\(\int \frac {e^{\text {sech}^{-1}(c x)} (d x)^m}{1-c^2 x^2} \, dx\) [71]

Optimal result

Integrand size = 24, antiderivative size = 104 \[ \int \frac {e^{\text {sech}^{-1}(c x)} (d x)^m}{1-c^2 x^2} \, dx=\frac {(d x)^m \sqrt {\frac {1}{1+c x}} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m}{2},\frac {2+m}{2},c^2 x^2\right )}{c m \sqrt {1-c x}}+\frac {(d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {2+m}{2},c^2 x^2\right )}{c m} \] Output:

(d*x)^m*(1/(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)*hypergeom([1/2, 1/2*m],[1+1/2 
*m],c^2*x^2)/c/m/(-c*x+1)^(1/2)+(d*x)^m*hypergeom([1, 1/2*m],[1+1/2*m],c^2 
*x^2)/c/m
 

Mathematica [A] (verified)

Time = 0.83 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.91 \[ \int \frac {e^{\text {sech}^{-1}(c x)} (d x)^m}{1-c^2 x^2} \, dx=\frac {(d x)^m \left (\frac {\sqrt {\frac {1-c x}{1+c x}} (1+c x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m}{2},1+\frac {m}{2},c^2 x^2\right )}{\sqrt {1-c^2 x^2}}+\operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},1+\frac {m}{2},c^2 x^2\right )\right )}{c m} \] Input:

Integrate[(E^ArcSech[c*x]*(d*x)^m)/(1 - c^2*x^2),x]
 

Output:

((d*x)^m*((Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*Hypergeometric2F1[1/2, m/2, 
 1 + m/2, c^2*x^2])/Sqrt[1 - c^2*x^2] + Hypergeometric2F1[1, m/2, 1 + m/2, 
 c^2*x^2]))/(c*m)
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.86, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6895, 278, 2044, 135, 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.

\(\displaystyle \int \frac {e^{\text {sech}^{-1}(c x)} (d x)^m}{1-c^2 x^2} \, dx\)

\(\Big \downarrow \) 6895

\(\displaystyle \frac {d \int \frac {(d x)^{m-1}}{1-c^2 x^2}dx}{c}+\frac {d \int \frac {(d x)^{m-1} \sqrt {\frac {1}{c x+1}}}{\sqrt {1-c x}}dx}{c}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {d \int \frac {(d x)^{m-1} \sqrt {\frac {1}{c x+1}}}{\sqrt {1-c x}}dx}{c}+\frac {(d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {m+2}{2},c^2 x^2\right )}{c m}\)

\(\Big \downarrow \) 2044

\(\displaystyle \frac {d \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {(d x)^{m-1}}{\sqrt {1-c x} \sqrt {c x+1}}dx}{c}+\frac {(d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {m+2}{2},c^2 x^2\right )}{c m}\)

\(\Big \downarrow \) 135

\(\displaystyle \frac {d \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {(d x)^{m-1}}{\sqrt {1-c^2 x^2}}dx}{c}+\frac {(d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {m+2}{2},c^2 x^2\right )}{c m}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {\sqrt {\frac {1}{c x+1}} \sqrt {c x+1} (d x)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m}{2},\frac {m+2}{2},c^2 x^2\right )}{c m}+\frac {(d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {m+2}{2},c^2 x^2\right )}{c m}\)

Input:

Int[(E^ArcSech[c*x]*(d*x)^m)/(1 - c^2*x^2),x]
 

Output:

((d*x)^m*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Hypergeometric2F1[1/2, m/2, (2 
 + m)/2, c^2*x^2])/(c*m) + ((d*x)^m*Hypergeometric2F1[1, m/2, (2 + m)/2, c 
^2*x^2])/(c*m)
 

Defintions of rubi rules used

rule 135

Int[((f_.)*(x_))^(p_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), 
x_] :> Int[(a*c + b*d*x^2)^m*(f*x)^p, x] /; FreeQ[{a, b, c, d, f, m, n, p}, 
 x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && GtQ[a, 0] && GtQ[c, 0]
 

rule 278

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])
 

rule 2044

Int[(u_.)*((c_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_))^(p_), x_Symbol] :> Simp[S 
imp[(c*(a + b*x^n)^q)^p/(a + b*x^n)^(p*q)]   Int[u*(a + b*x^n)^(p*q), x], x 
] /; FreeQ[{a, b, c, n, p, q}, x] && GeQ[a, 0]
 

rule 6895

Int[(E^ArcSech[(c_.)*(x_)]*((d_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^2), x_Sym 
bol] :> Simp[d/(a*c)   Int[(d*x)^(m - 1)*(Sqrt[1/(1 + c*x)]/Sqrt[1 - c*x]), 
 x], x] + Simp[d/c   Int[(d*x)^(m - 1)/(a + b*x^2), x], x] /; FreeQ[{a, b, 
c, d, m}, x] && EqQ[b + a*c^2, 0]
 

Maple [F]

\[\int \frac {\left (\frac {1}{c x}+\sqrt {\frac {1}{c x}-1}\, \sqrt {\frac {1}{c x}+1}\right ) \left (d x \right )^{m}}{-c^{2} x^{2}+1}d x\]

Input:

int((1/c/x+(1/c/x-1)^(1/2)*(1/c/x+1)^(1/2))*(d*x)^m/(-c^2*x^2+1),x)
 

Output:

int((1/c/x+(1/c/x-1)^(1/2)*(1/c/x+1)^(1/2))*(d*x)^m/(-c^2*x^2+1),x)
 

Fricas [F]

\[ \int \frac {e^{\text {sech}^{-1}(c x)} (d x)^m}{1-c^2 x^2} \, dx=\int { -\frac {\left (d x\right )^{m} {\left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )}}{c^{2} x^{2} - 1} \,d x } \] Input:

integrate((1/c/x+(1/c/x-1)^(1/2)*(1/c/x+1)^(1/2))*(d*x)^m/(-c^2*x^2+1),x, 
algorithm="fricas")
 

Output:

integral(-((d*x)^m*c*x*sqrt((c*x + 1)/(c*x))*sqrt(-(c*x - 1)/(c*x)) + (d*x 
)^m)/(c^3*x^3 - c*x), x)
 

Sympy [F]

\[ \int \frac {e^{\text {sech}^{-1}(c x)} (d x)^m}{1-c^2 x^2} \, dx=- \frac {\int \frac {\left (d x\right )^{m}}{c^{2} x^{3} - x}\, dx + \int \frac {c x \left (d x\right )^{m} \sqrt {-1 + \frac {1}{c x}} \sqrt {1 + \frac {1}{c x}}}{c^{2} x^{3} - x}\, dx}{c} \] Input:

integrate((1/c/x+(1/c/x-1)**(1/2)*(1/c/x+1)**(1/2))*(d*x)**m/(-c**2*x**2+1 
),x)
 

Output:

-(Integral((d*x)**m/(c**2*x**3 - x), x) + Integral(c*x*(d*x)**m*sqrt(-1 + 
1/(c*x))*sqrt(1 + 1/(c*x))/(c**2*x**3 - x), x))/c
 

Maxima [F]

\[ \int \frac {e^{\text {sech}^{-1}(c x)} (d x)^m}{1-c^2 x^2} \, dx=\int { -\frac {\left (d x\right )^{m} {\left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )}}{c^{2} x^{2} - 1} \,d x } \] Input:

integrate((1/c/x+(1/c/x-1)^(1/2)*(1/c/x+1)^(1/2))*(d*x)^m/(-c^2*x^2+1),x, 
algorithm="maxima")
 

Output:

-d^m*integrate(sqrt(c*x + 1)*sqrt(-c*x + 1)*x^m/(c^3*x^3 - c*x), x) - d^m* 
integrate(1/2*x^m/(c*x + 1), x) - d^m*integrate(1/2*x^m/(c*x - 1), x) + d^ 
m*x^m/(c*m)
 

Giac [F]

\[ \int \frac {e^{\text {sech}^{-1}(c x)} (d x)^m}{1-c^2 x^2} \, dx=\int { -\frac {\left (d x\right )^{m} {\left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )}}{c^{2} x^{2} - 1} \,d x } \] Input:

integrate((1/c/x+(1/c/x-1)^(1/2)*(1/c/x+1)^(1/2))*(d*x)^m/(-c^2*x^2+1),x, 
algorithm="giac")
 

Output:

integrate(-(d*x)^m*(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c*x))/(c^2*x^ 
2 - 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{\text {sech}^{-1}(c x)} (d x)^m}{1-c^2 x^2} \, dx=-\int \frac {\left (\sqrt {\frac {1}{c\,x}-1}\,\sqrt {\frac {1}{c\,x}+1}+\frac {1}{c\,x}\right )\,{\left (d\,x\right )}^m}{c^2\,x^2-1} \,d x \] Input:

int(-(((1/(c*x) - 1)^(1/2)*(1/(c*x) + 1)^(1/2) + 1/(c*x))*(d*x)^m)/(c^2*x^ 
2 - 1),x)
 

Output:

-int((((1/(c*x) - 1)^(1/2)*(1/(c*x) + 1)^(1/2) + 1/(c*x))*(d*x)^m)/(c^2*x^ 
2 - 1), x)
 

Reduce [F]

\[ \int \frac {e^{\text {sech}^{-1}(c x)} (d x)^m}{1-c^2 x^2} \, dx=-\frac {d^{m} \left (\int \frac {x^{m}}{c^{2} x^{3}-x}d x +\int \frac {x^{m} \sqrt {c x +1}\, \sqrt {-c x +1}}{c^{2} x^{3}-x}d x \right )}{c} \] Input:

int((1/c/x+(1/c/x-1)^(1/2)*(1/c/x+1)^(1/2))*(d*x)^m/(-c^2*x^2+1),x)
 

Output:

( - d**m*(int(x**m/(c**2*x**3 - x),x) + int((x**m*sqrt(c*x + 1)*sqrt( - c* 
x + 1))/(c**2*x**3 - x),x)))/c