Integrand size = 24, antiderivative size = 189 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {2 \left (35 a-\frac {27}{x}\right )}{105 a^2 c^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}+\frac {2}{7 c \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \left (a+\frac {1}{x}\right )}+\frac {144 \sqrt {c-\frac {c}{a^2 x^2}} x}{35 c^4}-\frac {\left (19 a-\frac {14}{x}\right ) x}{35 a c \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}-\frac {2 \left (9 a-\frac {7}{x}\right ) x}{7 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {c}}\right )}{a c^{7/2}} \] Output:
2/105*(35*a-27/x)/a^2/c^2/(c-c/a^2/x^2)^(3/2)+2/7/c/(c-c/a^2/x^2)^(5/2)/(a +1/x)+144/35*(c-c/a^2/x^2)^(1/2)*x/c^4-1/35*(19*a-14/x)*x/a/c/(c-c/a^2/x^2 )^(5/2)-2/7*(9*a-7/x)*x/a/c^3/(c-c/a^2/x^2)^(1/2)-2*arctanh((c-c/a^2/x^2)^ (1/2)/c^(1/2))/a/c^(7/2)
Time = 0.21 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.69 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {432+654 a x-636 a^2 x^2-1226 a^3 x^3+74 a^4 x^4+562 a^5 x^5+105 a^6 x^6-210 (-1+a x) (1+a x)^3 \sqrt {-1+a^2 x^2} \log \left (a x+\sqrt {-1+a^2 x^2}\right )}{105 a^2 \sqrt {c-\frac {c}{a^2 x^2}} x (-1+a x) (c+a c x)^3} \] Input:
Integrate[1/(E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^(7/2)),x]
Output:
(432 + 654*a*x - 636*a^2*x^2 - 1226*a^3*x^3 + 74*a^4*x^4 + 562*a^5*x^5 + 1 05*a^6*x^6 - 210*(-1 + a*x)*(1 + a*x)^3*Sqrt[-1 + a^2*x^2]*Log[a*x + Sqrt[ -1 + a^2*x^2]])/(105*a^2*Sqrt[c - c/(a^2*x^2)]*x*(-1 + a*x)*(c + a*c*x)^3)
Time = 1.42 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6717, 6709, 570, 529, 25, 2166, 2345, 2345, 27, 455, 223}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 6717 |
| \(\displaystyle -\int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 6709 |
| \(\displaystyle -\frac {\left (1-a^2 x^2\right )^{7/2} \int \frac {x^7}{(a x+1)^2 \left (1-a^2 x^2\right )^{5/2}}dx}{x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}\) |
| \(\Big \downarrow \) 570 |
| \(\displaystyle -\frac {\left (1-a^2 x^2\right )^{7/2} \int \frac {x^7 (1-a x)^2}{\left (1-a^2 x^2\right )^{9/2}}dx}{x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}\) |
| \(\Big \downarrow \) 529 |
| \(\displaystyle -\frac {\left (1-a^2 x^2\right )^{7/2} \left (\frac {(1-a x)^2}{7 a^8 \left (1-a^2 x^2\right )^{7/2}}-\frac {1}{7} \int -\frac {(1-a x) \left (\frac {7 x^6}{a}-\frac {7 x^5}{a^2}+\frac {7 x^4}{a^3}-\frac {7 x^3}{a^4}+\frac {7 x^2}{a^5}-\frac {7 x}{a^6}+\frac {2}{a^7}\right )}{\left (1-a^2 x^2\right )^{7/2}}dx\right )}{x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\left (1-a^2 x^2\right )^{7/2} \left (\frac {1}{7} \int \frac {(1-a x) \left (\frac {7 x^6}{a}-\frac {7 x^5}{a^2}+\frac {7 x^4}{a^3}-\frac {7 x^3}{a^4}+\frac {7 x^2}{a^5}-\frac {7 x}{a^6}+\frac {2}{a^7}\right )}{\left (1-a^2 x^2\right )^{7/2}}dx+\frac {(1-a x)^2}{7 a^8 \left (1-a^2 x^2\right )^{7/2}}\right )}{x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}\) |
| \(\Big \downarrow \) 2166 |
| \(\displaystyle -\frac {\left (1-a^2 x^2\right )^{7/2} \left (\frac {1}{7} \left (-\frac {1}{5} \int \frac {-\frac {35 x^5}{a^2}+\frac {70 x^4}{a^3}-\frac {105 x^3}{a^4}+\frac {140 x^2}{a^5}-\frac {175 x}{a^6}+\frac {34}{a^7}}{\left (1-a^2 x^2\right )^{5/2}}dx-\frac {44 (1-a x)}{5 a^8 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(1-a x)^2}{7 a^8 \left (1-a^2 x^2\right )^{7/2}}\right )}{x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle -\frac {\left (1-a^2 x^2\right )^{7/2} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {-\frac {105 x^3}{a^4}+\frac {210 x^2}{a^5}-\frac {420 x}{a^6}+\frac {142}{a^7}}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {315-244 a x}{3 a^8 \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {44 (1-a x)}{5 a^8 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(1-a x)^2}{7 a^8 \left (1-a^2 x^2\right )^{7/2}}\right )}{x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle -\frac {\left (1-a^2 x^2\right )^{7/2} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (-\int \frac {105 (2-a x)}{a^7 \sqrt {1-a^2 x^2}}dx-\frac {525-352 a x}{a^8 \sqrt {1-a^2 x^2}}\right )+\frac {315-244 a x}{3 a^8 \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {44 (1-a x)}{5 a^8 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(1-a x)^2}{7 a^8 \left (1-a^2 x^2\right )^{7/2}}\right )}{x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\left (1-a^2 x^2\right )^{7/2} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (-\frac {105 \int \frac {2-a x}{\sqrt {1-a^2 x^2}}dx}{a^7}-\frac {525-352 a x}{a^8 \sqrt {1-a^2 x^2}}\right )+\frac {315-244 a x}{3 a^8 \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {44 (1-a x)}{5 a^8 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(1-a x)^2}{7 a^8 \left (1-a^2 x^2\right )^{7/2}}\right )}{x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle -\frac {\left (1-a^2 x^2\right )^{7/2} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (-\frac {105 \left (2 \int \frac {1}{\sqrt {1-a^2 x^2}}dx+\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a^7}-\frac {525-352 a x}{a^8 \sqrt {1-a^2 x^2}}\right )+\frac {315-244 a x}{3 a^8 \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {44 (1-a x)}{5 a^8 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(1-a x)^2}{7 a^8 \left (1-a^2 x^2\right )^{7/2}}\right )}{x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {\left (1-a^2 x^2\right )^{7/2} \left (\frac {(1-a x)^2}{7 a^8 \left (1-a^2 x^2\right )^{7/2}}+\frac {1}{7} \left (\frac {1}{5} \left (\frac {315-244 a x}{3 a^8 \left (1-a^2 x^2\right )^{3/2}}+\frac {1}{3} \left (-\frac {525-352 a x}{a^8 \sqrt {1-a^2 x^2}}-\frac {105 \left (\frac {\sqrt {1-a^2 x^2}}{a}+\frac {2 \arcsin (a x)}{a}\right )}{a^7}\right )\right )-\frac {44 (1-a x)}{5 a^8 \left (1-a^2 x^2\right )^{5/2}}\right )\right )}{x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}\) |
Input:
Int[1/(E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^(7/2)),x]
Output:
-(((1 - a^2*x^2)^(7/2)*((1 - a*x)^2/(7*a^8*(1 - a^2*x^2)^(7/2)) + ((-44*(1 - a*x))/(5*a^8*(1 - a^2*x^2)^(5/2)) + ((315 - 244*a*x)/(3*a^8*(1 - a^2*x^ 2)^(3/2)) + (-((525 - 352*a*x)/(a^8*Sqrt[1 - a^2*x^2])) - (105*(Sqrt[1 - a ^2*x^2]/a + (2*ArcSin[a*x])/a))/a^7)/3)/5)/7))/((c - c/(a^2*x^2))^(7/2)*x^ 7))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m, a*d + b*c*x, x], R = PolynomialRem ainder[x^m, a*d + b*c*x, x]}, Simp[(-c)*R*(c + d*x)^n*((a + b*x^2)^(p + 1)/ (2*a*d*(p + 1))), x] + Simp[c/(2*a*(p + 1)) Int[(c + d*x)^(n - 1)*(a + b* x^2)^(p + 1)*ExpandToSum[2*a*d*(p + 1)*Qx + R*(n + 2*p + 2), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 1] && LtQ[p, -1] && EqQ[b* c^2 + a*d^2, 0]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c^(2*n)/a^n Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I LtQ[n, -1] && !(IGtQ[m, 0] && ILtQ[m + n, 0] && !GtQ[p, 1])
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{Qx = PolynomialQuotient[Pq, a*e + b*d*x, x], R = PolynomialRemainde r[Pq, a*e + b*d*x, x]}, Simp[(-d)*R*(d + e*x)^m*((a + b*x^2)^(p + 1)/(2*a*e *(p + 1))), x] + Simp[d/(2*a*(p + 1)) Int[(d + e*x)^(m - 1)*(a + b*x^2)^( p + 1)*ExpandToSum[2*a*e*(p + 1)*Qx + R*(m + 2*p + 2), x], x], x]] /; FreeQ [{a, b, d, e}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 0] && ILtQ[p + 1/2, 0] && GtQ[m, 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[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] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbo l] :> Simp[x^(2*p)*((c + d/x^2)^p/(1 - a^2*x^2)^p) Int[u*((1 + a*x)^n/(x^ (2*p)*(1 - a^2*x^2)^(n/2 - p))), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[p] && IntegerQ[n/2] && !GtQ[c, 0]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2) Int[ u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
Leaf count of result is larger than twice the leaf count of optimal. \(369\) vs. \(2(163)=326\).
Time = 0.21 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.96
| method | result | size |
| risch | \(\frac {a^{2} x^{2}-1}{a^{2} c^{3} x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}+\frac {\left (-\frac {2 \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{a^{7} \sqrt {a^{2} c}}+\frac {3061 \sqrt {\left (x +\frac {1}{a}\right )^{2} a^{2} c -2 \left (x +\frac {1}{a}\right ) a c}}{840 a^{9} c \left (x +\frac {1}{a}\right )}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +2 \left (x -\frac {1}{a}\right ) a c}}{48 a^{10} c \left (x -\frac {1}{a}\right )^{2}}-\frac {7 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +2 \left (x -\frac {1}{a}\right ) a c}}{24 a^{9} c \left (x -\frac {1}{a}\right )}-\frac {\sqrt {\left (x +\frac {1}{a}\right )^{2} a^{2} c -2 \left (x +\frac {1}{a}\right ) a c}}{28 a^{12} c \left (x +\frac {1}{a}\right )^{4}}+\frac {39 \sqrt {\left (x +\frac {1}{a}\right )^{2} a^{2} c -2 \left (x +\frac {1}{a}\right ) a c}}{140 a^{11} c \left (x +\frac {1}{a}\right )^{3}}-\frac {1753 \sqrt {\left (x +\frac {1}{a}\right )^{2} a^{2} c -2 \left (x +\frac {1}{a}\right ) a c}}{1680 a^{10} c \left (x +\frac {1}{a}\right )^{2}}\right ) a^{6} \sqrt {c \left (a^{2} x^{2}-1\right )}}{c^{3} x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}\) | \(370\) |
| default | \(-\frac {\left (-105 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} c^{\frac {7}{2}} a^{7} x^{7}+96 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} c^{\frac {7}{2}} a^{6} x^{6}-553 x^{6} c^{\frac {7}{2}} a^{6} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}}+96 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} c^{\frac {7}{2}} a^{5} x^{5}+392 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} c^{\frac {7}{2}} a^{5} x^{5}-240 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} c^{\frac {7}{2}} a^{4} x^{4}+1540 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} c^{\frac {7}{2}} a^{4} x^{4}+210 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) a^{6} c x -240 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} c^{\frac {7}{2}} a^{3} x^{3}-350 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} c^{\frac {7}{2}} a^{3} x^{3}+210 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) a^{5} c +180 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} c^{\frac {7}{2}} a^{2} x^{2}-1470 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} c^{\frac {7}{2}} a^{2} x^{2}+180 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} c^{\frac {7}{2}} a x +42 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} c^{\frac {7}{2}} a x -30 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} c^{\frac {7}{2}}+462 c^{\frac {7}{2}} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}}\right ) \left (a x -1\right )}{105 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} x^{7} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {7}{2}} a^{8} c^{\frac {7}{2}}}\) | \(572\) |
Input:
int((a*x-1)/(a*x+1)/(c-c/a^2/x^2)^(7/2),x,method=_RETURNVERBOSE)
Output:
1/a^2*(a^2*x^2-1)/c^3/x/(c*(a^2*x^2-1)/a^2/x^2)^(1/2)+(-2/a^7*ln(a^2*c*x/( a^2*c)^(1/2)+(a^2*c*x^2-c)^(1/2))/(a^2*c)^(1/2)+3061/840/a^9/c/(x+1/a)*((x +1/a)^2*a^2*c-2*(x+1/a)*a*c)^(1/2)-1/48/a^10/c/(x-1/a)^2*((x-1/a)^2*a^2*c+ 2*(x-1/a)*a*c)^(1/2)-7/24/a^9/c/(x-1/a)*((x-1/a)^2*a^2*c+2*(x-1/a)*a*c)^(1 /2)-1/28/a^12/c/(x+1/a)^4*((x+1/a)^2*a^2*c-2*(x+1/a)*a*c)^(1/2)+39/140/a^1 1/c/(x+1/a)^3*((x+1/a)^2*a^2*c-2*(x+1/a)*a*c)^(1/2)-1753/1680/a^10/c/(x+1/ a)^2*((x+1/a)^2*a^2*c-2*(x+1/a)*a*c)^(1/2))*a^6/c^3/x/(c*(a^2*x^2-1)/a^2/x ^2)^(1/2)*(c*(a^2*x^2-1))^(1/2)
Time = 0.19 (sec) , antiderivative size = 495, normalized size of antiderivative = 2.62 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\left [\frac {105 \, {\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt {c} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (105 \, a^{7} x^{7} + 562 \, a^{6} x^{6} + 74 \, a^{5} x^{5} - 1226 \, a^{4} x^{4} - 636 \, a^{3} x^{3} + 654 \, a^{2} x^{2} + 432 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{105 \, {\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}}, \frac {210 \, {\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + {\left (105 \, a^{7} x^{7} + 562 \, a^{6} x^{6} + 74 \, a^{5} x^{5} - 1226 \, a^{4} x^{4} - 636 \, a^{3} x^{3} + 654 \, a^{2} x^{2} + 432 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{105 \, {\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}}\right ] \] Input:
integrate((a*x-1)/(a*x+1)/(c-c/a^2/x^2)^(7/2),x, algorithm="fricas")
Output:
[1/105*(105*(a^6*x^6 + 2*a^5*x^5 - a^4*x^4 - 4*a^3*x^3 - a^2*x^2 + 2*a*x + 1)*sqrt(c)*log(2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2* x^2)) - c) + (105*a^7*x^7 + 562*a^6*x^6 + 74*a^5*x^5 - 1226*a^4*x^4 - 636* a^3*x^3 + 654*a^2*x^2 + 432*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^7*c^4 *x^6 + 2*a^6*c^4*x^5 - a^5*c^4*x^4 - 4*a^4*c^4*x^3 - a^3*c^4*x^2 + 2*a^2*c ^4*x + a*c^4), 1/105*(210*(a^6*x^6 + 2*a^5*x^5 - a^4*x^4 - 4*a^3*x^3 - a^2 *x^2 + 2*a*x + 1)*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a ^2*x^2))/(a^2*c*x^2 - c)) + (105*a^7*x^7 + 562*a^6*x^6 + 74*a^5*x^5 - 1226 *a^4*x^4 - 636*a^3*x^3 + 654*a^2*x^2 + 432*a*x)*sqrt((a^2*c*x^2 - c)/(a^2* x^2)))/(a^7*c^4*x^6 + 2*a^6*c^4*x^5 - a^5*c^4*x^4 - 4*a^4*c^4*x^3 - a^3*c^ 4*x^2 + 2*a^2*c^4*x + a*c^4)]
\[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\int \frac {a x - 1}{\left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{\frac {7}{2}} \left (a x + 1\right )}\, dx \] Input:
integrate((a*x-1)/(a*x+1)/(c-c/a**2/x**2)**(7/2),x)
Output:
Integral((a*x - 1)/((-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(7/2)*(a*x + 1)), x )
\[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\int { \frac {a x - 1}{{\left (a x + 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((a*x-1)/(a*x+1)/(c-c/a^2/x^2)^(7/2),x, algorithm="maxima")
Output:
integrate((a*x - 1)/((a*x + 1)*(c - c/(a^2*x^2))^(7/2)), x)
Exception generated. \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x-1)/(a*x+1)/(c-c/a^2/x^2)^(7/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\int \frac {a\,x-1}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{7/2}\,\left (a\,x+1\right )} \,d x \] Input:
int((a*x - 1)/((c - c/(a^2*x^2))^(7/2)*(a*x + 1)),x)
Output:
int((a*x - 1)/((c - c/(a^2*x^2))^(7/2)*(a*x + 1)), x)
Time = 0.18 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.96 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {\sqrt {c}\, \left (420 \sqrt {a^{2} x^{2}-1}\, a^{6} x^{6}+2248 \sqrt {a^{2} x^{2}-1}\, a^{5} x^{5}+296 \sqrt {a^{2} x^{2}-1}\, a^{4} x^{4}-4904 \sqrt {a^{2} x^{2}-1}\, a^{3} x^{3}-2544 \sqrt {a^{2} x^{2}-1}\, a^{2} x^{2}+2616 \sqrt {a^{2} x^{2}-1}\, a x +1728 \sqrt {a^{2} x^{2}-1}-840 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right ) a^{6} x^{6}-1680 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right ) a^{5} x^{5}+840 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right ) a^{4} x^{4}+3360 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right ) a^{3} x^{3}+840 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right ) a^{2} x^{2}-1680 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right ) a x -840 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right )-463 a^{6} x^{6}-926 a^{5} x^{5}+463 a^{4} x^{4}+1852 a^{3} x^{3}+463 a^{2} x^{2}-926 a x -463\right )}{420 a \,c^{4} \left (a^{6} x^{6}+2 a^{5} x^{5}-a^{4} x^{4}-4 a^{3} x^{3}-a^{2} x^{2}+2 a x +1\right )} \] Input:
int((a*x-1)/(a*x+1)/(c-c/a^2/x^2)^(7/2),x)
Output:
(sqrt(c)*(420*sqrt(a**2*x**2 - 1)*a**6*x**6 + 2248*sqrt(a**2*x**2 - 1)*a** 5*x**5 + 296*sqrt(a**2*x**2 - 1)*a**4*x**4 - 4904*sqrt(a**2*x**2 - 1)*a**3 *x**3 - 2544*sqrt(a**2*x**2 - 1)*a**2*x**2 + 2616*sqrt(a**2*x**2 - 1)*a*x + 1728*sqrt(a**2*x**2 - 1) - 840*log(sqrt(a**2*x**2 - 1) + a*x)*a**6*x**6 - 1680*log(sqrt(a**2*x**2 - 1) + a*x)*a**5*x**5 + 840*log(sqrt(a**2*x**2 - 1) + a*x)*a**4*x**4 + 3360*log(sqrt(a**2*x**2 - 1) + a*x)*a**3*x**3 + 840 *log(sqrt(a**2*x**2 - 1) + a*x)*a**2*x**2 - 1680*log(sqrt(a**2*x**2 - 1) + a*x)*a*x - 840*log(sqrt(a**2*x**2 - 1) + a*x) - 463*a**6*x**6 - 926*a**5* x**5 + 463*a**4*x**4 + 1852*a**3*x**3 + 463*a**2*x**2 - 926*a*x - 463))/(4 20*a*c**4*(a**6*x**6 + 2*a**5*x**5 - a**4*x**4 - 4*a**3*x**3 - a**2*x**2 + 2*a*x + 1))