Integrand size = 24, antiderivative size = 117 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} x}{c \sqrt {c-\frac {c}{a x}}}-\frac {2 \sqrt {c-\frac {c}{a x}} x}{c^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {3 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a c^{3/2}} \] Output:
3*(1-1/a^2/x^2)^(1/2)*x/c/(c-c/a/x)^(1/2)-2*(c-c/a/x)^(1/2)*x/c^2/(1-1/a^2 /x^2)^(1/2)-3*arctanh(c^(1/2)*(1-1/a^2/x^2)^(1/2)/(c-c/a/x)^(1/2))/a/c^(3/ 2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.06 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.55 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\frac {2 \left (1-\frac {1}{a x}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},2,\frac {1}{2},1+\frac {1}{a x}\right )}{a \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{3/2}} \] Input:
Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^(3/2)),x]
Output:
(2*(1 - 1/(a*x))^(3/2)*Hypergeometric2F1[-1/2, 2, 1/2, 1 + 1/(a*x)])/(a*Sq rt[1 + 1/(a*x)]*(c - c/(a*x))^(3/2))
Time = 0.65 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6731, 578, 579, 573, 219}
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.
| \(\displaystyle \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6731 |
| \(\displaystyle -\frac {\int \frac {\left (c-\frac {c}{a x}\right )^{3/2} x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{c^3}\) |
| \(\Big \downarrow \) 578 |
| \(\displaystyle -\frac {3 c \int \frac {\sqrt {c-\frac {c}{a x}} x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {2 c x \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a^2 x^2}}}}{c^3}\) |
| \(\Big \downarrow \) 579 |
| \(\displaystyle -\frac {3 c \left (-\frac {\int \frac {\sqrt {c-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{2 a}-\frac {c x \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )+\frac {2 c x \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a^2 x^2}}}}{c^3}\) |
| \(\Big \downarrow \) 573 |
| \(\displaystyle -\frac {3 c \left (\frac {c \int \frac {1}{1-\frac {c}{x^2}}d\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}}{a}-\frac {c x \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )+\frac {2 c x \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a^2 x^2}}}}{c^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {3 c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a}-\frac {c x \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )+\frac {2 c x \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a^2 x^2}}}}{c^3}\) |
Input:
Int[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^(3/2)),x]
Output:
-(((2*c*Sqrt[c - c/(a*x)]*x)/Sqrt[1 - 1/(a^2*x^2)] + 3*c*(-((c*Sqrt[1 - 1/ (a^2*x^2)]*x)/Sqrt[c - c/(a*x)]) + (Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a ^2*x^2)])/Sqrt[c - c/(a*x)]])/a))/c^3)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[Sqrt[(c_) + (d_.)*(x_)]/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp[-2*c Subst[Int[1/(a - c*x^2), x], x, Sqrt[a + b*x^2]/Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && 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)*(e*x)^(m + 1)*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1) /(a*e*(p + 1))), x] + Simp[c*((m - n + 2)/(a*(p + 1))) Int[(e*x)^m*(c + d *x)^(n - 1)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + p, 0] && LtQ[p, -1] && RationalQ[m]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-d^2)*(e*x)^(m + 1)*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*c*e*(m + 1))), x] - Simp[d*((n - m - 2)/(c*e*(m + 1))) Int[(e*x)^(m + 1)*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + p, 0] && LtQ[m, -1] && (IntegerQ[2*p] | | IntegerQ[m])
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> S imp[-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[2*p]
Time = 0.11 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.27
| method | result | size |
| default | \(-\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (-2 a^{\frac {3}{2}} x \sqrt {x \left (a x +1\right )}+3 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a x -6 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+3 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right )\right )}{2 \left (a x -1\right )^{2} \sqrt {a}\, c^{2} \sqrt {x \left (a x +1\right )}}\) | \(149\) |
| risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a c \sqrt {\frac {c \left (a x -1\right )}{a x}}}+\frac {\left (-\frac {3 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{2 a^{2} \sqrt {a^{2} c}}+\frac {2 \sqrt {\left (x +\frac {1}{a}\right )^{2} a^{2} c -\left (x +\frac {1}{a}\right ) a c}}{a^{4} c \left (x +\frac {1}{a}\right )}\right ) a \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) a c x}}{c x \sqrt {\frac {c \left (a x -1\right )}{a x}}}\) | \(181\) |
Input:
int(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/2*((a*x-1)/(a*x+1))^(3/2)/(a*x-1)^2*(a*x+1)*(c*(a*x-1)/a/x)^(1/2)*x/a^( 1/2)/c^2*(-2*a^(3/2)*x*(x*(a*x+1))^(1/2)+3*ln(1/2*(2*(x*(a*x+1))^(1/2)*a^( 1/2)+2*a*x+1)/a^(1/2))*a*x-6*(x*(a*x+1))^(1/2)*a^(1/2)+3*ln(1/2*(2*(x*(a*x +1))^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2)))/(x*(a*x+1))^(1/2)
Time = 0.19 (sec) , antiderivative size = 311, normalized size of antiderivative = 2.66 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (a^{2} x^{2} + 3 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} c^{2} x - a c^{2}\right )}}, \frac {3 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (a^{2} x^{2} + 3 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} c^{2} x - a c^{2}\right )}}\right ] \] Input:
integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^(3/2),x, algorithm="fricas")
Output:
[1/4*(3*(a*x - 1)*sqrt(c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^3*x^3 + 3*a ^2*x^2 + a*x)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(a^2*x^2 + 3*a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^2*c^2*x - a*c^2), 1/2*(3*(a*x - 1)*sqrt(-c)*arctan(2*(a^2* x^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(2*a ^2*c*x^2 - a*c*x - c)) + 2*(a^2*x^2 + 3*a*x)*sqrt((a*x - 1)/(a*x + 1))*sqr t((a*c*x - c)/(a*x)))/(a^2*c^2*x - a*c^2)]
Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(((a*x-1)/(a*x+1))**(3/2)/(c-c/a/x)**(3/2),x)
Output:
Timed out
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^(3/2),x, algorithm="maxima")
Output:
integrate(((a*x - 1)/(a*x + 1))^(3/2)/(c - c/(a*x))^(3/2), x)
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^(3/2),x, algorithm="giac")
Output:
integrate(((a*x - 1)/(a*x + 1))^(3/2)/(c - c/(a*x))^(3/2), x)
Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\int \frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{{\left (c-\frac {c}{a\,x}\right )}^{3/2}} \,d x \] Input:
int(((a*x - 1)/(a*x + 1))^(3/2)/(c - c/(a*x))^(3/2),x)
Output:
int(((a*x - 1)/(a*x + 1))^(3/2)/(c - c/(a*x))^(3/2), x)
Time = 0.15 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.53 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\frac {\sqrt {c}\, \left (-12 \sqrt {a x +1}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}\right )+9 \sqrt {a x +1}+4 \sqrt {x}\, \sqrt {a}\, a x +12 \sqrt {x}\, \sqrt {a}\right )}{4 \sqrt {a x +1}\, a \,c^{2}} \] Input:
int(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^(3/2),x)
Output:
(sqrt(c)*( - 12*sqrt(a*x + 1)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a)) + 9*sqr t(a*x + 1) + 4*sqrt(x)*sqrt(a)*a*x + 12*sqrt(x)*sqrt(a)))/(4*sqrt(a*x + 1) *a*c**2)