\(\int (d x)^m (a+b \text {csch}^{-1}(c x)) \, dx\) [41]

Optimal result

Integrand size = 14, antiderivative size = 67 \[ \int (d x)^m \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {(d x)^{1+m} \left (a+b \text {csch}^{-1}(c x)\right )}{d (1+m)}+\frac {b (d x)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {m}{2},1-\frac {m}{2},-\frac {1}{c^2 x^2}\right )}{c m (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.21 \[ \int (d x)^m \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {x (d x)^m \left ((1+m) \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b c \sqrt {1+\frac {1}{c^2 x^2}} x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},-c^2 x^2\right )}{\sqrt {1+c^2 x^2}}\right )}{(1+m)^2} \] Input:

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

Output:

(x*(d*x)^m*((1 + m)*(a + b*ArcCsch[c*x]) + (b*c*Sqrt[1 + 1/(c^2*x^2)]*x*Hy 
pergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -(c^2*x^2)])/Sqrt[1 + c^2*x^2]) 
)/(1 + m)^2
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6838, 862, 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.

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

\(\Big \downarrow \) 6838

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

\(\Big \downarrow \) 862

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

\(\Big \downarrow \) 278

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

Input:

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

Output:

((d*x)^(1 + m)*(a + b*ArcCsch[c*x]))/(d*(1 + m)) + (b*(d*x)^m*Hypergeometr 
ic2F1[1/2, -1/2*m, 1 - m/2, -(1/(c^2*x^2))])/(c*m*(1 + m))
 

Defintions of rubi rules used

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 862

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-c^ 
(-1))*(c*x)^(m + 1)*(1/x)^(m + 1)   Subst[Int[(a + b/x^n)^p/x^(m + 2), x], 
x, 1/x], x] /; FreeQ[{a, b, c, m, p}, x] && ILtQ[n, 0] &&  !RationalQ[m]
 

rule 6838

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Si 
mp[(d*x)^(m + 1)*((a + b*ArcCsch[c*x])/(d*(m + 1))), x] + Simp[b*(d/(c*(m + 
 1)))   Int[(d*x)^(m - 1)/Sqrt[1 + 1/(c^2*x^2)], x], x] /; FreeQ[{a, b, c, 
d, m}, x] && NeQ[m, -1]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((d*x)^m*(a+b*arccsch(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^2*x^2 + 1) + m + 1), x) - (d^m*x*x^m*log(x) - d^m*x*x^m*log(sqrt(c^2* 
x^2 + 1) + 1))/(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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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