Integrand size = 27, antiderivative size = 172 \[ \int e^{2 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\frac {x^{1+m} \sqrt {c-a^2 c x^2}}{2+m}-\frac {c (3+2 m) x^{1+m} \sqrt {1-a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )}{(1+m) (2+m) \sqrt {c-a^2 c x^2}}-\frac {2 a c x^{2+m} \sqrt {1-a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},a^2 x^2\right )}{(2+m) \sqrt {c-a^2 c x^2}} \]
-c*(3+2*m)*x^(1+m)*hypergeom([1/2, 1/2+1/2*m],[3/2+1/2*m],a^2*x^2)*(-a^2*x ^2+1)^(1/2)/(m^2+3*m+2)/(-a^2*c*x^2+c)^(1/2)-2*a*c*x^(2+m)*hypergeom([1/2, 1+1/2*m],[2+1/2*m],a^2*x^2)*(-a^2*x^2+1)^(1/2)/(2+m)/(-a^2*c*x^2+c)^(1/2) +x^(1+m)*(-a^2*c*x^2+c)^(1/2)/(2+m)
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 0.30 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.75 \[ \int e^{2 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\frac {x^{1+m} \left (\frac {2 \sqrt {1-a x} \sqrt {-c (1+a x)} \operatorname {AppellF1}\left (1+m,\frac {1}{2},-\frac {1}{2},2+m,a x,-a x\right )}{\sqrt {-1+a x} \sqrt {1+a x}}+\frac {\sqrt {c-a^2 c x^2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )}{\sqrt {1-a^2 x^2}}\right )}{1+m} \]
(x^(1 + m)*((2*Sqrt[1 - a*x]*Sqrt[-(c*(1 + a*x))]*AppellF1[1 + m, 1/2, -1/ 2, 2 + m, a*x, -(a*x)])/(Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (Sqrt[c - a^2*c*x ^2]*Hypergeometric2F1[-1/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/Sqrt[1 - a^2*x ^2]))/(1 + m)
Time = 0.57 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6717, 6701, 559, 25, 27, 557, 279, 278}
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.
\(\displaystyle \int x^m \sqrt {c-a^2 c x^2} e^{2 \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6717 |
\(\displaystyle -\int e^{2 \text {arctanh}(a x)} x^m \sqrt {c-a^2 c x^2}dx\) |
\(\Big \downarrow \) 6701 |
\(\displaystyle -c \int \frac {x^m (a x+1)^2}{\sqrt {c-a^2 c x^2}}dx\) |
\(\Big \downarrow \) 559 |
\(\displaystyle -c \left (-\frac {\int -\frac {a^2 c x^m (2 m+2 a (m+2) x+3)}{\sqrt {c-a^2 c x^2}}dx}{a^2 c (m+2)}-\frac {x^{m+1} \sqrt {c-a^2 c x^2}}{c (m+2)}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -c \left (\frac {\int \frac {a^2 c x^m (2 m+2 a (m+2) x+3)}{\sqrt {c-a^2 c x^2}}dx}{a^2 c (m+2)}-\frac {x^{m+1} \sqrt {c-a^2 c x^2}}{c (m+2)}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -c \left (\frac {\int \frac {x^m (2 m+2 a (m+2) x+3)}{\sqrt {c-a^2 c x^2}}dx}{m+2}-\frac {x^{m+1} \sqrt {c-a^2 c x^2}}{c (m+2)}\right )\) |
\(\Big \downarrow \) 557 |
\(\displaystyle -c \left (\frac {2 a (m+2) \int \frac {x^{m+1}}{\sqrt {c-a^2 c x^2}}dx+(2 m+3) \int \frac {x^m}{\sqrt {c-a^2 c x^2}}dx}{m+2}-\frac {x^{m+1} \sqrt {c-a^2 c x^2}}{c (m+2)}\right )\) |
\(\Big \downarrow \) 279 |
\(\displaystyle -c \left (\frac {\frac {2 a (m+2) \sqrt {1-a^2 x^2} \int \frac {x^{m+1}}{\sqrt {1-a^2 x^2}}dx}{\sqrt {c-a^2 c x^2}}+\frac {(2 m+3) \sqrt {1-a^2 x^2} \int \frac {x^m}{\sqrt {1-a^2 x^2}}dx}{\sqrt {c-a^2 c x^2}}}{m+2}-\frac {x^{m+1} \sqrt {c-a^2 c x^2}}{c (m+2)}\right )\) |
\(\Big \downarrow \) 278 |
\(\displaystyle -c \left (\frac {\frac {(2 m+3) \sqrt {1-a^2 x^2} x^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},a^2 x^2\right )}{(m+1) \sqrt {c-a^2 c x^2}}+\frac {2 a \sqrt {1-a^2 x^2} x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},a^2 x^2\right )}{\sqrt {c-a^2 c x^2}}}{m+2}-\frac {x^{m+1} \sqrt {c-a^2 c x^2}}{c (m+2)}\right )\) |
-(c*(-((x^(1 + m)*Sqrt[c - a^2*c*x^2])/(c*(2 + m))) + (((3 + 2*m)*x^(1 + m )*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, a^2*x^2]) /((1 + m)*Sqrt[c - a^2*c*x^2]) + (2*a*x^(2 + m)*Sqrt[1 - a^2*x^2]*Hypergeo metric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/Sqrt[c - a^2*c*x^2])/(2 + m )))
3.8.31.3.1 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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p]) Int[(c*x)^m* (1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && !(ILtQ[p, 0] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[c Int[(e*x)^m*(a + b*x^2)^p, x], x] + Simp[d/e Int[(e*x)^( m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d^n*(e*x)^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*e^(n - 1)*( m + n + 2*p + 1))), x] + Simp[1/(b*(m + n + 2*p + 1)) Int[(e*x)^m*(a + b* x^2)^p*ExpandToSum[b*(m + n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1 )*x^n - a*d^n*(m + n - 1)*x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && IGtQ[n, 1] && !IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[c^(n/2) Int[x^m*(c + d*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ [c, 0]) && IGtQ[n/2, 0]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2) Int[ u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
\[\int \frac {\left (a x +1\right ) x^{m} \sqrt {-a^{2} c \,x^{2}+c}}{a x -1}d x\]
\[ \int e^{2 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x + 1\right )} x^{m}}{a x - 1} \,d x } \]
\[ \int e^{2 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\int \frac {x^{m} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}{a x - 1}\, dx \]
\[ \int e^{2 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x + 1\right )} x^{m}}{a x - 1} \,d x } \]
Exception generated. \[ \int e^{2 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\text {Exception raised: TypeError} \]
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
Timed out. \[ \int e^{2 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\int \frac {x^m\,\sqrt {c-a^2\,c\,x^2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \]