\(\int \frac {e^{\text {sech}^{-1}(c x)} (d x)^m}{1-c^2 x^2} \, dx\) [88]

Optimal result

Integrand size = 24, antiderivative size = 89 \[ \int \frac {e^{\text {sech}^{-1}(c x)} (d x)^m}{1-c^2 x^2} \, dx=\frac {(d x)^m \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m}{2},\frac {2+m}{2},c^2 x^2\right )}{c m}+\frac {(d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {2+m}{2},c^2 x^2\right )}{c m} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6476, 1972, 126, 371} \[ \int \frac {e^{\text {sech}^{-1}(c x)} (d x)^m}{1-c^2 x^2} \, dx=\frac {\sqrt {\frac {1}{c x+1}} \sqrt {c x+1} (d x)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m}{2},\frac {m+2}{2},c^2 x^2\right )}{c m}+\frac {(d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {m+2}{2},c^2 x^2\right )}{c m} \]

[In]

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

[Out]

((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)

Rule 126

Int[((f_.)*(x_))^(p_.)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> 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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 1972

Int[(u_.)*((c_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_))^(p_), x_Symbol] :> Dist[Simp[(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 6476

Int[(E^ArcSech[(c_.)*(x_)]*((d_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Dist[d/(a*c), Int[(d*x)^(m
- 1)*(Sqrt[1/(1 + c*x)]/Sqrt[1 - c*x]), x], x] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {d \int \frac {(d x)^{-1+m} \sqrt {\frac {1}{1+c x}}}{\sqrt {1-c x}} \, dx}{c}+\frac {d \int \frac {(d x)^{-1+m}}{1-c^2 x^2} \, dx}{c} \\ & = \frac {(d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {2+m}{2},c^2 x^2\right )}{c m}+\frac {\left (d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {(d x)^{-1+m}}{\sqrt {1-c x} \sqrt {1+c x}} \, dx}{c} \\ & = \frac {(d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {2+m}{2},c^2 x^2\right )}{c m}+\frac {\left (d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {(d x)^{-1+m}}{\sqrt {1-c^2 x^2}} \, dx}{c} \\ & = \frac {(d x)^m \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m}{2},\frac {2+m}{2},c^2 x^2\right )}{c m}+\frac {(d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {2+m}{2},c^2 x^2\right )}{c m} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.07 \[ \int \frac {e^{\text {sech}^{-1}(c x)} (d x)^m}{1-c^2 x^2} \, dx=\frac {(d x)^m \left (\frac {\sqrt {\frac {1-c x}{1+c x}} (1+c x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m}{2},1+\frac {m}{2},c^2 x^2\right )}{\sqrt {1-c^2 x^2}}+\operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},1+\frac {m}{2},c^2 x^2\right )\right )}{c m} \]

[In]

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

[Out]

((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)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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]

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

[In]

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

[Out]

-(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]

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

[In]

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

[Out]

-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]

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

[In]

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

[Out]

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

[In]

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)

[Out]

-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)