Integrand size = 27, antiderivative size = 407 \[ \int \frac {e^{n \text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {x^3 (1-a x)^{\frac {1}{2} (-3-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{a c^2 (3+n) \sqrt {c-a^2 c x^2}}-\frac {3 (2-n) (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{a^4 c^2 \left (9-n^2\right ) \sqrt {c-a^2 c x^2}}-\frac {3 x (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{a^3 c^2 (3+n) \sqrt {c-a^2 c x^2}}+\frac {3 \left (1+2 n-n^2\right ) (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{a^4 c^2 (3-n) (1+n) (3+n) \sqrt {c-a^2 c x^2}}-\frac {3 \left (1+2 n-n^2\right ) (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{a^4 c^2 \left (9-10 n^2+n^4\right ) \sqrt {c-a^2 c x^2}} \]
x^3*(-a*x+1)^(-3/2-1/2*n)*(a*x+1)^(-3/2+1/2*n)*(-a^2*x^2+1)^(1/2)/a/c^2/(3 +n)/(-a^2*c*x^2+c)^(1/2)-3*(2-n)*(-a*x+1)^(-1/2-1/2*n)*(a*x+1)^(-3/2+1/2*n )*(-a^2*x^2+1)^(1/2)/a^4/c^2/(-n^2+9)/(-a^2*c*x^2+c)^(1/2)-3*x*(-a*x+1)^(- 1/2-1/2*n)*(a*x+1)^(-3/2+1/2*n)*(-a^2*x^2+1)^(1/2)/a^3/c^2/(3+n)/(-a^2*c*x ^2+c)^(1/2)+3*(-n^2+2*n+1)*(-a*x+1)^(-1/2-1/2*n)*(a*x+1)^(-1/2+1/2*n)*(-a^ 2*x^2+1)^(1/2)/a^4/c^2/(-n^3-n^2+9*n+9)/(-a^2*c*x^2+c)^(1/2)-3*(-n^2+2*n+1 )*(-a*x+1)^(1/2-1/2*n)*(a*x+1)^(-1/2+1/2*n)*(-a^2*x^2+1)^(1/2)/a^4/c^2/(n^ 4-10*n^2+9)/(-a^2*c*x^2+c)^(1/2)
Time = 0.15 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.28 \[ \int \frac {e^{n \text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {(1-a x)^{\frac {1}{2} (-3-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2} \left (6-6 a n x+3 a^2 \left (-3+n^2\right ) x^2-a^3 n \left (-7+n^2\right ) x^3\right )}{a^4 c^2 \left (9-10 n^2+n^4\right ) \sqrt {c-a^2 c x^2}} \]
-(((1 - a*x)^((-3 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2]*(6 - 6* a*n*x + 3*a^2*(-3 + n^2)*x^2 - a^3*n*(-7 + n^2)*x^3))/(a^4*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2]))
Time = 0.66 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.68, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6703, 6700, 105, 101, 25, 88, 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.
\(\displaystyle \int \frac {x^3 e^{n \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 6703 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {e^{n \text {arctanh}(a x)} x^3}{\left (1-a^2 x^2\right )^{5/2}}dx}{c^2 \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int x^3 (1-a x)^{\frac {1}{2} (-n-5)} (a x+1)^{\frac {n-5}{2}}dx}{c^2 \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {x^3 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{a (n+3)}-\frac {3 \int x^2 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-5}{2}}dx}{a (n+3)}\right )}{c^2 \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 101 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {x^3 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{a (n+3)}-\frac {3 \left (\frac {\int -(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-5}{2}} (1-a (1-n) x)dx}{a^2}+\frac {x (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{a^2}\right )}{a (n+3)}\right )}{c^2 \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {x^3 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{a (n+3)}-\frac {3 \left (\frac {x (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-3}{2}}}{a^2}-\frac {\int (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-5}{2}} (1-a (1-n) x)dx}{a^2}\right )}{a (n+3)}\right )}{c^2 \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 88 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {x^3 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{a (n+3)}-\frac {3 \left (\frac {x (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-3}{2}}}{a^2}-\frac {\frac {\left (-n^2+2 n+1\right ) \int (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}dx}{3-n}-\frac {(2-n) (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-3}{2}}}{a (3-n)}}{a^2}\right )}{a (n+3)}\right )}{c^2 \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {x^3 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{a (n+3)}-\frac {3 \left (\frac {x (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-3}{2}}}{a^2}-\frac {\frac {\left (-n^2+2 n+1\right ) \left (\frac {\int (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-3}{2}}dx}{n+1}+\frac {(a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{a (n+1)}\right )}{3-n}-\frac {(2-n) (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-3}{2}}}{a (3-n)}}{a^2}\right )}{a (n+3)}\right )}{c^2 \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {x^3 (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{a (n+3)}-\frac {3 \left (\frac {x (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-3}{2}}}{a^2}-\frac {\frac {\left (-n^2+2 n+1\right ) \left (\frac {(1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-1}{2}}}{a (n+1)}-\frac {(1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-1}{2}}}{a (1-n) (n+1)}\right )}{3-n}-\frac {(2-n) (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-3}{2}}}{a (3-n)}}{a^2}\right )}{a (n+3)}\right )}{c^2 \sqrt {c-a^2 c x^2}}\) |
(Sqrt[1 - a^2*x^2]*((x^3*(1 - a*x)^((-3 - n)/2)*(1 + a*x)^((-3 + n)/2))/(a *(3 + n)) - (3*((x*(1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-3 + n)/2))/a^2 - (- (((2 - n)*(1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-3 + n)/2))/(a*(3 - n))) + (( 1 + 2*n - n^2)*(((1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-1 + n)/2))/(a*(1 + n) ) - ((1 - a*x)^((1 - n)/2)*(1 + a*x)^((-1 + n)/2))/(a*(1 - n)*(1 + n))))/( 3 - n))/a^2))/(a*(3 + n))))/(c^2*Sqrt[c - a^2*c*x^2])
3.14.48.3.1 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[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && !RationalQ[p] && SumSimpl erQ[p, 1]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
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[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[c^p Int[x^m*(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_), x_ Symbol] :> Simp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPar t[p]) Int[x^m*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ[c, 0]) && !I ntegerQ[n/2]
Time = 0.25 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.23
method | result | size |
gosper | \(-\frac {\left (a x -1\right ) \left (a x +1\right ) \left (a^{3} n^{3} x^{3}-7 a^{3} x^{3} n -3 a^{2} n^{2} x^{2}+9 a^{2} x^{2}+6 a n x -6\right ) {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{a^{4} \left (n^{4}-10 n^{2}+9\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\) | \(93\) |
-(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*arctanh(a*x))/a^4/(n^4-10*n^2+9)/(-a^2*c*x^2+c)^(5/2)
Time = 0.26 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.43 \[ \int \frac {e^{n \text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {\sqrt {-a^{2} c x^{2} + c} {\left ({\left (a^{3} n^{3} - 7 \, a^{3} n\right )} x^{3} + 6 \, a n x - 3 \, {\left (a^{2} n^{2} - 3 \, a^{2}\right )} x^{2} - 6\right )} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{4} c^{3} n^{4} - 10 \, a^{4} c^{3} n^{2} + 9 \, a^{4} c^{3} + {\left (a^{8} c^{3} n^{4} - 10 \, a^{8} c^{3} n^{2} + 9 \, a^{8} c^{3}\right )} x^{4} - 2 \, {\left (a^{6} c^{3} n^{4} - 10 \, a^{6} c^{3} n^{2} + 9 \, a^{6} c^{3}\right )} x^{2}} \]
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)
\[ \int \frac {e^{n \text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^{3} e^{n \operatorname {atanh}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {e^{n \text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{3} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
Exception generated. \[ \int \frac {e^{n \text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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 = 3.80 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.40 \[ \int \frac {e^{n \text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {{\left (a\,x+1\right )}^{n/2}\,\left (\frac {6}{a^6\,c^2\,\left (n^4-10\,n^2+9\right )}-\frac {6\,n\,x}{a^5\,c^2\,\left (n^4-10\,n^2+9\right )}+\frac {x^2\,\left (3\,n^2-9\right )}{a^4\,c^2\,\left (n^4-10\,n^2+9\right )}-\frac {n\,x^3\,\left (n^2-7\right )}{a^3\,c^2\,\left (n^4-10\,n^2+9\right )}\right )}{{\left (1-a\,x\right )}^{n/2}\,\left (\frac {\sqrt {c-a^2\,c\,x^2}}{a^2}-x^2\,\sqrt {c-a^2\,c\,x^2}\right )} \]