\(\int (d x)^m (a+b \text {sech}^{-1}(c x)) \, dx\) [71]

Optimal result

Integrand size = 14, antiderivative size = 104 \[ \int (d x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {(d x)^{1+m} \left (a+b \text {sech}^{-1}(c x)\right )}{d (1+m)}+\frac {b (d x)^{1+m} \sqrt {\frac {1-c x}{1+c x}} (1+c x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{d (1+m)^2 \sqrt {1-c^2 x^2}} \] Output:

(d*x)^(1+m)*(a+b*arcsech(c*x))/d/(1+m)+b*(d*x)^(1+m)*((-c*x+1)/(c*x+1))^(1 
/2)*(c*x+1)*hypergeom([1/2, 1/2+1/2*m],[3/2+1/2*m],c^2*x^2)/d/(1+m)^2/(-c^ 
2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.93 \[ \int (d x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {x (d x)^m \left ((1+m) (-1+c x) \left (a+b \text {sech}^{-1}(c x)\right )-b \sqrt {\frac {1-c x}{1+c x}} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )\right )}{(1+m)^2 (-1+c x)} \] Input:

Integrate[(d*x)^m*(a + b*ArcSech[c*x]),x]
 

Output:

(x*(d*x)^m*((1 + m)*(-1 + c*x)*(a + b*ArcSech[c*x]) - b*Sqrt[(1 - c*x)/(1 
+ c*x)]*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2 
*x^2]))/((1 + m)^2*(-1 + c*x))
 

Rubi [A] (warning: unable to verify)

Time = 0.26 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.84, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6837, 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.

\(\displaystyle \int (d x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx\)

\(\Big \downarrow \) 6837

\(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {(d x)^m}{\sqrt {1-c x} \sqrt {c x+1}}dx}{m+1}+\frac {(d x)^{m+1} \left (a+b \text {sech}^{-1}(c x)\right )}{d (m+1)}\)

\(\Big \downarrow \) 135

\(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {(d x)^m}{\sqrt {1-c^2 x^2}}dx}{m+1}+\frac {(d x)^{m+1} \left (a+b \text {sech}^{-1}(c x)\right )}{d (m+1)}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {(d x)^{m+1} \left (a+b \text {sech}^{-1}(c x)\right )}{d (m+1)}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} (d x)^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},c^2 x^2\right )}{d (m+1)^2}\)

Input:

Int[(d*x)^m*(a + b*ArcSech[c*x]),x]
 

Output:

((d*x)^(1 + m)*(a + b*ArcSech[c*x]))/(d*(1 + m)) + (b*(d*x)^(1 + m)*Sqrt[( 
1 + c*x)^(-1)]*Sqrt[1 + c*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, 
c^2*x^2])/(d*(1 + m)^2)
 

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 6837

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Si 
mp[(d*x)^(m + 1)*((a + b*ArcSech[c*x])/(d*(m + 1))), x] + Simp[b*(Sqrt[1 + 
c*x]/(m + 1))*Sqrt[1/(1 + c*x)]   Int[(d*x)^m/(Sqrt[1 - c*x]*Sqrt[1 + c*x]) 
, x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
 

Maple [F]

\[\int \left (d x \right )^{m} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )d x\]

Input:

int((d*x)^m*(a+b*arcsech(c*x)),x)
 

Output:

int((d*x)^m*(a+b*arcsech(c*x)),x)
 

Fricas [F]

\[ \int (d x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} \left (d x\right )^{m} \,d x } \] Input:

integrate((d*x)^m*(a+b*arcsech(c*x)),x, algorithm="fricas")
 

Output:

integral((b*arcsech(c*x) + a)*(d*x)^m, x)
 

Sympy [F]

\[ \int (d x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int \left (d x\right )^{m} \left (a + b \operatorname {asech}{\left (c x \right )}\right )\, dx \] Input:

integrate((d*x)**m*(a+b*asech(c*x)),x)
 

Output:

Integral((d*x)**m*(a + b*asech(c*x)), x)
 

Maxima [F]

\[ \int (d x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} \left (d x\right )^{m} \,d x } \] Input:

integrate((d*x)^m*(a+b*arcsech(c*x)),x, algorithm="maxima")
 

Output:

(c^2*d^m*integrate(x^2*x^m/(c^2*(m + 1)*x^2 + (c^2*(m + 1)*x^2 - m - 1)*sq 
rt(c*x + 1)*sqrt(-c*x + 1) - m - 1), x) + (d^m*x*x^m*log(sqrt(c*x + 1)*sqr 
t(-c*x + 1) + 1) - d^m*x*x^m*log(x))/(m + 1) - integrate((c^2*d^m*(m + 1)* 
x^2*log(c) - d^m*(m + 1)*log(c) + d^m)*x^m/(c^2*(m + 1)*x^2 - m - 1), x))* 
b + (d*x)^(m + 1)*a/(d*(m + 1))
 

Giac [F]

\[ \int (d x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} \left (d x\right )^{m} \,d x } \] Input:

integrate((d*x)^m*(a+b*arcsech(c*x)),x, algorithm="giac")
 

Output:

integrate((b*arcsech(c*x) + a)*(d*x)^m, x)
 

Mupad [F(-1)]

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

int((d*x)^m*(a + b*acosh(1/(c*x))),x)
 

Output:

int((d*x)^m*(a + b*acosh(1/(c*x))), x)
 

Reduce [F]

\[ \int (d x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {d^{m} \left (x^{m} a x +\left (\int x^{m} \mathit {asech} \left (c x \right )d x \right ) b m +\left (\int x^{m} \mathit {asech} \left (c x \right )d x \right ) b \right )}{m +1} \] Input:

int((d*x)^m*(a+b*asech(c*x)),x)
 

Output:

(d**m*(x**m*a*x + int(x**m*asech(c*x),x)*b*m + int(x**m*asech(c*x),x)*b))/ 
(m + 1)