Integrand size = 20, antiderivative size = 137 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=-\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x^2}{2 \left (a-\frac {1}{x}\right ) (c-a c x)^{5/2}}+\frac {\left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}\right )}{2 \sqrt {2} a \left (\frac {1}{a x}\right )^{5/2} (c-a c x)^{5/2}} \] Output:
-1/2*a^2*(1-1/a/x)^(5/2)*(1+1/a/x)^(1/2)*x^2/(a-1/x)/(-a*c*x+c)^(5/2)+1/4* (1-1/a/x)^(5/2)*arctanh(2^(1/2)*(1/a/x)^(1/2)/(1+1/a/x)^(1/2))*2^(1/2)/a/( 1/a/x)^(5/2)/(-a*c*x+c)^(5/2)
Time = 0.11 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.85 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\frac {\sqrt {1-\frac {1}{a x}} x \left (-2 \sqrt {a} \sqrt {1+\frac {1}{a x}}+\sqrt {2} \sqrt {\frac {1}{x}} (-1+a x) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )\right )}{4 \sqrt {a} c^2 (-1+a x) \sqrt {c-a c x}} \] Input:
Integrate[1/(E^ArcCoth[a*x]*(c - a*c*x)^(5/2)),x]
Output:
(Sqrt[1 - 1/(a*x)]*x*(-2*Sqrt[a]*Sqrt[1 + 1/(a*x)] + Sqrt[2]*Sqrt[x^(-1)]* (-1 + a*x)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/( 4*Sqrt[a]*c^2*(-1 + a*x)*Sqrt[c - a*c*x])
Time = 0.46 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.85, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6727, 27, 105, 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.
\(\displaystyle \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 6727 |
\(\displaystyle -\frac {\left (1-\frac {1}{a x}\right )^{5/2} \int \frac {a^2 \sqrt {\frac {1}{x}}}{\left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/2} \int \frac {\sqrt {\frac {1}{x}}}{\left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle -\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}-\frac {1}{4} \int \frac {1}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}\right )}{\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle -\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}-\frac {1}{2} \int \frac {1}{a-\frac {2}{x^2}}d\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{a x}}}\right )}{\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{2 \sqrt {2} \sqrt {a}}\right )}{\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\) |
Input:
Int[1/(E^ArcCoth[a*x]*(c - a*c*x)^(5/2)),x]
Output:
-((a^2*(1 - 1/(a*x))^(5/2)*((Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)])/(2*(a - x^(-1 ))) - ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]/(2*Sqrt[ 2]*Sqrt[a])))/((x^(-1))^(5/2)*(c - a*c*x)^(5/2)))
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
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 = 123, normalized size of antiderivative = 0.90
method | result | size |
default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a c x -\sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c -2 \sqrt {-c \left (a x +1\right )}\, \sqrt {c}\right )}{4 c^{\frac {7}{2}} \left (a x -1\right )^{2} \sqrt {-c \left (a x +1\right )}\, a}\) | \(123\) |
Input:
int(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/4*((a*x-1)/(a*x+1))^(1/2)*(a*x+1)*(-c*(a*x-1))^(1/2)/c^(7/2)*(2^(1/2)*a rctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*a*c*x-2^(1/2)*arctan(1/2*(-c *(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*c-2*(-c*(a*x+1))^(1/2)*c^(1/2))/(a*x-1)^2 /(-c*(a*x+1))^(1/2)/a
Time = 0.09 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.09 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\left [-\frac {\sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 4 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{8 \, {\left (a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right )}}, -\frac {\sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, {\left (a c x - c\right )}}\right ) - 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{4 \, {\left (a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right )}}\right ] \] Input:
integrate(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(5/2),x, algorithm="fricas")
Output:
[-1/8*(sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a*c*x + c)*(a*x + 1)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1)) - 3*c)/(a^2*x^2 - 2*a*x + 1)) - 4*sqrt(-a*c*x + c)*(a*x + 1)*sqrt((a*x - 1)/ (a*x + 1)))/(a^3*c^3*x^2 - 2*a^2*c^3*x + a*c^3), -1/4*(sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*(a*x + 1)*sqrt(c)*s qrt((a*x - 1)/(a*x + 1))/(a*c*x - c)) - 2*sqrt(-a*c*x + c)*(a*x + 1)*sqrt( (a*x - 1)/(a*x + 1)))/(a^3*c^3*x^2 - 2*a^2*c^3*x + a*c^3)]
Timed out. \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(((a*x-1)/(a*x+1))**(1/2)/(-a*c*x+c)**(5/2),x)
Output:
Timed out
\[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\int { \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{{\left (-a c x + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(5/2),x, algorithm="maxima")
Output:
integrate(sqrt((a*x - 1)/(a*x + 1))/(-a*c*x + c)^(5/2), x)
Exception generated. \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(5/2),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. \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\int \frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{{\left (c-a\,c\,x\right )}^{5/2}} \,d x \] Input:
int(((a*x - 1)/(a*x + 1))^(1/2)/(c - a*c*x)^(5/2),x)
Output:
int(((a*x - 1)/(a*x + 1))^(1/2)/(c - a*c*x)^(5/2), x)
Time = 0.14 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.51 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\frac {\sqrt {c}\, i \left (-2 \sqrt {a x +1}-\sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )\right ) a x +\sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )\right )\right )}{4 a \,c^{3} \left (a x -1\right )} \] Input:
int(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(5/2),x)
Output:
(sqrt(c)*i*( - 2*sqrt(a*x + 1) - sqrt(2)*log(tan(asin(sqrt( - a*x + 1)/sqr t(2))/2))*a*x + sqrt(2)*log(tan(asin(sqrt( - a*x + 1)/sqrt(2))/2))))/(4*a* c**3*(a*x - 1))