735 a+\frac {1417}{x}}{315 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {945 c^7+\frac {6615 c^7 x}{a}+\frac {8502 c^7 x^2}{a^2}}{x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{945 c^{11}} \\ & = \frac {16 \left (9 a-\frac {5}{x}\right )}{63 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {64 \left (a+\frac {1}{x}\right )}{9 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}-\frac {8 \left (21 a+\frac {41}{x}\right )}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {735 a+\frac {1417}{x}}{315 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {2205 a+\frac {3149}{x}}{315 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\text {Subst}\left (\int \frac {-945 c^7-\frac {6615 c^7 x}{a}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{945 c^{11}} \\ & = \frac {16 \left (9 a-\frac {5}{x}\right )}{63 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {64 \left (a+\frac {1}{x}\right )}{9 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}-\frac {8 \left (21 a+\frac {41}{x}\right )}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {735 a+\frac {1417}{x}}{315 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {2205 a+\frac {3149}{x}}{315 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 {7 \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 (9 a-\frac {5}{x}\right )}{63 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {64 \left (a+\frac {1}{x}\right )}{9 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}-\frac {8 \left (21 a+\frac {41}{x}\right )}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {735 a+\frac {1417}{x}}{315 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {2205 a+\frac {3149}{x}}{315 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 {7 \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 (9 a-\frac {5}{x}\right )}{63 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {64 \left (a+\frac {1}{x}\right )}{9 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}-\frac {8 \left (21 a+\frac {41}{x}\right )}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {735 a+\frac {1417}{x}}{315 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {2205 a+\frac {3149}{x}}{315 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 {(7 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 (9 a-\frac {5}{x}\right )}{63 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {64 \left (a+\frac {1}{x}\right )}{9 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}-\frac {8 \left (21 a+\frac {41}{x}\right )}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {735 a+\frac {1417}{x}}{315 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {2205 a+\frac {3149}{x}}{315 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 {7 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.59 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {-3464+11651 a x-10232 a^2 x^2-5567 a^3 x^3+13241 a^4 x^4-6224 a^5 x^5+315 a^6 x^6+2205 a \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{315 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^4} \]

[In]

Integrate[E^(3*ArcCoth[a*x])/(c - c/(a*x))^4,x]

[Out]

(-3464 + 11651*a*x - 10232*a^2*x^2 - 5567*a^3*x^3 + 13241*a^4*x^4 - 6224*a^5*x^5 + 315*a^6*x^6 + 2205*a*Sqrt[1
- 1/(a^2*x^2)]*x*(-1 + a*x)^4*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(315*a^2*c^4*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)
^4)

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.50

method	result	size
risch	\(\frac {a x -1}{a \,c^{4} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {7 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{4} \sqrt {a^{2}}}-\frac {4 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{9 a^{10} \left (x -\frac {1}{a}\right )^{5}}-\frac {164 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{63 a^{9} \left (x -\frac {1}{a}\right )^{4}}-\frac {697 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{105 a^{8} \left (x -\frac {1}{a}\right )^{3}}-\frac {3226 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{315 a^{7} \left (x -\frac {1}{a}\right )^{2}}-\frac {4964 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{315 a^{6} \left (x -\frac {1}{a}\right )}\right ) a^{4} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{4} \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\)	\(305\)
default	\(-\frac {-2205 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{6} x^{6}-2205 \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^{7} x^{6}+1890 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}+13230 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}+13230 \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}-6376 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}-33075 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{4} x^{4}-33075 \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}+8646 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}+44100 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}+44100 \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}-5349 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -33075 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}-33075 \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}+1259 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+13230 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x +13230 \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 -2205 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}-2205 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 )}{315 a \left (a x -1\right )^{4} \sqrt {a^{2}}\, c^{4} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\)	\(622\)

[In]

int(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^4,x,method=_RETURNVERBOSE)

[Out]

1/a*(a*x-1)/c^4/((a*x-1)/(a*x+1))^(1/2)+(7/a^4*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2))/(a^2)^(1/2)-4/9/a^10/(x
-1/a)^5*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2)-164/63/a^9/(x-1/a)^4*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2)-697/105/a^8/(
x-1/a)^3*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2)-3226/315/a^7/(x-1/a)^2*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2)-4964/315/a
^6/(x-1/a)*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2))*a^4/c^4/(a*x+1)/((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.18 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {2205 \, {\left (a^{5} x^{5} - 5 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - 10 \, a^{2} x^{2} + 5 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 2205 \, {\left (a^{5} x^{5} - 5 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - 10 \, a^{2} x^{2} + 5 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (315 \, a^{6} x^{6} - 6224 \, a^{5} x^{5} + 13241 \, a^{4} x^{4} - 5567 \, a^{3} x^{3} - 10232 \, a^{2} x^{2} + 11651 \, a x - 3464\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{315 \, {\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right )}} \]

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^4,x, algorithm="fricas")

[Out]

1/315*(2205*(a^5*x^5 - 5*a^4*x^4 + 10*a^3*x^3 - 10*a^2*x^2 + 5*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 2
205*(a^5*x^5 - 5*a^4*x^4 + 10*a^3*x^3 - 10*a^2*x^2 + 5*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (315*a^6*
x^6 - 6224*a^5*x^5 + 13241*a^4*x^4 - 5567*a^3*x^3 - 10232*a^2*x^2 + 11651*a*x - 3464)*sqrt((a*x - 1)/(a*x + 1)
))/(a^6*c^4*x^5 - 5*a^5*c^4*x^4 + 10*a^4*c^4*x^3 - 10*a^3*c^4*x^2 + 5*a^2*c^4*x - a*c^4)

Sympy [F]

\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {a^{4} \int \frac {x^{4}}{\frac {a^{5} x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {5 a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {10 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {10 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {5 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx}{c^{4}} \]

[In]

integrate(1/((a*x-1)/(a*x+1))**(3/2)/(c-c/a/x)**4,x)

[Out]

a**4*Integral(x**4/(a**5*x**5*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - 5*a**4*x**4*sqrt(a*x/(a*x + 1) - 1
/(a*x + 1))/(a*x + 1) + 10*a**3*x**3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - 10*a**2*x**2*sqrt(a*x/(a*x
+ 1) - 1/(a*x + 1))/(a*x + 1) + 5*a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - sqrt(a*x/(a*x + 1) - 1/(a*
x + 1))/(a*x + 1)), x)/c**4

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.91 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {1}{1260} \, a {\left (\frac {\frac {235 \, {\left (a x - 1\right )}}{a x + 1} + \frac {801 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {2289 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + \frac {11760 \, {\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} - \frac {17640 \, {\left (a x - 1\right )}^{5}}{{\left (a x + 1\right )}^{5}} + 35}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} - a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}}} + \frac {8820 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac {8820 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{4}}\right )} \]

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^4,x, algorithm="maxima")

[Out]

1/1260*a*((235*(a*x - 1)/(a*x + 1) + 801*(a*x - 1)^2/(a*x + 1)^2 + 2289*(a*x - 1)^3/(a*x + 1)^3 + 11760*(a*x -
1)^4/(a*x + 1)^4 - 17640*(a*x - 1)^5/(a*x + 1)^5 + 35)/(a^2*c^4*((a*x - 1)/(a*x + 1))^(11/2) - a^2*c^4*((a*x
- 1)/(a*x + 1))^(9/2)) + 8820*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^4) - 8820*log(sqrt((a*x - 1)/(a*x + 1)
) - 1)/(a^2*c^4))

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.31 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=-\frac {7 \, \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{c^{4} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1}}{a c^{4} \mathrm {sgn}\left (a x + 1\right )} \]

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^4,x, algorithm="giac")

[Out]

-7*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))/(c^4*abs(a)*sgn(a*x + 1)) + sqrt(a^2*x^2 - 1)/(a*c^4*sgn(a*x + 1))

Mupad [B] (verification not implemented)

Time = 3.93 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.75 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {14\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^4}-\frac {\frac {89\,{\left (a\,x-1\right )}^2}{35\,{\left (a\,x+1\right )}^2}+\frac {109\,{\left (a\,x-1\right )}^3}{15\,{\left (a\,x+1\right )}^3}+\frac {112\,{\left (a\,x-1\right )}^4}{3\,{\left (a\,x+1\right )}^4}-\frac {56\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}+\frac {47\,\left (a\,x-1\right )}{63\,\left (a\,x+1\right )}+\frac {1}{9}}{4\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}-4\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}} \]

[In]

int(1/((c - c/(a*x))^4*((a*x - 1)/(a*x + 1))^(3/2)),x)

[Out]

(14*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(a*c^4) - ((89*(a*x - 1)^2)/(35*(a*x + 1)^2) + (109*(a*x - 1)^3)/(15*(
a*x + 1)^3) + (112*(a*x - 1)^4)/(3*(a*x + 1)^4) - (56*(a*x - 1)^5)/(a*x + 1)^5 + (47*(a*x - 1))/(63*(a*x + 1))
+ 1/9)/(4*a*c^4*((a*x - 1)/(a*x + 1))^(9/2) - 4*a*c^4*((a*x - 1)/(a*x + 1))^(11/2))