Integrand size = 27, antiderivative size = 330 \[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {a \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {3 a \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac {6 a \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{(3+n) \left (1-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {6 a \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{\left (9-10 n^2+n^4\right ) \left (c-a^2 c x^2\right )^{5/2}} \]
-a*(1-1/a^2/x^2)^(5/2)*(1-1/a/x)^(-3/2-1/2*n)*(1+1/a/x)^(-3/2+1/2*n)*x^5/( 3+n)/(-a^2*c*x^2+c)^(5/2)-3*a*(1-1/a^2/x^2)^(5/2)*(1-1/a/x)^(-1/2-1/2*n)*( 1+1/a/x)^(-3/2+1/2*n)*x^5/(n^2+4*n+3)/(-a^2*c*x^2+c)^(5/2)+6*a*(1-1/a^2/x^ 2)^(5/2)*(1-1/a/x)^(1/2-1/2*n)*(1+1/a/x)^(-3/2+1/2*n)*x^5/(-n^3-3*n^2+n+3) /(-a^2*c*x^2+c)^(5/2)-6*a*(1-1/a^2/x^2)^(5/2)*(1-1/a/x)^(3/2-1/2*n)*(1+1/a /x)^(-3/2+1/2*n)*x^5/(n^4-10*n^2+9)/(-a^2*c*x^2+c)^(5/2)
Time = 0.96 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.33 \[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {e^{n \coth ^{-1}(a x)} \left (3 \left (10-2 n^2-9 a n x+a n^3 x\right )-6 \left (-1+n^2\right ) \cosh \left (2 \coth ^{-1}(a x)\right )+a n \left (-1+n^2\right ) \sqrt {1-\frac {1}{a^2 x^2}} x \cosh \left (3 \coth ^{-1}(a x)\right )\right )}{4 a^4 c^2 \left (9-10 n^2+n^4\right ) \sqrt {c-a^2 c x^2}} \]
-1/4*(E^(n*ArcCoth[a*x])*(3*(10 - 2*n^2 - 9*a*n*x + a*n^3*x) - 6*(-1 + n^2 )*Cosh[2*ArcCoth[a*x]] + a*n*(-1 + n^2)*Sqrt[1 - 1/(a^2*x^2)]*x*Cosh[3*Arc Coth[a*x]]))/(a^4*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2])
Time = 0.62 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.72, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6746, 6749, 55, 55, 55, 48}
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.
| \(\displaystyle \int \frac {x^3 e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6746 |
| \(\displaystyle \frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \int \frac {e^{n \coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^2}dx}{\left (c-a^2 c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 6749 |
| \(\displaystyle -\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \int \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-5)} \left (1+\frac {1}{a x}\right )^{\frac {n-5}{2}}d\frac {1}{x}}{\left (c-a^2 c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {3 \int \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)} \left (1+\frac {1}{a x}\right )^{\frac {n-5}{2}}d\frac {1}{x}}{n+3}+\frac {a \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{n+3}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {3 \left (\frac {2 \int \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)} \left (1+\frac {1}{a x}\right )^{\frac {n-5}{2}}d\frac {1}{x}}{n+1}+\frac {a \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{n+1}\right )}{n+3}+\frac {a \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{n+3}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {3 \left (\frac {2 \left (-\frac {\int \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {n-5}{2}}d\frac {1}{x}}{1-n}-\frac {a \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}}}{1-n}\right )}{n+1}+\frac {a \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{n+1}\right )}{n+3}+\frac {a \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{n+3}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {a \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{n+3}+\frac {3 \left (\frac {a \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{n+1}+\frac {2 \left (\frac {a \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}}}{(1-n) (3-n)}-\frac {a \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}}}{1-n}\right )}{n+1}\right )}{n+3}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\) |
-((((3*((2*(-((a*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((-3 + n)/2))/(1 - n)) + (a*(1 - 1/(a*x))^((3 - n)/2)*(1 + 1/(a*x))^((-3 + n)/2))/((1 - n)* (3 - n))))/(1 + n) + (a*(1 - 1/(a*x))^((-1 - n)/2)*(1 + 1/(a*x))^((-3 + n) /2))/(1 + n)))/(3 + n) + (a*(1 - 1/(a*x))^((-3 - n)/2)*(1 + 1/(a*x))^((-3 + n)/2))/(3 + n))*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(c - a^2*c*x^2)^(5/2))
3.8.57.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[(c + d*x^2)^p/(x^(2*p)*(1 - 1/(a^2*x^2))^p) Int[u*x^(2*p)*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n/2] && !IntegerQ[p]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_)^2)^(p_.)*(x_)^(m_.), x _Symbol] :> Simp[-c^p Subst[Int[(1 - x/a)^(p - n/2)*((1 + x/a)^(p + n/2)/ x^(m + 2)), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && !IntegersQ[2*p, p + n/2] && IntegerQ[m]
Time = 0.54 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.28
| method | result | size |
| gosper | \(-\frac {\left (a x -1\right ) \left (a x +1\right ) \left (a^{3} n^{3} x^{3}-7 a^{3} x^{3} n -3 a^{2} n^{2} x^{2}+9 a^{2} x^{2}+6 a n x -6\right ) {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{a^{4} \left (n^{4}-10 n^{2}+9\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\) | \(93\) |
-(a*x-1)*(a*x+1)*(a^3*n^3*x^3-7*a^3*n*x^3-3*a^2*n^2*x^2+9*a^2*x^2+6*a*n*x- 6)*exp(n*arccoth(a*x))/a^4/(n^4-10*n^2+9)/(-a^2*c*x^2+c)^(5/2)
Time = 0.27 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.53 \[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {\sqrt {-a^{2} c x^{2} + c} {\left ({\left (a^{3} n^{3} - 7 \, a^{3} n\right )} x^{3} + 6 \, a n x - 3 \, {\left (a^{2} n^{2} - 3 \, a^{2}\right )} x^{2} - 6\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{4} c^{3} n^{4} - 10 \, a^{4} c^{3} n^{2} + 9 \, a^{4} c^{3} + {\left (a^{8} c^{3} n^{4} - 10 \, a^{8} c^{3} n^{2} + 9 \, a^{8} c^{3}\right )} x^{4} - 2 \, {\left (a^{6} c^{3} n^{4} - 10 \, a^{6} c^{3} n^{2} + 9 \, a^{6} c^{3}\right )} x^{2}} \]
sqrt(-a^2*c*x^2 + c)*((a^3*n^3 - 7*a^3*n)*x^3 + 6*a*n*x - 3*(a^2*n^2 - 3*a ^2)*x^2 - 6)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^4*c^3*n^4 - 10*a^4*c^3*n^2 + 9*a^4*c^3 + (a^8*c^3*n^4 - 10*a^8*c^3*n^2 + 9*a^8*c^3)*x^4 - 2*(a^6*c^3*n ^4 - 10*a^6*c^3*n^2 + 9*a^6*c^3)*x^2)
\[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^{3} e^{n \operatorname {acoth}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
Exception generated. \[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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
Time = 4.22 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.53 \[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}\,\left (\frac {6}{a^6\,c^2\,\left (n^4-10\,n^2+9\right )}-\frac {6\,n\,x}{a^5\,c^2\,\left (n^4-10\,n^2+9\right )}+\frac {x^2\,\left (3\,n^2-9\right )}{a^4\,c^2\,\left (n^4-10\,n^2+9\right )}-\frac {n\,x^3\,\left (n^2-7\right )}{a^3\,c^2\,\left (n^4-10\,n^2+9\right )}\right )}{\left (\frac {\sqrt {c-a^2\,c\,x^2}}{a^2}-x^2\,\sqrt {c-a^2\,c\,x^2}\right )\,{\left (\frac {a\,x-1}{a\,x}\right )}^{n/2}} \]