Integrand size = 24, antiderivative size = 95 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=-\frac {7}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {7}{a c \sqrt {c-\frac {c}{a x}}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}+\frac {7 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{3/2}} \]
-7/3/a/(c-c/a/x)^(3/2)+x/(c-c/a/x)^(3/2)+7*arctanh((c-c/a/x)^(1/2)/c^(1/2) )/a/c^(3/2)-7/a/c/(c-c/a/x)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.58 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\frac {x \left (3 a x-7 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},1-\frac {1}{a x}\right )\right )}{3 c \sqrt {c-\frac {c}{a x}} (-1+a x)} \]
(x*(3*a*x - 7*Hypergeometric2F1[-3/2, 1, -1/2, 1 - 1/(a*x)]))/(3*c*Sqrt[c - c/(a*x)]*(-1 + a*x))
Time = 0.44 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.16, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6717, 6683, 1035, 281, 899, 87, 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.
| \(\displaystyle \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6717 |
| \(\displaystyle -\int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 6683 |
| \(\displaystyle -\int \frac {a x+1}{\left (c-\frac {c}{a x}\right )^{3/2} (1-a x)}dx\) |
| \(\Big \downarrow \) 1035 |
| \(\displaystyle -\int \frac {a+\frac {1}{x}}{\left (\frac {1}{x}-a\right ) \left (c-\frac {c}{a x}\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 281 |
| \(\displaystyle \frac {c \int \frac {a+\frac {1}{x}}{\left (c-\frac {c}{a x}\right )^{5/2}}dx}{a}\) |
| \(\Big \downarrow \) 899 |
| \(\displaystyle -\frac {c \int \frac {\left (a+\frac {1}{x}\right ) x^2}{\left (c-\frac {c}{a x}\right )^{5/2}}d\frac {1}{x}}{a}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {c \left (\frac {7}{2} \int \frac {x}{\left (c-\frac {c}{a x}\right )^{5/2}}d\frac {1}{x}-\frac {a x}{c \left (c-\frac {c}{a x}\right )^{3/2}}\right )}{a}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {c \left (\frac {7}{2} \left (\frac {\int \frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}d\frac {1}{x}}{c}+\frac {2}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}\right )-\frac {a x}{c \left (c-\frac {c}{a x}\right )^{3/2}}\right )}{a}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {c \left (\frac {7}{2} \left (\frac {\frac {\int \frac {x}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{c}+\frac {2}{c \sqrt {c-\frac {c}{a x}}}}{c}+\frac {2}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}\right )-\frac {a x}{c \left (c-\frac {c}{a x}\right )^{3/2}}\right )}{a}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {c \left (\frac {7}{2} \left (\frac {\frac {2}{c \sqrt {c-\frac {c}{a x}}}-\frac {2 a \int \frac {1}{a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}}{c^2}}{c}+\frac {2}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}\right )-\frac {a x}{c \left (c-\frac {c}{a x}\right )^{3/2}}\right )}{a}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {c \left (\frac {7}{2} \left (\frac {\frac {2}{c \sqrt {c-\frac {c}{a x}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{c^{3/2}}}{c}+\frac {2}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}\right )-\frac {a x}{c \left (c-\frac {c}{a x}\right )^{3/2}}\right )}{a}\) |
-((c*(-((a*x)/(c*(c - c/(a*x))^(3/2))) + (7*(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))/2))/a)
3.5.52.3.1 Defintions of rubi rules used
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. \(200\) vs. \(2(81)=162\).
Time = 0.54 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.12
| method | result | size |
| risch | \(\frac {a x -1}{a c \sqrt {\frac {c \left (a x -1\right )}{a x}}}+\frac {\left (\frac {7 \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right )}{2 a^{2} \sqrt {a^{2} c}}-\frac {4 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+\left (x -\frac {1}{a}\right ) a c}}{3 a^{5} c \left (x -\frac {1}{a}\right )^{2}}-\frac {22 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+\left (x -\frac {1}{a}\right ) a c}}{3 a^{4} c \left (x -\frac {1}{a}\right )}\right ) a \sqrt {c \left (a x -1\right ) a x}}{c x \sqrt {\frac {c \left (a x -1\right )}{a x}}}\) | \(201\) |
| default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (42 a^{\frac {7}{2}} \sqrt {\left (a x -1\right ) x}\, x^{3}+21 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{3} x^{3}-36 a^{\frac {5}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} x -126 a^{\frac {5}{2}} \sqrt {\left (a x -1\right ) x}\, x^{2}-63 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{2} x^{2}+28 a^{\frac {3}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}}+126 \sqrt {\left (a x -1\right ) x}\, a^{\frac {3}{2}} x +63 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a x -42 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}-21 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right )\right )}{6 \sqrt {\left (a x -1\right ) x}\, c^{2} \sqrt {a}\, \left (a x -1\right )^{3}}\) | \(260\) |
1/a*(a*x-1)/c/(c*(a*x-1)/a/x)^(1/2)+(7/2/a^2*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)-4/3/a^5/c/(x-1/a)^2*(a^2*c*( x-1/a)^2+(x-1/a)*a*c)^(1/2)-22/3/a^4/c/(x-1/a)*(a^2*c*(x-1/a)^2+(x-1/a)*a* c)^(1/2))*a/c/x/(c*(a*x-1)/a/x)^(1/2)*(c*(a*x-1)*a*x)^(1/2)
Time = 0.28 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.51 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\left [\frac {21 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) + 2 \, {\left (3 \, a^{3} x^{3} - 28 \, a^{2} x^{2} + 21 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{6 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}, -\frac {21 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) - {\left (3 \, a^{3} x^{3} - 28 \, a^{2} x^{2} + 21 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}\right ] \]
[1/6*(21*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*log(-2*a*c*x - 2*a*sqrt(c)*x*sqrt(( a*c*x - c)/(a*x)) + c) + 2*(3*a^3*x^3 - 28*a^2*x^2 + 21*a*x)*sqrt((a*c*x - c)/(a*x)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2), -1/3*(21*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - (3*a^3*x^3 - 2 8*a^2*x^2 + 21*a*x)*sqrt((a*c*x - c)/(a*x)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2)]
\[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\int \frac {a x + 1}{\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {3}{2}} \left (a x - 1\right )}\, dx \]
\[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\int { \frac {a x + 1}{{\left (a x - 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (81) = 162\).
Time = 0.41 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.58 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\frac {7 \, \log \left (c^{2} {\left | a \right |} {\left | c \right |}\right ) \mathrm {sgn}\left (x\right )}{10 \, a c^{\frac {3}{2}}} - \frac {7 \, \log \left ({\left | 2 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{5} \sqrt {c} {\left | a \right |} - 9 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{4} a c + 16 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{3} c^{\frac {3}{2}} {\left | a \right |} - 14 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{2} a c^{2} + 6 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} c^{\frac {5}{2}} {\left | a \right |} - a c^{3} \right |}\right ) \mathrm {sgn}\left (x\right )}{10 \, a c^{\frac {3}{2}}} + \frac {\sqrt {a^{2} c x^{2} - a c x} {\left | a \right |} \mathrm {sgn}\left (x\right )}{a^{2} c^{2}} \]
7/10*log(c^2*abs(a)*abs(c))*sgn(x)/(a*c^(3/2)) - 7/10*log(abs(2*(sqrt(a^2* c)*x - sqrt(a^2*c*x^2 - a*c*x))^5*sqrt(c)*abs(a) - 9*(sqrt(a^2*c)*x - sqrt (a^2*c*x^2 - a*c*x))^4*a*c + 16*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^ 3*c^(3/2)*abs(a) - 14*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^2*a*c^2 + 6*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))*c^(5/2)*abs(a) - a*c^3))*sgn(x )/(a*c^(3/2)) + sqrt(a^2*c*x^2 - a*c*x)*abs(a)*sgn(x)/(a^2*c^2)
Timed out. \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\int \frac {a\,x+1}{{\left (c-\frac {c}{a\,x}\right )}^{3/2}\,\left (a\,x-1\right )} \,d x \]