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

Optimal. Leaf size=204 \[ \frac {16 \left (9 a-\frac {5}{x}\right )}{63 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {64 \left (a+\frac {1}{x}\right )}{9 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}-\frac {8 \left (21 a+\frac {41}{x}\right )}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {735 a+\frac {1417}{x}}{315 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {2205 a+\frac {3149}{x}}{315 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^4}+\frac {7 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^4} \]

[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

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Rubi [A]  time = 0.64, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6177, 852, 1805, 807, 266, 63, 208} \[ \frac {16 \left (9 a-\frac {5}{x}\right )}{63 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {64 \left (a+\frac {1}{x}\right )}{9 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}-\frac {8 \left (21 a+\frac {41}{x}\right )}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {735 a+\frac {1417}{x}}{315 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {2205 a+\frac {3149}{x}}{315 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^4}+\frac {7 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^4} \]

Antiderivative was successfully verified.

[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 63

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 208

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

Rule 266

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 807

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 852

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 1805

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 6177

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*} \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx &=-\left (c^3 \operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 \operatorname {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 \operatorname {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) \operatorname {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 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^4}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 120, normalized size = 0.59 \[ \frac {315 a^6 x^6-6224 a^5 x^5+13241 a^4 x^4-5567 a^3 x^3-10232 a^2 x^2+2205 a x \sqrt {1-\frac {1}{a^2 x^2}} (a x-1)^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )+11651 a x-3464}{315 a^2 c^4 x \sqrt {1-\frac {1}{a^2 x^2}} (a x-1)^4} \]

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.58, size = 240, normalized size = 1.18 \[ \frac {2205 \, {\left (a^{5} x^{5} - 5 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - 10 \, a^{2} x^{2} + 5 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 2205 \, {\left (a^{5} x^{5} - 5 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - 10 \, a^{2} x^{2} + 5 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (315 \, a^{6} x^{6} - 6224 \, a^{5} x^{5} + 13241 \, a^{4} x^{4} - 5567 \, a^{3} x^{3} - 10232 \, a^{2} x^{2} + 11651 \, a x - 3464\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{315 \, {\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.18, size = 198, normalized size = 0.97 \[ \frac {1}{1260} \, a {\left (\frac {8820 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac {8820 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c^{4}} - \frac {{\left (a x + 1\right )}^{4} {\left (\frac {270 \, {\left (a x - 1\right )}}{a x + 1} + \frac {1071 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {3360 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + \frac {15120 \, {\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 35\right )}}{{\left (a x - 1\right )}^{4} a^{2} c^{4} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {2520 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{4} {\left (\frac {a x - 1}{a x + 1} - 1\right )}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1260*a*(8820*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^4) - 8820*log(abs(sqrt((a*x - 1)/(a*x + 1)) - 1))/(a^
2*c^4) - (a*x + 1)^4*(270*(a*x - 1)/(a*x + 1) + 1071*(a*x - 1)^2/(a*x + 1)^2 + 3360*(a*x - 1)^3/(a*x + 1)^3 +
15120*(a*x - 1)^4/(a*x + 1)^4 + 35)/((a*x - 1)^4*a^2*c^4*sqrt((a*x - 1)/(a*x + 1))) - 2520*sqrt((a*x - 1)/(a*x
+ 1))/(a^2*c^4*((a*x - 1)/(a*x + 1) - 1)))

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maple [B]  time = 0.07, size = 622, normalized size = 3.05 \[ -\frac {-2205 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{6} a^{6}-2205 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{6} a^{7}+1890 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{4} a^{4}+13230 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{5} a^{5}+13230 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{5} a^{6}-6376 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x^{3} a^{3}-33075 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{4} a^{4}-33075 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{4} a^{5}+8646 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x^{2} a^{2}+44100 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{3} a^{3}+44100 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{3} a^{4}-5349 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x a -33075 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{2} a^{2}-33075 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}+1259 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+13230 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a +13230 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}-2205 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}-2205 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\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}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/315*(-2205*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*x^6*a^6-2205*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/
(a^2)^(1/2))*x^6*a^7+1890*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)*x^4*a^4+13230*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2
)*x^5*a^5+13230*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^5*a^6-6376*(a^2)^(1/2)*((a*x-1)*
(a*x+1))^(3/2)*x^3*a^3-33075*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^4*a^4-33075*ln((a^2*x+((a*x-1)*(a*x+1))^(1/
2)*(a^2)^(1/2))/(a^2)^(1/2))*x^4*a^5+8646*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^2*a^2+44100*(a^2)^(1/2)*((a*x-
1)*(a*x+1))^(1/2)*x^3*a^3+44100*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^3*a^4-5349*(a^2)
^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x*a-33075*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*x^2*a^2-33075*ln((a^2*x+((a*x-1)*
(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^2*a^3+1259*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)+13230*((a*x-1)*(a*x+
1))^(1/2)*(a^2)^(1/2)*x*a+13230*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x*a^2-2205*((a*x-1
)*(a*x+1))^(1/2)*(a^2)^(1/2)-2205*a*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2)))/a/(a*x-1)^4/(
a^2)^(1/2)/c^4/((a*x-1)*(a*x+1))^(1/2)/(a*x+1)/((a*x-1)/(a*x+1))^(3/2)

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maxima [A]  time = 0.31, size = 185, normalized size = 0.91 \[ \frac {1}{1260} \, a {\left (\frac {\frac {235 \, {\left (a x - 1\right )}}{a x + 1} + \frac {801 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {2289 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + \frac {11760 \, {\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} - \frac {17640 \, {\left (a x - 1\right )}^{5}}{{\left (a x + 1\right )}^{5}} + 35}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} - a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}}} + \frac {8820 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac {8820 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 1.25, size = 153, normalized size = 0.75 \[ \frac {14\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^4}-\frac {\frac {89\,{\left (a\,x-1\right )}^2}{35\,{\left (a\,x+1\right )}^2}+\frac {109\,{\left (a\,x-1\right )}^3}{15\,{\left (a\,x+1\right )}^3}+\frac {112\,{\left (a\,x-1\right )}^4}{3\,{\left (a\,x+1\right )}^4}-\frac {56\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}+\frac {47\,\left (a\,x-1\right )}{63\,\left (a\,x+1\right )}+\frac {1}{9}}{4\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}-4\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a^{4} \int \frac {x^{4}}{\frac {a^{5} x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {5 a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {10 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {10 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {5 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx}{c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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