Integrand size = 20, antiderivative size = 121 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\frac {a \left (1-\frac {1}{a x}\right )^{5/2} x^2}{\sqrt {1+\frac {1}{a x}} (c-a c x)^{5/2}}-\frac {\left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}\right )}{\sqrt {2} a \left (\frac {1}{a x}\right )^{5/2} (c-a c x)^{5/2}} \] Output:
a*(1-1/a/x)^(5/2)*x^2/(1+1/a/x)^(1/2)/(-a*c*x+c)^(5/2)-1/2*(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 = 0.09 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.01 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\frac {\sqrt {1-\frac {1}{a x}} \left (2 \sqrt {\frac {1}{x}}-\sqrt {2} \sqrt {a} \sqrt {1+\frac {1}{a x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )\right )}{2 a c^2 \sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}} \sqrt {c-a c x}} \] Input:
Integrate[1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^(5/2)),x]
Output:
(Sqrt[1 - 1/(a*x)]*(2*Sqrt[x^(-1)] - Sqrt[2]*Sqrt[a]*Sqrt[1 + 1/(a*x)]*Arc Tanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(2*a*c^2*Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)]*Sqrt[c - a*c*x])
Time = 0.45 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.83, 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.
\(\displaystyle \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 6727 |
\(\displaystyle -\frac {\left (1-\frac {1}{a x}\right )^{5/2} \int \frac {a \sqrt {\frac {1}{x}}}{\left (a-\frac {1}{x}\right ) \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}}{\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {a \left (1-\frac {1}{a x}\right )^{5/2} \int \frac {\sqrt {\frac {1}{x}}}{\left (a-\frac {1}{x}\right ) \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}}{\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle -\frac {a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{2} a \int \frac {1}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}-\frac {\sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{a x}+1}}\right )}{\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle -\frac {a \left (1-\frac {1}{a x}\right )^{5/2} \left (a \int \frac {1}{a-\frac {2}{x^2}}d\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{a x}}}-\frac {\sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{a x}+1}}\right )}{\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{\sqrt {2}}-\frac {\sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{a x}+1}}\right )}{\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\) |
Input:
Int[1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^(5/2)),x]
Output:
-((a*(1 - 1/(a*x))^(5/2)*(-(Sqrt[x^(-1)]/Sqrt[1 + 1/(a*x)]) + (Sqrt[a]*Arc Tanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/Sqrt[2]))/((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 = 85, normalized size of antiderivative = 0.70
method | result | size |
default | \(-\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (\arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, \sqrt {-c \left (a x +1\right )}+2 \sqrt {c}\right )}{2 \left (a x -1\right )^{2} c^{\frac {7}{2}} a}\) | \(85\) |
Input:
int(((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c)^(5/2),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))^(1/2)/c^(7/2)* (arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*2^(1/2)*(-c*(a*x+1))^(1/2) +2*c^(1/2))/a
Time = 0.10 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.99 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\left [-\frac {\sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 4 \, \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{4 \, {\left (a^{2} c^{3} x - a c^{3}\right )}}, \frac {\sqrt {2} {\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, {\left (a c x - c\right )}}\right ) - 2 \, \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, {\left (a^{2} c^{3} x - a c^{3}\right )}}\right ] \] Input:
integrate(((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c)^(5/2),x, algorithm="fricas")
Output:
[-1/4*(sqrt(2)*(a*x - 1)*sqrt(-c)*log(-(a^2*c*x^2 + 2*a*c*x + 2*sqrt(2)*sq rt(-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)*sqrt((a*x - 1)/(a*x + 1)))/(a^2*c^3*x - a*c^3), 1/2*(sqrt(2)*(a*x - 1)*sqrt(c)*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*(a*x + 1)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))/(a*c*x - c)) - 2*sqrt(-a*c *x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a^2*c^3*x - a*c^3)]
Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(((a*x-1)/(a*x+1))**(3/2)/(-a*c*x+c)**(5/2),x)
Output:
Timed out
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (-a c x + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c)^(5/2),x, algorithm="maxima")
Output:
integrate(((a*x - 1)/(a*x + 1))^(3/2)/(-a*c*x + c)^(5/2), x)
Exception generated. \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(((a*x-1)/(a*x+1))^(3/2)/(-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. \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\int \frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{{\left (c-a\,c\,x\right )}^{5/2}} \,d x \] Input:
int(((a*x - 1)/(a*x + 1))^(3/2)/(c - a*c*x)^(5/2),x)
Output:
int(((a*x - 1)/(a*x + 1))^(3/2)/(c - a*c*x)^(5/2), x)
Time = 0.14 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.47 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\frac {\sqrt {c}\, i \left (\sqrt {a x +1}\, \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )\right )-\sqrt {a x +1}\, \sqrt {2}+2\right )}{2 \sqrt {a x +1}\, a \,c^{3}} \] Input:
int(((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c)^(5/2),x)
Output:
(sqrt(c)*i*(sqrt(a*x + 1)*sqrt(2)*log(tan(asin(sqrt( - a*x + 1)/sqrt(2))/2 )) - sqrt(a*x + 1)*sqrt(2) + 2))/(2*sqrt(a*x + 1)*a*c**3)