Integrand size = 24, antiderivative size = 116 \[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {8 \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} \sqrt {c-a^2 c x^2} \operatorname {Hypergeometric2F1}\left (3,\frac {3-n}{2},\frac {5-n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a^2 (3-n) \sqrt {1-\frac {1}{a^2 x^2}} x} \] Output:
8*(1-1/a/x)^(3/2-1/2*n)*(1+1/a/x)^(-3/2+1/2*n)*(-a^2*c*x^2+c)^(1/2)*hyperg eom([3, 3/2-1/2*n],[5/2-1/2*n],(a-1/x)/(a+1/x))/a^2/(3-n)/(1-1/a^2/x^2)^(1 /2)/x
Time = 0.87 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.87 \[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=-\frac {c e^{n \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} x \left (a \sqrt {1-\frac {1}{a^2 x^2}} x (n+a x)+2 e^{\coth ^{-1}(a x)} (-1+n) \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )}{2 \sqrt {c-a^2 c x^2}} \] Input:
Integrate[E^(n*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2],x]
Output:
-1/2*(c*E^(n*ArcCoth[a*x])*Sqrt[1 - 1/(a^2*x^2)]*x*(a*Sqrt[1 - 1/(a^2*x^2) ]*x*(n + a*x) + 2*E^ArcCoth[a*x]*(-1 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, E^(2*ArcCoth[a*x])]))/Sqrt[c - a^2*c*x^2]
Time = 0.79 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6746, 6749, 141}
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.
\(\displaystyle \int \sqrt {c-a^2 c x^2} e^{n \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6746 |
\(\displaystyle \frac {\sqrt {c-a^2 c x^2} \int e^{n \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} xdx}{x \sqrt {1-\frac {1}{a^2 x^2}}}\) |
\(\Big \downarrow \) 6749 |
\(\displaystyle -\frac {\sqrt {c-a^2 c x^2} \int \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {n+1}{2}} x^3d\frac {1}{x}}{x \sqrt {1-\frac {1}{a^2 x^2}}}\) |
\(\Big \downarrow \) 141 |
\(\displaystyle \frac {8 \sqrt {c-a^2 c x^2} \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \operatorname {Hypergeometric2F1}\left (3,\frac {3-n}{2},\frac {5-n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a^2 (3-n) x \sqrt {1-\frac {1}{a^2 x^2}}}\) |
Input:
Int[E^(n*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2],x]
Output:
(8*(1 - 1/(a*x))^((3 - n)/2)*(1 + 1/(a*x))^((-3 + n)/2)*Sqrt[c - a^2*c*x^2 ]*Hypergeometric2F1[3, (3 - n)/2, (5 - n)/2, (a - x^(-1))/(a + x^(-1))])/( a^2*(3 - n)*Sqrt[1 - 1/(a^2*x^2)]*x)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^( n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f ))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] || !Su mSimplerQ[p, 1]) && !ILtQ[m, 0]
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]
\[\int {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \sqrt {-a^{2} c \,x^{2}+c}d x\]
Input:
int(exp(n*arccoth(a*x))*(-a^2*c*x^2+c)^(1/2),x)
Output:
int(exp(n*arccoth(a*x))*(-a^2*c*x^2+c)^(1/2),x)
\[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\int { \sqrt {-a^{2} c x^{2} + c} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \] Input:
integrate(exp(n*arccoth(a*x))*(-a^2*c*x^2+c)^(1/2),x, algorithm="fricas")
Output:
integral(sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n), x)
\[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\int \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx \] Input:
integrate(exp(n*acoth(a*x))*(-a**2*c*x**2+c)**(1/2),x)
Output:
Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*exp(n*acoth(a*x)), x)
\[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\int { \sqrt {-a^{2} c x^{2} + c} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \] Input:
integrate(exp(n*arccoth(a*x))*(-a^2*c*x^2+c)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n), x)
Exception generated. \[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(exp(n*arccoth(a*x))*(-a^2*c*x^2+c)^(1/2),x, algorithm="giac")
Output:
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. \[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,\sqrt {c-a^2\,c\,x^2} \,d x \] Input:
int(exp(n*acoth(a*x))*(c - a^2*c*x^2)^(1/2),x)
Output:
int(exp(n*acoth(a*x))*(c - a^2*c*x^2)^(1/2), x)
\[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\sqrt {c}\, \left (\int e^{\mathit {acoth} \left (a x \right ) n} \sqrt {-a^{2} x^{2}+1}d x \right ) \] Input:
int(exp(n*acoth(a*x))*(-a^2*c*x^2+c)^(1/2),x)
Output:
sqrt(c)*int(e**(acoth(a*x)*n)*sqrt( - a**2*x**2 + 1),x)