Integrand size = 20, antiderivative size = 307 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=-\frac {a^5 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2} x^2}{8 \left (a-\frac {1}{x}\right )^4 (c-a c x)^{7/2}}-\frac {3 a^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x^3}{256 \left (a-\frac {1}{x}\right ) (c-a c x)^{7/2}}-\frac {a^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2} x^3}{128 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{7/2}}+\frac {a^5 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2} x^3}{32 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{7/2}}-\frac {3 a^{5/2} \left (1-\frac {1}{a x}\right )^{7/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{256 \sqrt {2} \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}} \]
-1/8*a^5*(1-1/a/x)^(7/2)*(1+1/a/x)^(5/2)*x^2/(a-1/x)^4/(-a*c*x+c)^(7/2)-1/ 128*a^4*(1-1/a/x)^(7/2)*(1+1/a/x)^(3/2)*x^3/(a-1/x)^2/(-a*c*x+c)^(7/2)+1/3 2*a^5*(1-1/a/x)^(7/2)*(1+1/a/x)^(5/2)*x^3/(a-1/x)^3/(-a*c*x+c)^(7/2)-3/512 *a^(5/2)*(1-1/a/x)^(7/2)*arctanh(2^(1/2)*(1/x)^(1/2)/a^(1/2)/(1+1/a/x)^(1/ 2))/(1/x)^(7/2)/(-a*c*x+c)^(7/2)*2^(1/2)-3/256*a^3*(1-1/a/x)^(7/2)*x^3*(1+ 1/a/x)^(1/2)/(a-1/x)/(-a*c*x+c)^(7/2)
Time = 0.22 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.48 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\frac {\sqrt {1-\frac {1}{a x}} \left (\frac {2 \sqrt {a} \sqrt {1+\frac {1}{a x}} \left (39+79 a x+13 a^2 x^2-3 a^3 x^3\right )}{\sqrt {\frac {1}{x}}}+3 \sqrt {2} (-1+a x)^4 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )\right )}{512 \sqrt {a} c^3 \sqrt {\frac {1}{x}} (-1+a x)^4 \sqrt {c-a c x}} \]
(Sqrt[1 - 1/(a*x)]*((2*Sqrt[a]*Sqrt[1 + 1/(a*x)]*(39 + 79*a*x + 13*a^2*x^2 - 3*a^3*x^3))/Sqrt[x^(-1)] + 3*Sqrt[2]*(-1 + a*x)^4*ArcTanh[(Sqrt[2]*Sqrt [x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(512*Sqrt[a]*c^3*Sqrt[x^(-1)]*(-1 + a*x)^4*Sqrt[c - a*c*x])
Time = 0.34 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.76, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6727, 27, 105, 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.
| \(\displaystyle \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 6727 |
| \(\displaystyle -\frac {\left (1-\frac {1}{a x}\right )^{7/2} \int \frac {a^5 \left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{3/2}}{\left (a-\frac {1}{x}\right )^5}d\frac {1}{x}}{\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^5 \left (1-\frac {1}{a x}\right )^{7/2} \int \frac {\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{3/2}}{\left (a-\frac {1}{x}\right )^5}d\frac {1}{x}}{\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {a^5 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {\left (\frac {1}{x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}{8 \left (a-\frac {1}{x}\right )^4}-\frac {3}{16} \int \frac {\left (1+\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{x}}}{\left (a-\frac {1}{x}\right )^4}d\frac {1}{x}\right )}{\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {a^5 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {\left (\frac {1}{x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}{8 \left (a-\frac {1}{x}\right )^4}-\frac {3}{16} \left (\frac {\sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{5/2}}{6 \left (a-\frac {1}{x}\right )^3}-\frac {1}{12} \int \frac {\left (1+\frac {1}{a x}\right )^{3/2}}{\left (a-\frac {1}{x}\right )^3 \sqrt {\frac {1}{x}}}d\frac {1}{x}\right )\right )}{\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {a^5 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {\left (\frac {1}{x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}{8 \left (a-\frac {1}{x}\right )^4}-\frac {3}{16} \left (\frac {1}{12} \left (-\frac {3 \int \frac {\sqrt {1+\frac {1}{a x}}}{\left (a-\frac {1}{x}\right )^2 \sqrt {\frac {1}{x}}}d\frac {1}{x}}{4 a}-\frac {\sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{3/2}}{2 a \left (a-\frac {1}{x}\right )^2}\right )+\frac {\sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{5/2}}{6 \left (a-\frac {1}{x}\right )^3}\right )\right )}{\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {a^5 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {\left (\frac {1}{x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}{8 \left (a-\frac {1}{x}\right )^4}-\frac {3}{16} \left (\frac {1}{12} \left (-\frac {3 \left (\frac {\int \frac {1}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}}{2 a}+\frac {\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}}{a \left (a-\frac {1}{x}\right )}\right )}{4 a}-\frac {\sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{3/2}}{2 a \left (a-\frac {1}{x}\right )^2}\right )+\frac {\sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{5/2}}{6 \left (a-\frac {1}{x}\right )^3}\right )\right )}{\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {a^5 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {\left (\frac {1}{x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}{8 \left (a-\frac {1}{x}\right )^4}-\frac {3}{16} \left (\frac {1}{12} \left (-\frac {3 \left (\frac {\int \frac {1}{a-\frac {2}{x^2}}d\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{a x}}}}{a}+\frac {\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}}{a \left (a-\frac {1}{x}\right )}\right )}{4 a}-\frac {\sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{3/2}}{2 a \left (a-\frac {1}{x}\right )^2}\right )+\frac {\sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{5/2}}{6 \left (a-\frac {1}{x}\right )^3}\right )\right )}{\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {a^5 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {\left (\frac {1}{x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}{8 \left (a-\frac {1}{x}\right )^4}-\frac {3}{16} \left (\frac {1}{12} \left (-\frac {3 \left (\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{\sqrt {2} a^{3/2}}+\frac {\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}}{a \left (a-\frac {1}{x}\right )}\right )}{4 a}-\frac {\sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{3/2}}{2 a \left (a-\frac {1}{x}\right )^2}\right )+\frac {\sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{5/2}}{6 \left (a-\frac {1}{x}\right )^3}\right )\right )}{\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}}\) |
-((a^5*(1 - 1/(a*x))^(7/2)*(((1 + 1/(a*x))^(5/2)*(x^(-1))^(3/2))/(8*(a - x ^(-1))^4) - (3*(((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]*Sqrt[x^(-1)])/( Sqrt[a]*Sqrt[1 + 1/(a*x)])]/(Sqrt[2]*a^(3/2))))/(4*a))/12))/16))/((x^(-1)) ^(7/2)*(c - a*c*x)^(7/2)))
3.3.51.3.1 Defintions of rubi rules used
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 = 0.44 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.91
| method | result | size |
| default | \(\frac {\sqrt {-c \left (a x -1\right )}\, \left (-3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a^{4} c \,x^{4}+12 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a^{3} c \,x^{3}+6 a^{3} x^{3} \sqrt {-c \left (a x +1\right )}\, \sqrt {c}-18 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a^{2} c \,x^{2}-26 a^{2} x^{2} \sqrt {-c \left (a x +1\right )}\, \sqrt {c}+12 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a c x -158 a x \sqrt {c}\, \sqrt {-c \left (a x +1\right )}-3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c -78 \sqrt {-c \left (a x +1\right )}\, \sqrt {c}\right )}{512 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x -1\right )^{3} \left (a x +1\right ) c^{\frac {9}{2}} \sqrt {-c \left (a x +1\right )}\, a}\) | \(278\) |
1/512*(-c*(a*x-1))^(1/2)*(-3*2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2) /c^(1/2))*a^4*c*x^4+12*2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/ 2))*a^3*c*x^3+6*a^3*x^3*(-c*(a*x+1))^(1/2)*c^(1/2)-18*2^(1/2)*arctan(1/2*( -c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*a^2*c*x^2-26*a^2*x^2*(-c*(a*x+1))^(1/2) *c^(1/2)+12*2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*a*c*x-1 58*a*x*c^(1/2)*(-c*(a*x+1))^(1/2)-3*2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)* 2^(1/2)/c^(1/2))*c-78*(-c*(a*x+1))^(1/2)*c^(1/2))/((a*x-1)/(a*x+1))^(3/2)/ (a*x-1)^3/(a*x+1)/c^(9/2)/(-c*(a*x+1))^(1/2)/a
Time = 0.27 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.46 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\left [-\frac {3 \, \sqrt {2} {\left (a^{5} x^{5} - 5 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - 10 \, a^{2} x^{2} + 5 \, a x - 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 4 \, {\left (3 \, a^{4} x^{4} - 10 \, a^{3} x^{3} - 92 \, a^{2} x^{2} - 118 \, a x - 39\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{1024 \, {\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right )}}, -\frac {3 \, \sqrt {2} {\left (a^{5} x^{5} - 5 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - 10 \, a^{2} x^{2} + 5 \, a x - 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - 2 \, {\left (3 \, a^{4} x^{4} - 10 \, a^{3} x^{3} - 92 \, a^{2} x^{2} - 118 \, a x - 39\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{512 \, {\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right )}}\right ] \]
[-1/1024*(3*sqrt(2)*(a^5*x^5 - 5*a^4*x^4 + 10*a^3*x^3 - 10*a^2*x^2 + 5*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^4*x^4 - 10*a^3*x^3 - 92*a^2*x^2 - 118*a*x - 39)*sqrt(-a*c*x + c)*sqr t((a*x - 1)/(a*x + 1)))/(a^6*c^4*x^5 - 5*a^5*c^4*x^4 + 10*a^4*c^4*x^3 - 10 *a^3*c^4*x^2 + 5*a^2*c^4*x - a*c^4), -1/512*(3*sqrt(2)*(a^5*x^5 - 5*a^4*x^ 4 + 10*a^3*x^3 - 10*a^2*x^2 + 5*a*x - 1)*sqrt(c)*arctan(sqrt(2)*sqrt(-a*c* x + c)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))/(a*c*x - c)) - 2*(3*a^4*x^4 - 10* a^3*x^3 - 92*a^2*x^2 - 118*a*x - 39)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a^6*c^4*x^5 - 5*a^5*c^4*x^4 + 10*a^4*c^4*x^3 - 10*a^3*c^4*x^2 + 5* a^2*c^4*x - a*c^4)]
Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\int { \frac {1}{{\left (-a c x + c\right )}^{\frac {7}{2}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
Time = 0.34 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.42 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\frac {\frac {3 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right )}{c^{\frac {5}{2}}} - \frac {2 \, {\left (3 \, {\left (a c x + c\right )}^{3} \sqrt {-a c x - c} - 22 \, {\left (a c x + c\right )}^{2} \sqrt {-a c x - c} c + 44 \, {\left (-a c x - c\right )}^{\frac {3}{2}} c^{2} + 24 \, \sqrt {-a c x - c} c^{3}\right )}}{{\left (a c x - c\right )}^{4} c^{2}}}{512 \, a {\left | c \right |}} \]
1/512*(3*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-a*c*x - c)/sqrt(c))/c^(5/2) - 2* (3*(a*c*x + c)^3*sqrt(-a*c*x - c) - 22*(a*c*x + c)^2*sqrt(-a*c*x - c)*c + 44*(-a*c*x - c)^(3/2)*c^2 + 24*sqrt(-a*c*x - c)*c^3)/((a*c*x - c)^4*c^2))/ (a*abs(c))
Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\int \frac {1}{{\left (c-a\,c\,x\right )}^{7/2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]