\(\int (d x)^m (a+b \text {arccosh}(c x)) \, dx\) [165]

Optimal result

Integrand size = 14, antiderivative size = 93 \[ \int (d x)^m (a+b \text {arccosh}(c x)) \, dx=\frac {(d x)^{1+m} (a+b \text {arccosh}(c x))}{d (1+m)}-\frac {b c (d x)^{2+m} \sqrt {1-c x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{d^2 (1+m) (2+m) \sqrt {-1+c x}} \] Output:

(d*x)^(1+m)*(a+b*arccosh(c*x))/d/(1+m)-b*c*(d*x)^(2+m)*(-c*x+1)^(1/2)*hype 
rgeom([1/2, 1+1/2*m],[2+1/2*m],c^2*x^2)/d^2/(1+m)/(2+m)/(c*x-1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.94 \[ \int (d x)^m (a+b \text {arccosh}(c x)) \, dx=\frac {x (d x)^m \left (a+b \text {arccosh}(c x)-\frac {b c x \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{(2+m) \sqrt {-1+c x} \sqrt {1+c x}}\right )}{1+m} \] Input:

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

Output:

(x*(d*x)^m*(a + b*ArcCosh[c*x] - (b*c*x*Sqrt[1 - c^2*x^2]*Hypergeometric2F 
1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c* 
x])))/(1 + m)
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.14, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6298, 136, 279, 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.

\(\displaystyle \int (d x)^m (a+b \text {arccosh}(c x)) \, dx\)

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 136

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

\(\Big \downarrow \) 279

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

\(\Big \downarrow \) 278

\(\displaystyle \frac {(d x)^{m+1} (a+b \text {arccosh}(c x))}{d (m+1)}-\frac {b c \sqrt {1-c^2 x^2} (d x)^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{d^2 (m+1) (m+2) \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

((d*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(d*(1 + m)) - (b*c*(d*x)^(2 + m)*Sqrt 
[1 - c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(d^2* 
(1 + m)*(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
 

Defintions of rubi rules used

rule 136

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

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 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP 
art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(c*x)^m* 
(1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && 
!(ILtQ[p, 0] || GtQ[a, 0])
 

rule 6298

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
 :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* 
(n/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + 
 c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& NeQ[m, -1]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-(c^2*d^m*integrate(x^2*x^m/(c^2*(m + 1)*x^2 - m - 1), x) - c*d^m*integrat 
e(x*x^m/(c^3*(m + 1)*x^3 - c*(m + 1)*x + (c^2*(m + 1)*x^2 - m - 1)*sqrt(c* 
x + 1)*sqrt(c*x - 1)), x) - d^m*x*x^m*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1 
))/(m + 1))*b + (d*x)^(m + 1)*a/(d*(m + 1))
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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