\(\int (f x)^m (d-c^2 d x^2) (a+b \text {arccosh}(c x)) \, dx\) [142]

Optimal result

Integrand size = 25, antiderivative size = 171 \[ \int (f x)^m \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {b c d (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x}}{f^2 (3+m)^2}+\frac {d (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}-\frac {c^2 d (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}-\frac {b c d (7+3 m) (f x)^{2+m} \sqrt {1-c x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{f^2 (1+m) (2+m) (3+m)^2 \sqrt {-1+c x}} \] Output:

b*c*d*(f*x)^(2+m)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/f^2/(3+m)^2+d*(f*x)^(1+m)*(a 
+b*arccosh(c*x))/f/(1+m)-c^2*d*(f*x)^(3+m)*(a+b*arccosh(c*x))/f^3/(3+m)-b* 
c*d*(7+3*m)*(f*x)^(2+m)*(-c*x+1)^(1/2)*hypergeom([1/2, 1+1/2*m],[2+1/2*m], 
c^2*x^2)/f^2/(1+m)/(2+m)/(3+m)^2/(c*x-1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.12 \[ \int (f x)^m \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=d x (f x)^m \left (\frac {a+b \text {arccosh}(c x)}{1+m}-\frac {c^2 x^2 (a+b \text {arccosh}(c x))}{3+m}-\frac {b c x \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{\left (2+3 m+m^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 x^3 \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4+m}{2},\frac {6+m}{2},c^2 x^2\right )}{\left (12+7 m+m^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}\right ) \] Input:

Integrate[(f*x)^m*(d - c^2*d*x^2)*(a + b*ArcCosh[c*x]),x]
 

Output:

d*x*(f*x)^m*((a + b*ArcCosh[c*x])/(1 + m) - (c^2*x^2*(a + b*ArcCosh[c*x])) 
/(3 + m) - (b*c*x*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + 
 m)/2, c^2*x^2])/((2 + 3*m + m^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*x 
^3*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, c^2*x^2] 
)/((12 + 7*m + m^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6336, 27, 960, 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.

\(\displaystyle \int \left (d-c^2 d x^2\right ) (f x)^m (a+b \text {arccosh}(c x)) \, dx\)

\(\Big \downarrow \) 6336

\(\displaystyle -b c \int \frac {d (f x)^{m+1} \left (-c^2 (m+1) x^2+m+3\right )}{f \left (m^2+4 m+3\right ) \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {c^2 d (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {d (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c d \int \frac {(f x)^{m+1} \left (-c^2 (m+1) x^2+m+3\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx}{f \left (m^2+4 m+3\right )}-\frac {c^2 d (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {d (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}\)

\(\Big \downarrow \) 960

\(\displaystyle -\frac {b c d \left (\frac {(3 m+7) \int \frac {(f x)^{m+1}}{\sqrt {c x-1} \sqrt {c x+1}}dx}{m+3}-\frac {(m+1) \sqrt {c x-1} \sqrt {c x+1} (f x)^{m+2}}{f (m+3)}\right )}{f \left (m^2+4 m+3\right )}-\frac {c^2 d (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {d (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}\)

\(\Big \downarrow \) 136

\(\displaystyle -\frac {b c d \left (\frac {(3 m+7) \sqrt {c^2 x^2-1} \int \frac {(f x)^{m+1}}{\sqrt {c^2 x^2-1}}dx}{(m+3) \sqrt {c x-1} \sqrt {c x+1}}-\frac {(m+1) \sqrt {c x-1} \sqrt {c x+1} (f x)^{m+2}}{f (m+3)}\right )}{f \left (m^2+4 m+3\right )}-\frac {c^2 d (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {d (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}\)

\(\Big \downarrow \) 279

\(\displaystyle -\frac {b c d \left (\frac {(3 m+7) \sqrt {1-c^2 x^2} \int \frac {(f x)^{m+1}}{\sqrt {1-c^2 x^2}}dx}{(m+3) \sqrt {c x-1} \sqrt {c x+1}}-\frac {(m+1) \sqrt {c x-1} \sqrt {c x+1} (f x)^{m+2}}{f (m+3)}\right )}{f \left (m^2+4 m+3\right )}-\frac {c^2 d (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {d (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}\)

\(\Big \downarrow \) 278

\(\displaystyle -\frac {c^2 d (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {d (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}-\frac {b c d \left (\frac {(3 m+7) \sqrt {1-c^2 x^2} (f x)^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{f (m+2) (m+3) \sqrt {c x-1} \sqrt {c x+1}}-\frac {(m+1) \sqrt {c x-1} \sqrt {c x+1} (f x)^{m+2}}{f (m+3)}\right )}{f \left (m^2+4 m+3\right )}\)

Input:

Int[(f*x)^m*(d - c^2*d*x^2)*(a + b*ArcCosh[c*x]),x]
 

Output:

(d*(f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(f*(1 + m)) - (c^2*d*(f*x)^(3 + m)* 
(a + b*ArcCosh[c*x]))/(f^3*(3 + m)) - (b*c*d*(-(((1 + m)*(f*x)^(2 + m)*Sqr 
t[-1 + c*x]*Sqrt[1 + c*x])/(f*(3 + m))) + ((7 + 3*m)*(f*x)^(2 + m)*Sqrt[1 
- c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(f*(2 + 
m)*(3 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/(f*(3 + 4*m + m^2))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 960

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.) 
*(x_)^(non2_.))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^( 
m + 1)*(a1 + b1*x^(n/2))^(p + 1)*((a2 + b2*x^(n/2))^(p + 1)/(b1*b2*e*(m + n 
*(p + 1) + 1))), x] - Simp[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/ 
(b1*b2*(m + n*(p + 1) + 1))   Int[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n 
/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] && EqQ[non2, 
 n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]
 

rule 6336

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ 
)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Simp 
[(a + b*ArcCosh[c*x])   u, x] - Simp[b*c   Int[SimplifyIntegrand[u/(Sqrt[1 
+ c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && E 
qQ[c^2*d + e, 0] && IGtQ[p, 0]
 

Maple [F]

\[\int \left (f x \right )^{m} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )d x\]

Input:

int((f*x)^m*(-c^2*d*x^2+d)*(a+b*arccosh(c*x)),x)
 

Output:

int((f*x)^m*(-c^2*d*x^2+d)*(a+b*arccosh(c*x)),x)
 

Fricas [F]

\[ \int (f x)^m \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\int { -{\left (c^{2} d x^{2} - d\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \] Input:

integrate((f*x)^m*(-c^2*d*x^2+d)*(a+b*arccosh(c*x)),x, algorithm="fricas")
 

Output:

integral(-(a*c^2*d*x^2 - a*d + (b*c^2*d*x^2 - b*d)*arccosh(c*x))*(f*x)^m, 
x)
 

Sympy [F]

\[ \int (f x)^m \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=- d \left (\int \left (- a \left (f x\right )^{m}\right )\, dx + \int \left (- b \left (f x\right )^{m} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int a c^{2} x^{2} \left (f x\right )^{m}\, dx + \int b c^{2} x^{2} \left (f x\right )^{m} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \] Input:

integrate((f*x)**m*(-c**2*d*x**2+d)*(a+b*acosh(c*x)),x)
 

Output:

-d*(Integral(-a*(f*x)**m, x) + Integral(-b*(f*x)**m*acosh(c*x), x) + Integ 
ral(a*c**2*x**2*(f*x)**m, x) + Integral(b*c**2*x**2*(f*x)**m*acosh(c*x), x 
))
 

Maxima [F]

\[ \int (f x)^m \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\int { -{\left (c^{2} d x^{2} - d\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \] Input:

integrate((f*x)^m*(-c^2*d*x^2+d)*(a+b*arccosh(c*x)),x, algorithm="maxima")
 

Output:

-a*c^2*d*f^m*x^3*x^m/(m + 3) - (b*c^2*d*f^m*(m + 1)*x^3 - b*d*f^m*(m + 3)* 
x)*x^m*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(m^2 + 4*m + 3) + (f*x)^(m + 
 1)*a*d/(f*(m + 1)) - integrate((b*c^3*d*f^m*(m + 1)*x^3 - b*c*d*f^m*(m + 
3)*x)*x^m/((m^2 + 4*m + 3)*c^3*x^3 - (m^2 + 4*m + 3)*c*x + ((m^2 + 4*m + 3 
)*c^2*x^2 - m^2 - 4*m - 3)*sqrt(c*x + 1)*sqrt(c*x - 1)), x) + integrate((b 
*c^4*d*f^m*(m + 1)*x^4 - b*c^2*d*f^m*(m + 3)*x^2)*x^m/((m^2 + 4*m + 3)*c^2 
*x^2 - m^2 - 4*m - 3), x)
 

Giac [F(-2)]

Exception generated. \[ \int (f x)^m \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \] Input:

integrate((f*x)^m*(-c^2*d*x^2+d)*(a+b*arccosh(c*x)),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int (f x)^m \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (d-c^2\,d\,x^2\right )\,{\left (f\,x\right )}^m \,d x \] Input:

int((a + b*acosh(c*x))*(d - c^2*d*x^2)*(f*x)^m,x)
 

Output:

int((a + b*acosh(c*x))*(d - c^2*d*x^2)*(f*x)^m, x)
 

Reduce [F]

\[ \int (f x)^m \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {f^{m} d \left (-x^{m} a \,c^{2} m \,x^{3}-x^{m} a \,c^{2} x^{3}+x^{m} a m x +3 x^{m} a x -\left (\int x^{m} \mathit {acosh} \left (c x \right ) x^{2}d x \right ) b \,c^{2} m^{2}-4 \left (\int x^{m} \mathit {acosh} \left (c x \right ) x^{2}d x \right ) b \,c^{2} m -3 \left (\int x^{m} \mathit {acosh} \left (c x \right ) x^{2}d x \right ) b \,c^{2}+\left (\int x^{m} \mathit {acosh} \left (c x \right )d x \right ) b \,m^{2}+4 \left (\int x^{m} \mathit {acosh} \left (c x \right )d x \right ) b m +3 \left (\int x^{m} \mathit {acosh} \left (c x \right )d x \right ) b \right )}{m^{2}+4 m +3} \] Input:

int((f*x)^m*(-c^2*d*x^2+d)*(a+b*acosh(c*x)),x)
                                                                                    
                                                                                    
 

Output:

(f**m*d*( - x**m*a*c**2*m*x**3 - x**m*a*c**2*x**3 + x**m*a*m*x + 3*x**m*a* 
x - int(x**m*acosh(c*x)*x**2,x)*b*c**2*m**2 - 4*int(x**m*acosh(c*x)*x**2,x 
)*b*c**2*m - 3*int(x**m*acosh(c*x)*x**2,x)*b*c**2 + int(x**m*acosh(c*x),x) 
*b*m**2 + 4*int(x**m*acosh(c*x),x)*b*m + 3*int(x**m*acosh(c*x),x)*b))/(m** 
2 + 4*m + 3)