Integrand size = 24, antiderivative size = 145 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=-\frac {11}{7 a \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {11}{5 a c \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {11}{3 a c^2 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {11}{a c^3 \sqrt {c-\frac {c}{a x}}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{7/2}}+\frac {11 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{7/2}} \] Output:
-11/7/a/(c-c/a/x)^(7/2)-11/5/a/c/(c-c/a/x)^(5/2)-11/3/a/c^2/(c-c/a/x)^(3/2 )-11/a/c^3/(c-c/a/x)^(1/2)+x/(c-c/a/x)^(7/2)+11*arctanh((c-c/a/x)^(1/2)/c^ (1/2))/a/c^(7/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.32 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\frac {7 x-\frac {11 \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},1,-\frac {5}{2},1-\frac {1}{a x}\right )}{a}}{7 \left (c-\frac {c}{a x}\right )^{7/2}} \] Input:
Integrate[E^(2*ArcCoth[a*x])/(c - c/(a*x))^(7/2),x]
Output:
(7*x - (11*Hypergeometric2F1[-7/2, 1, -5/2, 1 - 1/(a*x)])/a)/(7*(c - c/(a* x))^(7/2))
Time = 0.83 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6717, 6683, 1035, 281, 899, 87, 61, 61, 61, 61, 73, 221}
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.
| \(\displaystyle \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 6717 |
| \(\displaystyle -\int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 6683 |
| \(\displaystyle -\int \frac {a x+1}{\left (c-\frac {c}{a x}\right )^{7/2} (1-a x)}dx\) |
| \(\Big \downarrow \) 1035 |
| \(\displaystyle -\int \frac {a+\frac {1}{x}}{\left (\frac {1}{x}-a\right ) \left (c-\frac {c}{a x}\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 281 |
| \(\displaystyle \frac {c \int \frac {a+\frac {1}{x}}{\left (c-\frac {c}{a x}\right )^{9/2}}dx}{a}\) |
| \(\Big \downarrow \) 899 |
| \(\displaystyle -\frac {c \int \frac {\left (a+\frac {1}{x}\right ) x^2}{\left (c-\frac {c}{a x}\right )^{9/2}}d\frac {1}{x}}{a}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {c \left (\frac {11}{2} \int \frac {x}{\left (c-\frac {c}{a x}\right )^{9/2}}d\frac {1}{x}-\frac {a x}{c \left (c-\frac {c}{a x}\right )^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {c \left (\frac {11}{2} \left (\frac {\int \frac {x}{\left (c-\frac {c}{a x}\right )^{7/2}}d\frac {1}{x}}{c}+\frac {2}{7 c \left (c-\frac {c}{a x}\right )^{7/2}}\right )-\frac {a x}{c \left (c-\frac {c}{a x}\right )^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {c \left (\frac {11}{2} \left (\frac {\frac {\int \frac {x}{\left (c-\frac {c}{a x}\right )^{5/2}}d\frac {1}{x}}{c}+\frac {2}{5 c \left (c-\frac {c}{a x}\right )^{5/2}}}{c}+\frac {2}{7 c \left (c-\frac {c}{a x}\right )^{7/2}}\right )-\frac {a x}{c \left (c-\frac {c}{a x}\right )^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {c \left (\frac {11}{2} \left (\frac {\frac {\frac {\int \frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}d\frac {1}{x}}{c}+\frac {2}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}}{c}+\frac {2}{5 c \left (c-\frac {c}{a x}\right )^{5/2}}}{c}+\frac {2}{7 c \left (c-\frac {c}{a x}\right )^{7/2}}\right )-\frac {a x}{c \left (c-\frac {c}{a x}\right )^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {c \left (\frac {11}{2} \left (\frac {\frac {\frac {\frac {\int \frac {x}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{c}+\frac {2}{c \sqrt {c-\frac {c}{a x}}}}{c}+\frac {2}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}}{c}+\frac {2}{5 c \left (c-\frac {c}{a x}\right )^{5/2}}}{c}+\frac {2}{7 c \left (c-\frac {c}{a x}\right )^{7/2}}\right )-\frac {a x}{c \left (c-\frac {c}{a x}\right )^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {c \left (\frac {11}{2} \left (\frac {\frac {\frac {\frac {2}{c \sqrt {c-\frac {c}{a x}}}-\frac {2 a \int \frac {1}{a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}}{c^2}}{c}+\frac {2}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}}{c}+\frac {2}{5 c \left (c-\frac {c}{a x}\right )^{5/2}}}{c}+\frac {2}{7 c \left (c-\frac {c}{a x}\right )^{7/2}}\right )-\frac {a x}{c \left (c-\frac {c}{a x}\right )^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {c \left (\frac {11}{2} \left (\frac {\frac {\frac {\frac {2}{c \sqrt {c-\frac {c}{a x}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{c^{3/2}}}{c}+\frac {2}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}}{c}+\frac {2}{5 c \left (c-\frac {c}{a x}\right )^{5/2}}}{c}+\frac {2}{7 c \left (c-\frac {c}{a x}\right )^{7/2}}\right )-\frac {a x}{c \left (c-\frac {c}{a x}\right )^{7/2}}\right )}{a}\) |
Input:
Int[E^(2*ArcCoth[a*x])/(c - c/(a*x))^(7/2),x]
Output:
-((c*(-((a*x)/(c*(c - c/(a*x))^(7/2))) + (11*(2/(7*c*(c - c/(a*x))^(7/2)) + (2/(5*c*(c - c/(a*x))^(5/2)) + (2/(3*c*(c - c/(a*x))^(3/2)) + (2/(c*Sqrt [c - c/(a*x)]) - (2*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/c^(3/2))/c)/c)/c)) /2))/a)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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]
Int[(u_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_ Symbol] :> Simp[(b/d)^p Int[u*(c + d*x^n)^(p + q), x], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && EqQ[b*c - a*d, 0] && IntegerQ[p] && !(IntegerQ[q] & & SimplerQ[a + b*x^n, c + d*x^n])
Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol ] :> -Subst[Int[(a + b/x^n)^p*((c + d/x^n)^q/x^2), x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]
Int[((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_.))^(p_.)*((e_) + (f_.)*(x_)^(n_.))^(r_.), x_Symbol] :> Int[x^(n*(p + r))*(b + a/x^n)^p*(c + d/x^n)^q*(f + e/x^n)^r, x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && EqQ[ mn, -n] && IntegerQ[p] && IntegerQ[r]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Int[u*(c + d/x)^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c, d, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[p] && IntegerQ[n/2] && !G tQ[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. \(290\) vs. \(2(123)=246\).
Time = 0.18 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.01
| method | result | size |
| risch | \(\frac {a x -1}{a \,c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}}+\frac {\left (\frac {11 \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right )}{2 a^{4} \sqrt {a^{2} c}}-\frac {4 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +\left (x -\frac {1}{a}\right ) a c}}{7 a^{9} c \left (x -\frac {1}{a}\right )^{4}}-\frac {102 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +\left (x -\frac {1}{a}\right ) a c}}{35 a^{8} c \left (x -\frac {1}{a}\right )^{3}}-\frac {712 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +\left (x -\frac {1}{a}\right ) a c}}{105 a^{7} c \left (x -\frac {1}{a}\right )^{2}}-\frac {1516 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +\left (x -\frac {1}{a}\right ) a c}}{105 a^{6} c \left (x -\frac {1}{a}\right )}\right ) a^{3} \sqrt {c \left (a x -1\right ) a x}}{c^{3} x \sqrt {\frac {c \left (a x -1\right )}{a x}}}\) | \(291\) |
| default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (-2310 \sqrt {x \left (a x -1\right )}\, a^{\frac {11}{2}} x^{5}+2100 \left (x \left (a x -1\right )\right )^{\frac {3}{2}} a^{\frac {9}{2}} x^{3}-1155 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{5} x^{5}+11550 \sqrt {x \left (a x -1\right )}\, a^{\frac {9}{2}} x^{4}-5368 \left (x \left (a x -1\right )\right )^{\frac {3}{2}} a^{\frac {7}{2}} x^{2}+5775 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{4} x^{4}-23100 \sqrt {x \left (a x -1\right )}\, a^{\frac {7}{2}} x^{3}+4928 \left (x \left (a x -1\right )\right )^{\frac {3}{2}} a^{\frac {5}{2}} x -11550 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{3} x^{3}+23100 \sqrt {x \left (a x -1\right )}\, a^{\frac {5}{2}} x^{2}-1540 a^{\frac {3}{2}} \left (x \left (a x -1\right )\right )^{\frac {3}{2}}+11550 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{2} x^{2}-11550 \sqrt {x \left (a x -1\right )}\, a^{\frac {3}{2}} x -5775 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a x +2310 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+1155 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right )\right )}{210 \sqrt {x \left (a x -1\right )}\, c^{4} \sqrt {a}\, \left (a x -1\right )^{5}}\) | \(396\) |
Input:
int(1/(a*x-1)*(a*x+1)/(c-c/a/x)^(7/2),x,method=_RETURNVERBOSE)
Output:
1/a*(a*x-1)/c^3/(c*(a*x-1)/a/x)^(1/2)+(11/2/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)-4/7/a^9/c/(x-1/a)^4*((x-1 /a)^2*a^2*c+(x-1/a)*a*c)^(1/2)-102/35/a^8/c/(x-1/a)^3*((x-1/a)^2*a^2*c+(x- 1/a)*a*c)^(1/2)-712/105/a^7/c/(x-1/a)^2*((x-1/a)^2*a^2*c+(x-1/a)*a*c)^(1/2 )-1516/105/a^6/c/(x-1/a)*((x-1/a)^2*a^2*c+(x-1/a)*a*c)^(1/2))*a^3/c^3/x/(c *(a*x-1)/a/x)^(1/2)*(c*(a*x-1)*a*x)^(1/2)
Time = 0.12 (sec) , antiderivative size = 355, normalized size of antiderivative = 2.45 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\left [\frac {1155 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt {c} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) + 2 \, {\left (105 \, a^{5} x^{5} - 1936 \, a^{4} x^{4} + 4466 \, a^{3} x^{3} - 3850 \, a^{2} x^{2} + 1155 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{210 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}}, -\frac {1155 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {a \sqrt {-c} x \sqrt {\frac {a c x - c}{a x}}}{a c x - c}\right ) - {\left (105 \, a^{5} x^{5} - 1936 \, a^{4} x^{4} + 4466 \, a^{3} x^{3} - 3850 \, a^{2} x^{2} + 1155 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{105 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}}\right ] \] Input:
integrate(1/(a*x-1)*(a*x+1)/(c-c/a/x)^(7/2),x, algorithm="fricas")
Output:
[1/210*(1155*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt(c)*log(-2* a*c*x - 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) + 2*(105*a^5*x^5 - 1936 *a^4*x^4 + 4466*a^3*x^3 - 3850*a^2*x^2 + 1155*a*x)*sqrt((a*c*x - c)/(a*x)) )/(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), -1/ 105*(1155*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt(-c)*arctan(a* sqrt(-c)*x*sqrt((a*c*x - c)/(a*x))/(a*c*x - c)) - (105*a^5*x^5 - 1936*a^4* x^4 + 4466*a^3*x^3 - 3850*a^2*x^2 + 1155*a*x)*sqrt((a*c*x - c)/(a*x)))/(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)]
\[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\int \frac {a x + 1}{\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {7}{2}} \left (a x - 1\right )}\, dx \] Input:
integrate(1/(a*x-1)*(a*x+1)/(c-c/a/x)**(7/2),x)
Output:
Integral((a*x + 1)/((-c*(-1 + 1/(a*x)))**(7/2)*(a*x - 1)), x)
\[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\int { \frac {a x + 1}{{\left (a x - 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/(a*x-1)*(a*x+1)/(c-c/a/x)^(7/2),x, algorithm="maxima")
Output:
integrate((a*x + 1)/((a*x - 1)*(c - c/(a*x))^(7/2)), x)
Leaf count of result is larger than twice the leaf count of optimal. 391 vs. \(2 (123) = 246\).
Time = 0.34 (sec) , antiderivative size = 391, normalized size of antiderivative = 2.70 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\frac {11 \, \log \left (c^{4} {\left | a \right |} {\left | c \right |}\right ) \mathrm {sgn}\left (x\right )}{18 \, a c^{\frac {7}{2}}} - \frac {11 \, \log \left ({\left | 2 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{9} \sqrt {c} {\left | a \right |} - 17 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{8} a c + 64 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{7} c^{\frac {3}{2}} {\left | a \right |} - 140 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{6} a c^{2} + 196 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{5} c^{\frac {5}{2}} {\left | a \right |} - 182 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{4} a c^{3} + 112 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{3} c^{\frac {7}{2}} {\left | a \right |} - 44 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{2} a c^{4} + 10 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} c^{\frac {9}{2}} {\left | a \right |} - a c^{5} \right |}\right ) \mathrm {sgn}\left (x\right )}{18 \, a c^{\frac {7}{2}}} + \frac {\sqrt {a^{2} c x^{2} - a c x} {\left | a \right |} \mathrm {sgn}\left (x\right )}{a^{2} c^{4}} \] Input:
integrate(1/(a*x-1)*(a*x+1)/(c-c/a/x)^(7/2),x, algorithm="giac")
Output:
11/18*log(c^4*abs(a)*abs(c))*sgn(x)/(a*c^(7/2)) - 11/18*log(abs(2*(sqrt(a^ 2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^9*sqrt(c)*abs(a) - 17*(sqrt(a^2*c)*x - s qrt(a^2*c*x^2 - a*c*x))^8*a*c + 64*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x ))^7*c^(3/2)*abs(a) - 140*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^6*a*c^ 2 + 196*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^5*c^(5/2)*abs(a) - 182*( sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^4*a*c^3 + 112*(sqrt(a^2*c)*x - sq rt(a^2*c*x^2 - a*c*x))^3*c^(7/2)*abs(a) - 44*(sqrt(a^2*c)*x - sqrt(a^2*c*x ^2 - a*c*x))^2*a*c^4 + 10*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))*c^(9/2 )*abs(a) - a*c^5))*sgn(x)/(a*c^(7/2)) + sqrt(a^2*c*x^2 - a*c*x)*abs(a)*sgn (x)/(a^2*c^4)
Timed out. \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\int \frac {a\,x+1}{{\left (c-\frac {c}{a\,x}\right )}^{7/2}\,\left (a\,x-1\right )} \,d x \] Input:
int((a*x + 1)/((c - c/(a*x))^(7/2)*(a*x - 1)),x)
Output:
int((a*x + 1)/((c - c/(a*x))^(7/2)*(a*x - 1)), x)
Time = 0.16 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.63 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\frac {\sqrt {c}\, \left (1155 \sqrt {a x -1}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}\right ) a^{3} x^{3}-3465 \sqrt {a x -1}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}\right ) a^{2} x^{2}+3465 \sqrt {a x -1}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}\right ) a x -1155 \sqrt {a x -1}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}\right )+751 \sqrt {a x -1}\, a^{3} x^{3}-2253 \sqrt {a x -1}\, a^{2} x^{2}+2253 \sqrt {a x -1}\, a x -751 \sqrt {a x -1}+105 \sqrt {x}\, \sqrt {a}\, a^{4} x^{4}-1936 \sqrt {x}\, \sqrt {a}\, a^{3} x^{3}+4466 \sqrt {x}\, \sqrt {a}\, a^{2} x^{2}-3850 \sqrt {x}\, \sqrt {a}\, a x +1155 \sqrt {x}\, \sqrt {a}\right )}{105 \sqrt {a x -1}\, a \,c^{4} \left (a^{3} x^{3}-3 a^{2} x^{2}+3 a x -1\right )} \] Input:
int(1/(a*x-1)*(a*x+1)/(c-c/a/x)^(7/2),x)
Output:
(sqrt(c)*(1155*sqrt(a*x - 1)*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a))*a**3*x** 3 - 3465*sqrt(a*x - 1)*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a))*a**2*x**2 + 34 65*sqrt(a*x - 1)*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a))*a*x - 1155*sqrt(a*x - 1)*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a)) + 751*sqrt(a*x - 1)*a**3*x**3 - 2253*sqrt(a*x - 1)*a**2*x**2 + 2253*sqrt(a*x - 1)*a*x - 751*sqrt(a*x - 1) + 105*sqrt(x)*sqrt(a)*a**4*x**4 - 1936*sqrt(x)*sqrt(a)*a**3*x**3 + 4466*sq rt(x)*sqrt(a)*a**2*x**2 - 3850*sqrt(x)*sqrt(a)*a*x + 1155*sqrt(x)*sqrt(a)) )/(105*sqrt(a*x - 1)*a*c**4*(a**3*x**3 - 3*a**2*x**2 + 3*a*x - 1))