Integrand size = 20, antiderivative size = 251 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x^2}{16 \left (a-\frac {1}{x}\right ) (c-a c x)^{5/2}}+\frac {a^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2} x^2}{24 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{5/2}}-\frac {a^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^2}{6 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}+\frac {\left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}\right )}{16 \sqrt {2} a \left (\frac {1}{a x}\right )^{5/2} (c-a c x)^{5/2}} \] Output:
1/16*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)+1/24 *a^3*(1-1/a/x)^(5/2)*(1+1/a/x)^(3/2)*x^2/(a-1/x)^2/(-a*c*x+c)^(5/2)-1/6*a^ 4*(1-1/a/x)^(5/2)*(1+1/a/x)^(5/2)*x^2/(a-1/x)^3/(-a*c*x+c)^(5/2)+1/32*(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 = 0.17 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.57 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=-\frac {\sqrt {1-\frac {1}{a x}} \left (2 \sqrt {a} \sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}} \left (7+22 a x+3 a^2 x^2\right )-\frac {3 \sqrt {2} (-1+a x)^3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{x}\right )}{96 \sqrt {a} c^2 \left (\frac {1}{x}\right )^{3/2} (-1+a x)^3 \sqrt {c-a c x}} \] Input:
Integrate[E^(3*ArcCoth[a*x])/(c - a*c*x)^(5/2),x]
Output:
-1/96*(Sqrt[1 - 1/(a*x)]*(2*Sqrt[a]*Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)]*(7 + 22 *a*x + 3*a^2*x^2) - (3*Sqrt[2]*(-1 + a*x)^3*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)]) /(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/x))/(Sqrt[a]*c^2*(x^(-1))^(3/2)*(-1 + a*x)^ 3*Sqrt[c - a*c*x])
Time = 0.56 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.78, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6727, 27, 105, 105, 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 |
| \(\displaystyle -\frac {\left (1-\frac {1}{a x}\right )^{5/2} \int \frac {a^4 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{x}}}{\left (a-\frac {1}{x}\right )^4}d\frac {1}{x}}{\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^4 \left (1-\frac {1}{a x}\right )^{5/2} \int \frac {\left (1+\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{x}}}{\left (a-\frac {1}{x}\right )^4}d\frac {1}{x}}{\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {a^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {\sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{5/2}}{6 \left (a-\frac {1}{x}\right )^3}-\frac {1}{12} \int \frac {\left (1+\frac {1}{a x}\right )^{3/2}}{\left (a-\frac {1}{x}\right )^3 \sqrt {\frac {1}{x}}}d\frac {1}{x}\right )}{\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {a^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{12} \left (-\frac {3 \int \frac {\sqrt {1+\frac {1}{a x}}}{\left (a-\frac {1}{x}\right )^2 \sqrt {\frac {1}{x}}}d\frac {1}{x}}{4 a}-\frac {\sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{3/2}}{2 a \left (a-\frac {1}{x}\right )^2}\right )+\frac {\sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{5/2}}{6 \left (a-\frac {1}{x}\right )^3}\right )}{\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {a^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{12} \left (-\frac {3 \left (\frac {\int \frac {1}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}}{2 a}+\frac {\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}}{a \left (a-\frac {1}{x}\right )}\right )}{4 a}-\frac {\sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{3/2}}{2 a \left (a-\frac {1}{x}\right )^2}\right )+\frac {\sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{5/2}}{6 \left (a-\frac {1}{x}\right )^3}\right )}{\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {a^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{12} \left (-\frac {3 \left (\frac {\int \frac {1}{a-\frac {2}{x^2}}d\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{a x}}}}{a}+\frac {\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}}{a \left (a-\frac {1}{x}\right )}\right )}{4 a}-\frac {\sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{3/2}}{2 a \left (a-\frac {1}{x}\right )^2}\right )+\frac {\sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{5/2}}{6 \left (a-\frac {1}{x}\right )^3}\right )}{\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {a^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{12} \left (-\frac {3 \left (\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{\sqrt {2} a^{3/2}}+\frac {\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}}{a \left (a-\frac {1}{x}\right )}\right )}{4 a}-\frac {\sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{3/2}}{2 a \left (a-\frac {1}{x}\right )^2}\right )+\frac {\sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{5/2}}{6 \left (a-\frac {1}{x}\right )^3}\right )}{\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\) |
Input:
Int[E^(3*ArcCoth[a*x])/(c - a*c*x)^(5/2),x]
Output:
-((a^4*(1 - 1/(a*x))^(5/2)*(((1 + 1/(a*x))^(5/2)*Sqrt[x^(-1)])/(6*(a - x^( -1))^3) + (-1/2*((1 + 1/(a*x))^(3/2)*Sqrt[x^(-1)])/(a*(a - x^(-1))^2) - (3 *((Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)])/(a*(a - x^(-1))) + ArcTanh[(Sqrt[2]*Sqr t[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]/(Sqrt[2]*a^(3/2))))/(4*a))/12))/(( 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 = 0.15 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.90
| method | result | size |
| default | \(-\frac {\sqrt {-c \left (a x -1\right )}\, \left (3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a^{3} c \,x^{3}-9 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a^{2} c \,x^{2}-6 a^{2} x^{2} \sqrt {-c \left (a x +1\right )}\, \sqrt {c}+9 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a c x -44 a x \sqrt {-c \left (a x +1\right )}\, \sqrt {c}-3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c -14 \sqrt {-c \left (a x +1\right )}\, \sqrt {c}\right )}{96 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x -1\right )^{2} \left (a x +1\right ) c^{\frac {7}{2}} \sqrt {-c \left (a x +1\right )}\, a}\) | \(226\) |
Input:
int(1/((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/96/((a*x-1)/(a*x+1))^(3/2)/(a*x-1)^2/(a*x+1)*(-c*(a*x-1))^(1/2)/c^(7/2) *(3*2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*a^3*c*x^3-9*2^( 1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*a^2*c*x^2-6*a^2*x^2*(- c*(a*x+1))^(1/2)*c^(1/2)+9*2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c ^(1/2))*a*c*x-44*a*x*(-c*(a*x+1))^(1/2)*c^(1/2)-3*2^(1/2)*arctan(1/2*(-c*( a*x+1))^(1/2)*2^(1/2)/c^(1/2))*c-14*(-c*(a*x+1))^(1/2)*c^(1/2))/(-c*(a*x+1 ))^(1/2)/a
Time = 0.10 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.59 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\left [-\frac {3 \, \sqrt {2} {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 4 \, {\left (3 \, a^{3} x^{3} + 25 \, a^{2} x^{2} + 29 \, a x + 7\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{192 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}}, -\frac {3 \, \sqrt {2} {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, {\left (a c x - c\right )}}\right ) - 2 \, {\left (3 \, a^{3} x^{3} + 25 \, a^{2} x^{2} + 29 \, a x + 7\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{96 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}}\right ] \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c)^(5/2),x, algorithm="fricas" )
Output:
[-1/192*(3*sqrt(2)*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*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*(3*a^3*x^3 + 2 5*a^2*x^2 + 29*a*x + 7)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a^5*c ^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^3), -1/96*(3*sq rt(2)*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt(c)*arctan(1/2*sqr t(2)*sqrt(-a*c*x + c)*(a*x + 1)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))/(a*c*x - c)) - 2*(3*a^3*x^3 + 25*a^2*x^2 + 29*a*x + 7)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*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(1/((a*x-1)/(a*x+1))**(3/2)/(-a*c*x+c)**(5/2),x)
Output:
Timed out
\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\int { \frac {1}{{\left (-a c x + c\right )}^{\frac {5}{2}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c)^(5/2),x, algorithm="maxima" )
Output:
integrate(1/((-a*c*x + c)^(5/2)*((a*x - 1)/(a*x + 1))^(3/2)), x)
Time = 0.16 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.42 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\frac {\frac {3 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right )}{c^{\frac {3}{2}}} - \frac {2 \, {\left (3 \, {\left (a c x + c\right )}^{2} \sqrt {-a c x - c} - 16 \, {\left (-a c x - c\right )}^{\frac {3}{2}} c - 12 \, \sqrt {-a c x - c} c^{2}\right )}}{{\left (a c x - c\right )}^{3} c}}{96 \, a {\left | c \right |}} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c)^(5/2),x, algorithm="giac")
Output:
1/96*(3*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-a*c*x - c)/sqrt(c))/c^(3/2) - 2*( 3*(a*c*x + c)^2*sqrt(-a*c*x - c) - 16*(-a*c*x - c)^(3/2)*c - 12*sqrt(-a*c* x - c)*c^2)/((a*c*x - c)^3*c))/(a*abs(c))
Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\int \frac {1}{{\left (c-a\,c\,x\right )}^{5/2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \] Input:
int(1/((c - a*c*x)^(5/2)*((a*x - 1)/(a*x + 1))^(3/2)),x)
Output:
int(1/((c - a*c*x)^(5/2)*((a*x - 1)/(a*x + 1))^(3/2)), x)
Time = 0.15 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.66 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\frac {\sqrt {c}\, i \left (-6 \sqrt {a x +1}\, a^{2} x^{2}-44 \sqrt {a x +1}\, a x -14 \sqrt {a x +1}-3 \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )\right ) a^{3} x^{3}+9 \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )\right ) a^{2} x^{2}-9 \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )\right ) a x +3 \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )\right )\right )}{96 a \,c^{3} \left (a^{3} x^{3}-3 a^{2} x^{2}+3 a x -1\right )} \] Input:
int(1/((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c)^(5/2),x)
Output:
(sqrt(c)*i*( - 6*sqrt(a*x + 1)*a**2*x**2 - 44*sqrt(a*x + 1)*a*x - 14*sqrt( a*x + 1) - 3*sqrt(2)*log(tan(asin(sqrt( - a*x + 1)/sqrt(2))/2))*a**3*x**3 + 9*sqrt(2)*log(tan(asin(sqrt( - a*x + 1)/sqrt(2))/2))*a**2*x**2 - 9*sqrt( 2)*log(tan(asin(sqrt( - a*x + 1)/sqrt(2))/2))*a*x + 3*sqrt(2)*log(tan(asin (sqrt( - a*x + 1)/sqrt(2))/2))))/(96*a*c**3*(a**3*x**3 - 3*a**2*x**2 + 3*a *x - 1))