Integrand size = 14, antiderivative size = 60 \[ \int e^{-4 \coth ^{-1}(a x)} (c x)^m \, dx=\frac {(c x)^{1+m}}{c (1+m)}+\frac {4 (c x)^{1+m}}{c (1+a x)}-\frac {4 (c x)^{1+m} \operatorname {Hypergeometric2F1}(1,1+m,2+m,-a x)}{c} \] Output:
(c*x)^(1+m)/c/(1+m)+4*(c*x)^(1+m)/c/(a*x+1)-4*(c*x)^(1+m)*hypergeom([1, 1+ m],[2+m],-a*x)/c
Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.85 \[ \int e^{-4 \coth ^{-1}(a x)} (c x)^m \, dx=-\frac {x (c x)^m (-5-4 m-a x+4 (1+m) (1+a x) \operatorname {Hypergeometric2F1}(1,1+m,2+m,-a x))}{(1+m) (1+a x)} \] Input:
Integrate[(c*x)^m/E^(4*ArcCoth[a*x]),x]
Output:
-((x*(c*x)^m*(-5 - 4*m - a*x + 4*(1 + m)*(1 + a*x)*Hypergeometric2F1[1, 1 + m, 2 + m, -(a*x)]))/((1 + m)*(1 + a*x)))
Time = 0.51 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6717, 6676, 100, 27, 90, 74}
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.
\(\displaystyle \int e^{-4 \coth ^{-1}(a x)} (c x)^m \, dx\) |
\(\Big \downarrow \) 6717 |
\(\displaystyle \int e^{-4 \text {arctanh}(a x)} (c x)^mdx\) |
\(\Big \downarrow \) 6676 |
\(\displaystyle \int \frac {(1-a x)^2 (c x)^m}{(a x+1)^2}dx\) |
\(\Big \downarrow \) 100 |
\(\displaystyle \frac {4 (c x)^{m+1}}{c (a x+1)}-\frac {\int \frac {a^2 c (c x)^m (4 m-a x+3)}{a x+1}dx}{a^2 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4 (c x)^{m+1}}{c (a x+1)}-\int \frac {(c x)^m (4 m-a x+3)}{a x+1}dx\) |
\(\Big \downarrow \) 90 |
\(\displaystyle -4 (m+1) \int \frac {(c x)^m}{a x+1}dx+\frac {4 (c x)^{m+1}}{c (a x+1)}+\frac {(c x)^{m+1}}{c (m+1)}\) |
\(\Big \downarrow \) 74 |
\(\displaystyle -\frac {4 (c x)^{m+1} \operatorname {Hypergeometric2F1}(1,m+1,m+2,-a x)}{c}+\frac {4 (c x)^{m+1}}{c (a x+1)}+\frac {(c x)^{m+1}}{c (m+1)}\) |
Input:
Int[(c*x)^m/E^(4*ArcCoth[a*x]),x]
Output:
(c*x)^(1 + m)/(c*(1 + m)) + (4*(c*x)^(1 + m))/(c*(1 + a*x)) - (4*(c*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, -(a*x)])/c
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x )^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] && !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x) ^m*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c, m, n}, x] && !Int egerQ[(n - 1)/2]
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]
Result contains higher order function than in optimal. Order 9 vs. order 5.
Time = 0.33 (sec) , antiderivative size = 206, normalized size of antiderivative = 3.43
method | result | size |
meijerg | \(a^{-1-m} \left (x c \right )^{m} x^{-m} \left (\frac {x^{m} a^{m} \left (a^{2} m \,x^{2}-a m x -2 a x -m^{2}-3 m -2\right )}{\left (1+m \right ) m \left (a x +1\right )}+x^{m} a^{m} \left (2+m \right ) \operatorname {LerchPhi}\left (-a x , 1, m\right )\right )-2 a^{-1-m} \left (x c \right )^{m} x^{-m} \left (\frac {x^{m} a^{m} \left (a x +m +1\right )}{m \left (a x +1\right )}-x^{m} a^{m} \left (1+m \right ) \operatorname {LerchPhi}\left (-a x , 1, m\right )\right )+a^{-1-m} \left (x c \right )^{m} x^{-m} \left (\frac {x^{m} a^{m} \left (-1-m \right )}{\left (1+m \right ) \left (a x +1\right )}+x^{m} a^{m} m \operatorname {LerchPhi}\left (-a x , 1, m\right )\right )\) | \(206\) |
Input:
int((x*c)^m*(a*x-1)^2/(a*x+1)^2,x,method=_RETURNVERBOSE)
Output:
a^(-1-m)*(x*c)^m*x^(-m)*(x^m*a^m*(a^2*m*x^2-a*m*x-2*a*x-m^2-3*m-2)/(1+m)/m /(a*x+1)+x^m*a^m*(2+m)*LerchPhi(-a*x,1,m))-2*a^(-1-m)*(x*c)^m*x^(-m)*(x^m* a^m*(a*x+m+1)/m/(a*x+1)-x^m*a^m*(1+m)*LerchPhi(-a*x,1,m))+a^(-1-m)*(x*c)^m *x^(-m)*(1/(1+m)*x^m*a^m*(-1-m)/(a*x+1)+x^m*a^m*m*LerchPhi(-a*x,1,m))
\[ \int e^{-4 \coth ^{-1}(a x)} (c x)^m \, dx=\int { \frac {{\left (a x - 1\right )}^{2} \left (c x\right )^{m}}{{\left (a x + 1\right )}^{2}} \,d x } \] Input:
integrate((c*x)^m*(a*x-1)^2/(a*x+1)^2,x, algorithm="fricas")
Output:
integral((a^2*x^2 - 2*a*x + 1)*(c*x)^m/(a^2*x^2 + 2*a*x + 1), x)
\[ \int e^{-4 \coth ^{-1}(a x)} (c x)^m \, dx=\int \frac {\left (c x\right )^{m} \left (a x - 1\right )^{2}}{\left (a x + 1\right )^{2}}\, dx \] Input:
integrate((c*x)**m*(a*x-1)**2/(a*x+1)**2,x)
Output:
Integral((c*x)**m*(a*x - 1)**2/(a*x + 1)**2, x)
\[ \int e^{-4 \coth ^{-1}(a x)} (c x)^m \, dx=\int { \frac {{\left (a x - 1\right )}^{2} \left (c x\right )^{m}}{{\left (a x + 1\right )}^{2}} \,d x } \] Input:
integrate((c*x)^m*(a*x-1)^2/(a*x+1)^2,x, algorithm="maxima")
Output:
integrate((a*x - 1)^2*(c*x)^m/(a*x + 1)^2, x)
\[ \int e^{-4 \coth ^{-1}(a x)} (c x)^m \, dx=\int { \frac {{\left (a x - 1\right )}^{2} \left (c x\right )^{m}}{{\left (a x + 1\right )}^{2}} \,d x } \] Input:
integrate((c*x)^m*(a*x-1)^2/(a*x+1)^2,x, algorithm="giac")
Output:
integrate((a*x - 1)^2*(c*x)^m/(a*x + 1)^2, x)
Timed out. \[ \int e^{-4 \coth ^{-1}(a x)} (c x)^m \, dx=\int \frac {{\left (c\,x\right )}^m\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2} \,d x \] Input:
int(((c*x)^m*(a*x - 1)^2)/(a*x + 1)^2,x)
Output:
int(((c*x)^m*(a*x - 1)^2)/(a*x + 1)^2, x)
\[ \int e^{-4 \coth ^{-1}(a x)} (c x)^m \, dx=\frac {c^{m} \left (x^{m} a^{2} m^{2} x^{2}-x^{m} a^{2} m \,x^{2}-3 x^{m} a \,m^{2} x -x^{m} a m x +4 x^{m} a x +4 x^{m} m^{2}+8 x^{m} m +4 x^{m}-4 \left (\int \frac {x^{m}}{a^{2} m \,x^{3}-a^{2} x^{3}+2 a m \,x^{2}-2 a \,x^{2}+m x -x}d x \right ) a \,m^{4} x -4 \left (\int \frac {x^{m}}{a^{2} m \,x^{3}-a^{2} x^{3}+2 a m \,x^{2}-2 a \,x^{2}+m x -x}d x \right ) a \,m^{3} x +4 \left (\int \frac {x^{m}}{a^{2} m \,x^{3}-a^{2} x^{3}+2 a m \,x^{2}-2 a \,x^{2}+m x -x}d x \right ) a \,m^{2} x +4 \left (\int \frac {x^{m}}{a^{2} m \,x^{3}-a^{2} x^{3}+2 a m \,x^{2}-2 a \,x^{2}+m x -x}d x \right ) a m x -4 \left (\int \frac {x^{m}}{a^{2} m \,x^{3}-a^{2} x^{3}+2 a m \,x^{2}-2 a \,x^{2}+m x -x}d x \right ) m^{4}-4 \left (\int \frac {x^{m}}{a^{2} m \,x^{3}-a^{2} x^{3}+2 a m \,x^{2}-2 a \,x^{2}+m x -x}d x \right ) m^{3}+4 \left (\int \frac {x^{m}}{a^{2} m \,x^{3}-a^{2} x^{3}+2 a m \,x^{2}-2 a \,x^{2}+m x -x}d x \right ) m^{2}+4 \left (\int \frac {x^{m}}{a^{2} m \,x^{3}-a^{2} x^{3}+2 a m \,x^{2}-2 a \,x^{2}+m x -x}d x \right ) m \right )}{a m \left (a \,m^{2} x -a x +m^{2}-1\right )} \] Input:
int((c*x)^m*(a*x-1)^2/(a*x+1)^2,x)
Output:
(c**m*(x**m*a**2*m**2*x**2 - x**m*a**2*m*x**2 - 3*x**m*a*m**2*x - x**m*a*m *x + 4*x**m*a*x + 4*x**m*m**2 + 8*x**m*m + 4*x**m - 4*int(x**m/(a**2*m*x** 3 - a**2*x**3 + 2*a*m*x**2 - 2*a*x**2 + m*x - x),x)*a*m**4*x - 4*int(x**m/ (a**2*m*x**3 - a**2*x**3 + 2*a*m*x**2 - 2*a*x**2 + m*x - x),x)*a*m**3*x + 4*int(x**m/(a**2*m*x**3 - a**2*x**3 + 2*a*m*x**2 - 2*a*x**2 + m*x - x),x)* a*m**2*x + 4*int(x**m/(a**2*m*x**3 - a**2*x**3 + 2*a*m*x**2 - 2*a*x**2 + m *x - x),x)*a*m*x - 4*int(x**m/(a**2*m*x**3 - a**2*x**3 + 2*a*m*x**2 - 2*a* x**2 + m*x - x),x)*m**4 - 4*int(x**m/(a**2*m*x**3 - a**2*x**3 + 2*a*m*x**2 - 2*a*x**2 + m*x - x),x)*m**3 + 4*int(x**m/(a**2*m*x**3 - a**2*x**3 + 2*a *m*x**2 - 2*a*x**2 + m*x - x),x)*m**2 + 4*int(x**m/(a**2*m*x**3 - a**2*x** 3 + 2*a*m*x**2 - 2*a*x**2 + m*x - x),x)*m))/(a*m*(a*m**2*x - a*x + m**2 - 1))