Integrand size = 20, antiderivative size = 77 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx=-\frac {\sqrt {2} \left (1-\frac {1}{a x}\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}\right )}{a \left (\frac {1}{a x}\right )^{3/2} (c-a c x)^{3/2}} \] Output:
-(1-1/a/x)^(3/2)*arctanh(2^(1/2)*(1/a/x)^(1/2)/(1+1/a/x)^(1/2))*2^(1/2)/a/ (1/a/x)^(3/2)/(-a*c*x+c)^(3/2)
Time = 0.10 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx=-\frac {\sqrt {2} \sqrt {a} \left (1-\frac {1}{a x}\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}} \] Input:
Integrate[1/(E^ArcCoth[a*x]*(c - a*c*x)^(3/2)),x]
Output:
-((Sqrt[2]*Sqrt[a]*(1 - 1/(a*x))^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqr t[a]*Sqrt[1 + 1/(a*x)])])/((x^(-1))^(3/2)*(c - a*c*x)^(3/2)))
Time = 0.42 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6727, 27, 104, 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.
\(\displaystyle \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 6727 |
\(\displaystyle -\frac {\left (1-\frac {1}{a x}\right )^{3/2} \int \frac {a}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}}{\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {a \left (1-\frac {1}{a x}\right )^{3/2} \int \frac {1}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}}{\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle -\frac {2 a \left (1-\frac {1}{a x}\right )^{3/2} \int \frac {1}{a-\frac {2}{x^2}}d\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{a x}}}}{\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {\sqrt {2} \sqrt {a} \left (1-\frac {1}{a x}\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}}\) |
Input:
Int[1/(E^ArcCoth[a*x]*(c - a*c*x)^(3/2)),x]
Output:
-((Sqrt[2]*Sqrt[a]*(1 - 1/(a*x))^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqr t[a]*Sqrt[1 + 1/(a*x)])])/((x^(-1))^(3/2)*(c - a*c*x)^(3/2)))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
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[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_), x_Symbol] :> Si mp[(-(1/x)^p)*((c + d*x)^p/(1 + c/(d*x))^p) Subst[Int[((1 + c*(x/d))^p*(( 1 + x/a)^(n/2)/x^(p + 2)))/(1 - x/a)^(n/2), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && !IntegerQ[p]
Time = 0.14 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.01
method | result | size |
default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\left (a x -1\right ) \sqrt {-c \left (a x +1\right )}\, c^{\frac {3}{2}} a}\) | \(78\) |
Input:
int(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
-((a*x-1)/(a*x+1))^(1/2)*(a*x+1)*(-c*(a*x-1))^(1/2)*2^(1/2)*arctan(1/2*(-c *(a*x+1))^(1/2)*2^(1/2)/c^(1/2))/(a*x-1)/(-c*(a*x+1))^(1/2)/c^(3/2)/a
Time = 0.10 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.91 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx=\left [\frac {\sqrt {2} \sqrt {-\frac {1}{c}} \log \left (-\frac {a^{2} x^{2} + 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {-\frac {1}{c}} + 2 \, a x - 3}{a^{2} x^{2} - 2 \, a x + 1}\right )}{2 \, a c}, -\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, {\left (a x - 1\right )} \sqrt {c}}\right )}{a c^{\frac {3}{2}}}\right ] \] Input:
integrate(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(3/2),x, algorithm="fricas")
Output:
[1/2*sqrt(2)*sqrt(-1/c)*log(-(a^2*x^2 + 2*sqrt(2)*sqrt(-a*c*x + c)*(a*x + 1)*sqrt((a*x - 1)/(a*x + 1))*sqrt(-1/c) + 2*a*x - 3)/(a^2*x^2 - 2*a*x + 1) )/(a*c), -sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*(a*x + 1)*sqrt((a*x - 1)/(a*x + 1))/((a*x - 1)*sqrt(c)))/(a*c^(3/2))]
\[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx=\int \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{\left (- c \left (a x - 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(((a*x-1)/(a*x+1))**(1/2)/(-a*c*x+c)**(3/2),x)
Output:
Integral(sqrt((a*x - 1)/(a*x + 1))/(-c*(a*x - 1))**(3/2), x)
\[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx=\int { \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{{\left (-a c x + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(3/2),x, algorithm="maxima")
Output:
integrate(sqrt((a*x - 1)/(a*x + 1))/(-a*c*x + c)^(3/2), x)
Time = 0.12 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.83 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx=\frac {{\left (\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right )}{a \sqrt {c}} - \frac {\sqrt {2} \arctan \left (\frac {\sqrt {-c}}{\sqrt {c}}\right )}{a \sqrt {c}}\right )} {\left | c \right |} \mathrm {sgn}\left (a x + 1\right )}{c^{2}} \] Input:
integrate(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(3/2),x, algorithm="giac")
Output:
(sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-a*c*x - c)/sqrt(c))/(a*sqrt(c)) - sqrt(2 )*arctan(sqrt(-c)/sqrt(c))/(a*sqrt(c)))*abs(c)*sgn(a*x + 1)/c^2
Timed out. \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx=\int \frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{{\left (c-a\,c\,x\right )}^{3/2}} \,d x \] Input:
int(((a*x - 1)/(a*x + 1))^(1/2)/(c - a*c*x)^(3/2),x)
Output:
int(((a*x - 1)/(a*x + 1))^(1/2)/(c - a*c*x)^(3/2), x)
Time = 0.14 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.39 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx=-\frac {\sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )\right ) i}{a \,c^{2}} \] Input:
int(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(3/2),x)
Output:
( - sqrt(c)*sqrt(2)*log(tan(asin(sqrt( - a*x + 1)/sqrt(2))/2))*i)/(a*c**2)