Integrand size = 12, antiderivative size = 77 \[ \int e^{\coth ^{-1}(a x)} (c x)^m \, dx=\frac {x (c x)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-1-m),\frac {1-m}{2},\frac {1}{a^2 x^2}\right )}{1+m}+\frac {(c x)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {m}{2},1-\frac {m}{2},\frac {1}{a^2 x^2}\right )}{a m} \] Output:
x*(c*x)^m*hypergeom([1/2, -1/2-1/2*m],[1/2-1/2*m],1/a^2/x^2)/(1+m)+(c*x)^m *hypergeom([1/2, -1/2*m],[1-1/2*m],1/a^2/x^2)/a/m
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 0.43 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.68 \[ \int e^{\coth ^{-1}(a x)} (c x)^m \, dx=x (c x)^m \left (-\frac {\sqrt {1-\frac {1}{a^2 x^2}} \sqrt {-\frac {1}{a^2}+x^2} \operatorname {AppellF1}\left (m,-\frac {1}{2},\frac {1}{2},1+m,-a x,a x\right )}{m \sqrt {-1+a x} \sqrt {\frac {1+a x}{a^2}} \sqrt {1-a^2 x^2}}+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{2}-\frac {m}{2},\frac {1}{2}-\frac {m}{2},\frac {1}{a^2 x^2}\right )}{1+m}\right ) \] Input:
Integrate[E^ArcCoth[a*x]*(c*x)^m,x]
Output:
x*(c*x)^m*(-((Sqrt[1 - 1/(a^2*x^2)]*Sqrt[-a^(-2) + x^2]*AppellF1[m, -1/2, 1/2, 1 + m, -(a*x), a*x])/(m*Sqrt[-1 + a*x]*Sqrt[(1 + a*x)/a^2]*Sqrt[1 - a ^2*x^2])) + Hypergeometric2F1[-1/2, -1/2 - m/2, 1/2 - m/2, 1/(a^2*x^2)]/(1 + m))
Time = 0.47 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.26, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6722, 27, 557, 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.
\(\displaystyle \int e^{\coth ^{-1}(a x)} (c x)^m \, dx\) |
\(\Big \downarrow \) 6722 |
\(\displaystyle -\left (\frac {1}{x}\right )^m (c x)^m \int \frac {\left (a+\frac {1}{x}\right ) \left (\frac {1}{x}\right )^{-m-2}}{a \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^m (c x)^m \int \frac {\left (a+\frac {1}{x}\right ) \left (\frac {1}{x}\right )^{-m-2}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{a}\) |
\(\Big \downarrow \) 557 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^m (c x)^m \left (a \int \frac {\left (\frac {1}{x}\right )^{-m-2}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\int \frac {\left (\frac {1}{x}\right )^{-m-1}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )}{a}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^m (c x)^m \left (-\frac {a \left (\frac {1}{x}\right )^{-m-1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-m-1),\frac {1-m}{2},\frac {1}{a^2 x^2}\right )}{m+1}-\frac {\left (\frac {1}{x}\right )^{-m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {m}{2},1-\frac {m}{2},\frac {1}{a^2 x^2}\right )}{m}\right )}{a}\) |
Input:
Int[E^ArcCoth[a*x]*(c*x)^m,x]
Output:
-(((x^(-1))^m*(c*x)^m*(-((a*(x^(-1))^(-1 - m)*Hypergeometric2F1[1/2, (-1 - m)/2, (1 - m)/2, 1/(a^2*x^2)])/(1 + m)) - Hypergeometric2F1[1/2, -1/2*m, 1 - m/2, 1/(a^2*x^2)]/(m*(x^(-1))^m)))/a)
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[((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^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_.)*(x_))^(m_), x_Symbol] :> Simp[(-(c *x)^m)*(1/x)^m Subst[Int[(1 + x/a)^((n + 1)/2)/(x^(m + 2)*(1 - x/a)^((n - 1)/2)*Sqrt[1 - x^2/a^2]), x], x, 1/x], x] /; FreeQ[{a, c, m}, x] && Intege rQ[(n - 1)/2] && !IntegerQ[m]
\[\int \frac {\left (x c \right )^{m}}{\sqrt {\frac {a x -1}{a x +1}}}d x\]
Input:
int(1/((a*x-1)/(a*x+1))^(1/2)*(x*c)^m,x)
Output:
int(1/((a*x-1)/(a*x+1))^(1/2)*(x*c)^m,x)
\[ \int e^{\coth ^{-1}(a x)} (c x)^m \, dx=\int { \frac {\left (c x\right )^{m}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(c*x)^m,x, algorithm="fricas")
Output:
integral((a*x + 1)*(c*x)^m*sqrt((a*x - 1)/(a*x + 1))/(a*x - 1), x)
\[ \int e^{\coth ^{-1}(a x)} (c x)^m \, dx=\int \frac {\left (c x\right )^{m}}{\sqrt {\frac {a x - 1}{a x + 1}}}\, dx \] Input:
integrate(1/((a*x-1)/(a*x+1))**(1/2)*(c*x)**m,x)
Output:
Integral((c*x)**m/sqrt((a*x - 1)/(a*x + 1)), x)
\[ \int e^{\coth ^{-1}(a x)} (c x)^m \, dx=\int { \frac {\left (c x\right )^{m}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(c*x)^m,x, algorithm="maxima")
Output:
integrate((c*x)^m/sqrt((a*x - 1)/(a*x + 1)), x)
\[ \int e^{\coth ^{-1}(a x)} (c x)^m \, dx=\int { \frac {\left (c x\right )^{m}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(c*x)^m,x, algorithm="giac")
Output:
integrate((c*x)^m/sqrt((a*x - 1)/(a*x + 1)), x)
Timed out. \[ \int e^{\coth ^{-1}(a x)} (c x)^m \, dx=\int \frac {{\left (c\,x\right )}^m}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \] Input:
int((c*x)^m/((a*x - 1)/(a*x + 1))^(1/2),x)
Output:
int((c*x)^m/((a*x - 1)/(a*x + 1))^(1/2), x)
\[ \int e^{\coth ^{-1}(a x)} (c x)^m \, dx=c^{m} \left (\int \frac {x^{m} \sqrt {a x +1}}{\sqrt {a x -1}}d x \right ) \] Input:
int(1/((a*x-1)/(a*x+1))^(1/2)*(c*x)^m,x)
Output:
c**m*int((x**m*sqrt(a*x + 1))/sqrt(a*x - 1),x)