Integrand size = 24, antiderivative size = 295 \[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1+n}{2}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {2 n \sqrt {c-\frac {c}{a^2 x^2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} \operatorname {Hypergeometric2F1}\left (1,\frac {1-n}{2},\frac {3-n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (1-n) \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {2^{\frac {1+n}{2}} \sqrt {c-\frac {c}{a^2 x^2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1-n}{2},\frac {1-n}{2},\frac {3-n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{a (1-n) \sqrt {1-\frac {1}{a^2 x^2}}} \]
(1-1/a/x)^(1/2-1/2*n)*(1+1/a/x)^(1/2+1/2*n)*x*(c-c/a^2/x^2)^(1/2)/(1-1/a^2 /x^2)^(1/2)+2*n*(1-1/a/x)^(1/2-1/2*n)*(1+1/a/x)^(-1/2+1/2*n)*hypergeom([1, 1/2-1/2*n],[3/2-1/2*n],(a-1/x)/(a+1/x))*(c-c/a^2/x^2)^(1/2)/a/(1-n)/(1-1/ a^2/x^2)^(1/2)-2^(1/2+1/2*n)*(1-1/a/x)^(1/2-1/2*n)*hypergeom([1/2-1/2*n, 1 /2-1/2*n],[3/2-1/2*n],1/2*(a-1/x)/a)*(c-c/a^2/x^2)^(1/2)/a/(1-n)/(1-1/a^2/ x^2)^(1/2)
Time = 0.71 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.49 \[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=\frac {a e^{n \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \left (a (1+n) \sqrt {1-\frac {1}{a^2 x^2}} x+2 e^{\coth ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-e^{2 \coth ^{-1}(a x)}\right )+2 e^{\coth ^{-1}(a x)} n \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )}{(1+n) \left (-1+a^2 x^2\right )} \]
(a*E^(n*ArcCoth[a*x])*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a^2*x^2)]*x^2*(a*( 1 + n)*Sqrt[1 - 1/(a^2*x^2)]*x + 2*E^ArcCoth[a*x]*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -E^(2*ArcCoth[a*x])] + 2*E^ArcCoth[a*x]*n*Hypergeometri c2F1[1, (1 + n)/2, (3 + n)/2, E^(2*ArcCoth[a*x])]))/((1 + n)*(-1 + a^2*x^2 ))
Time = 0.49 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.81, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6751, 6748, 140, 79, 107, 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.
| \(\displaystyle \int \sqrt {c-\frac {c}{a^2 x^2}} e^{n \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6751 |
| \(\displaystyle \frac {\sqrt {c-\frac {c}{a^2 x^2}} \int e^{n \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}}dx}{\sqrt {1-\frac {1}{a^2 x^2}}}\) |
| \(\Big \downarrow \) 6748 |
| \(\displaystyle -\frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {n+1}{2}} x^2d\frac {1}{x}}{\sqrt {1-\frac {1}{a^2 x^2}}}\) |
| \(\Big \downarrow \) 140 |
| \(\displaystyle -\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (\int \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}-1} \left (1+\frac {1}{a x}\right )^{\frac {n-1}{2}} x^2d\frac {1}{x}-\frac {\int \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)} \left (1+\frac {1}{a x}\right )^{\frac {n-1}{2}}d\frac {1}{x}}{a^2}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\) |
| \(\Big \downarrow \) 79 |
| \(\displaystyle -\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (\int \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}-1} \left (1+\frac {1}{a x}\right )^{\frac {n-1}{2}} x^2d\frac {1}{x}+\frac {2^{\frac {n+1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1-n}{2},\frac {1-n}{2},\frac {3-n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{a (1-n)}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\) |
| \(\Big \downarrow \) 107 |
| \(\displaystyle -\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {n \int \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)} \left (1+\frac {1}{a x}\right )^{\frac {n-1}{2}} xd\frac {1}{x}}{a}+\frac {2^{\frac {n+1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1-n}{2},\frac {1-n}{2},\frac {3-n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{a (1-n)}+x \left (-\left (\frac {1}{a x}+1\right )^{\frac {n+1}{2}}\right ) \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\) |
| \(\Big \downarrow \) 141 |
| \(\displaystyle -\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (-\frac {2 n \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \operatorname {Hypergeometric2F1}\left (1,\frac {1-n}{2},\frac {3-n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (1-n)}+\frac {2^{\frac {n+1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1-n}{2},\frac {1-n}{2},\frac {3-n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{a (1-n)}+x \left (-\left (\frac {1}{a x}+1\right )^{\frac {n+1}{2}}\right ) \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\) |
-((Sqrt[c - c/(a^2*x^2)]*(-((1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1 + n)/2)*x) - (2*n*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((-1 + n)/2)*Hyper geometric2F1[1, (1 - n)/2, (3 - n)/2, (a - x^(-1))/(a + x^(-1))])/(a*(1 - n)) + (2^((1 + n)/2)*(1 - 1/(a*x))^((1 - n)/2)*Hypergeometric2F1[(1 - n)/2 , (1 - n)/2, (3 - n)/2, (a - x^(-1))/(2*a)])/(a*(1 - n))))/Sqrt[1 - 1/(a^2 *x^2)])
3.10.31.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*d^(m + n)*f^p Int[(a + b*x)^(m - 1)/(c + d*x)^m, x] , x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandToSum[(a + b*x )*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 0] || SumSimplerQ[m, -1] || !(GtQ[n, 0] || SumSimplerQ[n, -1]))
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_.))*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Simp[-c^p Subst[Int[(1 - x/a)^(p - n/2)*((1 + x/a)^(p + n/2)/x^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]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbo l] :> Simp[c^IntPart[p]*((c + d/x^2)^FracPart[p]/(1 - 1/(a^2*x^2))^FracPart [p]) Int[u*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[n/2] && !(IntegerQ[p] || GtQ[c, 0])
\[\int {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \sqrt {c -\frac {c}{a^{2} x^{2}}}d x\]
\[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=\int { \sqrt {c - \frac {c}{a^{2} x^{2}}} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
\[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=\int \sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx \]
\[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=\int { \sqrt {c - \frac {c}{a^{2} x^{2}}} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
Exception generated. \[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=\int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,\sqrt {c-\frac {c}{a^2\,x^2}} \,d x \]