Integrand size = 14, antiderivative size = 60 \[ \int e^{4 \text {arctanh}(a x)} (c x)^m \, dx=\frac {(c x)^{1+m}}{c (1+m)}+\frac {4 (c x)^{1+m}}{c (1-a x)}-\frac {4 (c x)^{1+m} \operatorname {Hypergeometric2F1}(1,1+m,2+m,a x)}{c} \] 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 = 0.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.83 \[ \int e^{4 \text {arctanh}(a x)} (c x)^m \, dx=-\frac {x (c x)^m (5+4 m-a x+4 (1+m) (-1+a x) \operatorname {Hypergeometric2F1}(1,1+m,2+m,a x))}{(1+m) (-1+a x)} \] Input:
Integrate[E^(4*ArcTanh[a*x])*(c*x)^m,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 = 0.24 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {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.
\(\displaystyle \int e^{4 \text {arctanh}(a x)} (c x)^m \, dx\) |
\(\Big \downarrow \) 6676 |
\(\displaystyle \int \frac {(a x+1)^2 (c x)^m}{(1-a x)^2}dx\) |
\(\Big \downarrow \) 100 |
\(\displaystyle \frac {4 (c x)^{m+1}}{c (1-a x)}-\frac {\int \frac {a^2 c (c x)^m (4 m+a x+3)}{1-a x}dx}{a^2 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4 (c x)^{m+1}}{c (1-a x)}-\int \frac {(c x)^m (4 m+a x+3)}{1-a x}dx\) |
\(\Big \downarrow \) 90 |
\(\displaystyle -4 (m+1) \int \frac {(c x)^m}{1-a x}dx+\frac {4 (c x)^{m+1}}{c (1-a x)}+\frac {(c x)^{m+1}}{c (m+1)}\) |
\(\Big \downarrow \) 74 |
\(\displaystyle -\frac {4 (c x)^{m+1} \operatorname {Hypergeometric2F1}(1,m+1,m+2,a x)}{c}+\frac {4 (c x)^{m+1}}{c (1-a x)}+\frac {(c x)^{m+1}}{c (m+1)}\) |
Input:
Int[E^(4*ArcTanh[a*x])*(c*x)^m,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]
Result contains higher order function than in optimal. Order 9 vs. order 5.
Time = 0.20 (sec) , antiderivative size = 511, normalized size of antiderivative = 8.52
method | result | size |
meijerg | \(\frac {\left (-a^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} \left (x c \right )^{m} x^{-m} \left (-\frac {x^{1+m} \left (-a^{2}\right )^{\frac {5}{2}+\frac {m}{2}} \left (2 a^{2} x^{2}-m -3\right )}{\left (-a^{2} x^{2}+1\right ) a^{4} \left (1+m \right )}-\frac {x^{1+m} \left (-a^{2}\right )^{\frac {5}{2}+\frac {m}{2}} \left (3+m \right ) \operatorname {LerchPhi}\left (a^{2} x^{2}, 1, \frac {m}{2}+\frac {1}{2}\right )}{2 a^{4}}\right )}{2}+\frac {2 \left (-a^{2}\right )^{-\frac {m}{2}} \left (x c \right )^{m} x^{-m} \left (-\frac {x^{m} \left (-a^{2}\right )^{\frac {m}{2}} \left (2 a^{2} x^{2}-m -2\right )}{\left (-a^{2} x^{2}+1\right ) m}-\frac {x^{m} \left (-a^{2}\right )^{\frac {m}{2}} \left (2+m \right ) \operatorname {LerchPhi}\left (a^{2} x^{2}, 1, \frac {m}{2}\right )}{2}\right )}{a}-3 \left (-a^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} \left (x c \right )^{m} x^{-m} \left (-\frac {x^{1+m} \left (-a^{2}\right )^{\frac {3}{2}+\frac {m}{2}} \left (-3-m \right )}{\left (3+m \right ) a^{2} \left (-a^{2} x^{2}+1\right )}-\frac {x^{1+m} \left (-a^{2}\right )^{\frac {3}{2}+\frac {m}{2}} \left (1+m \right ) \operatorname {LerchPhi}\left (a^{2} x^{2}, 1, \frac {m}{2}+\frac {1}{2}\right )}{2 a^{2}}\right )-\frac {2 \left (-a^{2}\right )^{-\frac {m}{2}} \left (x c \right )^{m} x^{-m} \left (\frac {x^{m} \left (-a^{2}\right )^{\frac {m}{2}} \left (-2-m \right )}{\left (2+m \right ) \left (-a^{2} x^{2}+1\right )}+\frac {x^{m} \left (-a^{2}\right )^{\frac {m}{2}} m \operatorname {LerchPhi}\left (a^{2} x^{2}, 1, \frac {m}{2}\right )}{2}\right )}{a}+\frac {\left (-a^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} \left (x c \right )^{m} x^{-m} \left (-\frac {2 x^{1+m} \left (-a^{2}\right )^{\frac {m}{2}+\frac {1}{2}} \left (-1-m \right )}{\left (1+m \right ) \left (-2 a^{2} x^{2}+2\right )}+\frac {2 x^{1+m} \left (-a^{2}\right )^{\frac {m}{2}+\frac {1}{2}} \left (-\frac {m^{2}}{4}+\frac {1}{4}\right ) \operatorname {LerchPhi}\left (a^{2} x^{2}, 1, \frac {m}{2}+\frac {1}{2}\right )}{1+m}\right )}{2}\) | \(511\) |
Input:
int((a*x+1)^4/(-a^2*x^2+1)^2*(x*c)^m,x,method=_RETURNVERBOSE)
Output:
1/2*(-a^2)^(-1/2-1/2*m)*(x*c)^m*x^(-m)*(-x^(1+m)*(-a^2)^(5/2+1/2*m)*(2*a^2 *x^2-m-3)/(-a^2*x^2+1)/a^4/(1+m)-1/2*x^(1+m)*(-a^2)^(5/2+1/2*m)*(3+m)/a^4* LerchPhi(a^2*x^2,1,1/2*m+1/2))+2/a*(-a^2)^(-1/2*m)*(x*c)^m*x^(-m)*(-x^m*(- a^2)^(1/2*m)*(2*a^2*x^2-m-2)/(-a^2*x^2+1)/m-1/2*x^m*(-a^2)^(1/2*m)*(2+m)*L erchPhi(a^2*x^2,1,1/2*m))-3*(-a^2)^(-1/2-1/2*m)*(x*c)^m*x^(-m)*(-1/(3+m)*x ^(1+m)*(-a^2)^(3/2+1/2*m)*(-3-m)/a^2/(-a^2*x^2+1)-1/2*x^(1+m)*(-a^2)^(3/2+ 1/2*m)*(1+m)/a^2*LerchPhi(a^2*x^2,1,1/2*m+1/2))-2/a*(-a^2)^(-1/2*m)*(x*c)^ m*x^(-m)*(1/(2+m)*x^m*(-a^2)^(1/2*m)*(-2-m)/(-a^2*x^2+1)+1/2*x^m*(-a^2)^(1 /2*m)*m*LerchPhi(a^2*x^2,1,1/2*m))+1/2*(-a^2)^(-1/2-1/2*m)*(x*c)^m*x^(-m)* (-2/(1+m)*x^(1+m)*(-a^2)^(1/2*m+1/2)*(-1-m)/(-2*a^2*x^2+2)+2/(1+m)*x^(1+m) *(-a^2)^(1/2*m+1/2)*(-1/4*m^2+1/4)*LerchPhi(a^2*x^2,1,1/2*m+1/2))
\[ \int e^{4 \text {arctanh}(a x)} (c x)^m \, dx=\int { \frac {{\left (a x + 1\right )}^{4} \left (c x\right )^{m}}{{\left (a^{2} x^{2} - 1\right )}^{2}} \,d x } \] Input:
integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c*x)^m,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)
\[ \int e^{4 \text {arctanh}(a x)} (c x)^m \, dx=\int \frac {\left (c x\right )^{m} \left (a x + 1\right )^{2}}{\left (a x - 1\right )^{2}}\, dx \] Input:
integrate((a*x+1)**4/(-a**2*x**2+1)**2*(c*x)**m,x)
Output:
Integral((c*x)**m*(a*x + 1)**2/(a*x - 1)**2, x)
\[ \int e^{4 \text {arctanh}(a x)} (c x)^m \, dx=\int { \frac {{\left (a x + 1\right )}^{4} \left (c x\right )^{m}}{{\left (a^{2} x^{2} - 1\right )}^{2}} \,d x } \] Input:
integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c*x)^m,x, algorithm="maxima")
Output:
integrate((a*x + 1)^4*(c*x)^m/(a^2*x^2 - 1)^2, x)
\[ \int e^{4 \text {arctanh}(a x)} (c x)^m \, dx=\int { \frac {{\left (a x + 1\right )}^{4} \left (c x\right )^{m}}{{\left (a^{2} x^{2} - 1\right )}^{2}} \,d x } \] Input:
integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c*x)^m,x, algorithm="giac")
Output:
integrate((a*x + 1)^4*(c*x)^m/(a^2*x^2 - 1)^2, x)
Timed out. \[ \int e^{4 \text {arctanh}(a x)} (c x)^m \, dx=\int \frac {{\left (c\,x\right )}^m\,{\left (a\,x+1\right )}^4}{{\left (a^2\,x^2-1\right )}^2} \,d x \] Input:
int(((c*x)^m*(a*x + 1)^4)/(a^2*x^2 - 1)^2,x)
Output:
int(((c*x)^m*(a*x + 1)^4)/(a^2*x^2 - 1)^2, x)
\[ \int e^{4 \text {arctanh}(a x)} (c x)^m \, dx=\frac {c^{m} \left (x^{m} a^{2} m^{2} x^{2}-x^{m} a^{2} m \,x^{2}+3 x^{m} a \,m^{2} x +x^{m} a m x -4 x^{m} a x +4 x^{m} m^{2}+8 x^{m} m +4 x^{m}+4 \left (\int \frac {x^{m}}{a^{2} m \,x^{3}-a^{2} x^{3}-2 a m \,x^{2}+2 a \,x^{2}+m x -x}d x \right ) a \,m^{4} x +4 \left (\int \frac {x^{m}}{a^{2} m \,x^{3}-a^{2} x^{3}-2 a m \,x^{2}+2 a \,x^{2}+m x -x}d x \right ) a \,m^{3} x -4 \left (\int \frac {x^{m}}{a^{2} m \,x^{3}-a^{2} x^{3}-2 a m \,x^{2}+2 a \,x^{2}+m x -x}d x \right ) a \,m^{2} x -4 \left (\int \frac {x^{m}}{a^{2} m \,x^{3}-a^{2} x^{3}-2 a m \,x^{2}+2 a \,x^{2}+m x -x}d x \right ) a m x -4 \left (\int \frac {x^{m}}{a^{2} m \,x^{3}-a^{2} x^{3}-2 a m \,x^{2}+2 a \,x^{2}+m x -x}d x \right ) m^{4}-4 \left (\int \frac {x^{m}}{a^{2} m \,x^{3}-a^{2} x^{3}-2 a m \,x^{2}+2 a \,x^{2}+m x -x}d x \right ) m^{3}+4 \left (\int \frac {x^{m}}{a^{2} m \,x^{3}-a^{2} x^{3}-2 a m \,x^{2}+2 a \,x^{2}+m x -x}d x \right ) m^{2}+4 \left (\int \frac {x^{m}}{a^{2} m \,x^{3}-a^{2} x^{3}-2 a m \,x^{2}+2 a \,x^{2}+m x -x}d x \right ) m \right )}{a m \left (a \,m^{2} x -a x -m^{2}+1\right )} \] Input:
int((a*x+1)^4/(-a^2*x^2+1)^2*(c*x)^m,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))