\(\int \frac {a+b \text {arccosh}(c x)}{x^4 \sqrt {d-c^2 d x^2}} \, dx\) [108]

Optimal result

Integrand size = 27, antiderivative size = 161 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \sqrt {d-c^2 d x^2}} \, dx=-\frac {b c \sqrt {d-c^2 d x^2}}{6 d x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 d x^3}-\frac {2 c^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 d x}+\frac {2 b c^3 \sqrt {d-c^2 d x^2} \log (x)}{3 d \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-1/6*b*c*(-c^2*d*x^2+d)^(1/2)/d/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/3*(-c^2* 
d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/d/x^3-2/3*c^2*(-c^2*d*x^2+d)^(1/2)*(a+b* 
arccosh(c*x))/d/x+2/3*b*c^3*(-c^2*d*x^2+d)^(1/2)*ln(x)/d/(c*x-1)^(1/2)/(c* 
x+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.08 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \sqrt {d-c^2 d x^2}} \, dx=-\frac {\sqrt {d-c^2 d x^2} \left (b c x+6 b c^3 x^3+2 a \sqrt {-1+c x} \sqrt {1+c x}+4 a c^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}+2 b \sqrt {-1+c x} \sqrt {1+c x} \left (1+2 c^2 x^2\right ) \text {arccosh}(c x)-4 b c^3 x^3 \log (-1+c x)-4 b c^3 x^3 \log \left (1+\frac {1}{-1+c x}\right )\right )}{6 d x^3 \sqrt {-1+c x} \sqrt {1+c x}} \] Input:

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

Output:

-1/6*(Sqrt[d - c^2*d*x^2]*(b*c*x + 6*b*c^3*x^3 + 2*a*Sqrt[-1 + c*x]*Sqrt[1 
 + c*x] + 4*a*c^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 2*b*Sqrt[-1 + c*x]*Sq 
rt[1 + c*x]*(1 + 2*c^2*x^2)*ArcCosh[c*x] - 4*b*c^3*x^3*Log[-1 + c*x] - 4*b 
*c^3*x^3*Log[1 + (-1 + c*x)^(-1)]))/(d*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6347, 15, 6332, 14}

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.

\(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{x^4 \sqrt {d-c^2 d x^2}} \, dx\)

\(\Big \downarrow \) 6347

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

\(\Big \downarrow \) 15

\(\displaystyle \frac {2}{3} c^2 \int \frac {a+b \text {arccosh}(c x)}{x^2 \sqrt {d-c^2 d x^2}}dx-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 d x^3}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{6 x^2 \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 6332

\(\displaystyle \frac {2}{3} c^2 \left (-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {1}{x}dx}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{d x}\right )-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 d x^3}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{6 x^2 \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 14

\(\displaystyle \frac {2}{3} c^2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{d x}-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \log (x)}{\sqrt {d-c^2 d x^2}}\right )-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 d x^3}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{6 x^2 \sqrt {d-c^2 d x^2}}\)

Input:

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

Output:

(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*x^2*Sqrt[d - c^2*d*x^2]) - (Sqrt[d - 
 c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(3*d*x^3) + (2*c^2*(-((Sqrt[d - c^2*d*x^ 
2]*(a + b*ArcCosh[c*x]))/(d*x)) - (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[x] 
)/Sqrt[d - c^2*d*x^2]))/3
 

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 6332

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + 
 b*ArcCosh[c*x])^n/(d*f*(m + 1))), x] + Simp[b*c*(n/(f*(m + 1)))*Simp[(d + 
e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]   Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2 
)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3 
, 0] && NeQ[m, -1]
 

rule 6347

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + 
b*ArcCosh[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1 
)))   Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x] + Simp 
[b*c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]   Int[( 
f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^ 
(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && 
 GtQ[n, 0] && ILtQ[m, -1]
 

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.19

method	result	size
default	\(a \left (-\frac {\sqrt {-c^{2} d \,x^{2}+d}}{3 d \,x^{3}}-\frac {2 c^{2} \sqrt {-c^{2} d \,x^{2}+d}}{3 d x}\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (4 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}+4 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-4 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}+2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+c x \right )}{6 d \,x^{3} \left (c^{2} x^{2}-1\right )}\)	\(192\)
parts	\(a \left (-\frac {\sqrt {-c^{2} d \,x^{2}+d}}{3 d \,x^{3}}-\frac {2 c^{2} \sqrt {-c^{2} d \,x^{2}+d}}{3 d x}\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (4 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}+4 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-4 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}+2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+c x \right )}{6 d \,x^{3} \left (c^{2} x^{2}-1\right )}\)	\(192\)

Input:

int((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

a*(-1/3/d/x^3*(-c^2*d*x^2+d)^(1/2)-2/3*c^2/d/x*(-c^2*d*x^2+d)^(1/2))-1/6*b 
*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(4*arccosh(c*x)*(c*x+1 
)^(1/2)*(c*x-1)^(1/2)*c^2*x^2+4*c^3*x^3*arccosh(c*x)-4*ln(1+(c*x+(c*x-1)^( 
1/2)*(c*x+1)^(1/2))^2)*x^3*c^3+2*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)+ 
c*x)/d/x^3/(c^2*x^2-1)
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 480, normalized size of antiderivative = 2.98 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\left [-\frac {2 \, {\left (2 \, b c^{4} x^{4} - b c^{2} x^{2} - b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, {\left (b c^{5} x^{5} - b c^{3} x^{3}\right )} \sqrt {-d} \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} + \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{4} - 1\right )} \sqrt {-d} - d}{c^{2} x^{4} - x^{2}}\right ) - \sqrt {-c^{2} d x^{2} + d} {\left (b c x^{3} - b c x\right )} \sqrt {c^{2} x^{2} - 1} + 2 \, {\left (2 \, a c^{4} x^{4} - a c^{2} x^{2} - a\right )} \sqrt {-c^{2} d x^{2} + d}}{6 \, {\left (c^{2} d x^{5} - d x^{3}\right )}}, \frac {4 \, {\left (b c^{5} x^{5} - b c^{3} x^{3}\right )} \sqrt {d} \arctan \left (\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{2} - 1\right )} \sqrt {d}}{c^{2} d x^{4} + {\left (c^{2} - 1\right )} d x^{2} - d}\right ) - 2 \, {\left (2 \, b c^{4} x^{4} - b c^{2} x^{2} - b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + \sqrt {-c^{2} d x^{2} + d} {\left (b c x^{3} - b c x\right )} \sqrt {c^{2} x^{2} - 1} - 2 \, {\left (2 \, a c^{4} x^{4} - a c^{2} x^{2} - a\right )} \sqrt {-c^{2} d x^{2} + d}}{6 \, {\left (c^{2} d x^{5} - d x^{3}\right )}}\right ] \] Input:

integrate((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas 
")
 

Output:

[-1/6*(2*(2*b*c^4*x^4 - b*c^2*x^2 - b)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt 
(c^2*x^2 - 1)) + 2*(b*c^5*x^5 - b*c^3*x^3)*sqrt(-d)*log((c^2*d*x^6 + c^2*d 
*x^2 - d*x^4 + sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*(x^4 - 1)*sqrt(-d) - 
 d)/(c^2*x^4 - x^2)) - sqrt(-c^2*d*x^2 + d)*(b*c*x^3 - b*c*x)*sqrt(c^2*x^2 
 - 1) + 2*(2*a*c^4*x^4 - a*c^2*x^2 - a)*sqrt(-c^2*d*x^2 + d))/(c^2*d*x^5 - 
 d*x^3), 1/6*(4*(b*c^5*x^5 - b*c^3*x^3)*sqrt(d)*arctan(sqrt(-c^2*d*x^2 + d 
)*sqrt(c^2*x^2 - 1)*(x^2 - 1)*sqrt(d)/(c^2*d*x^4 + (c^2 - 1)*d*x^2 - d)) - 
 2*(2*b*c^4*x^4 - b*c^2*x^2 - b)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x 
^2 - 1)) + sqrt(-c^2*d*x^2 + d)*(b*c*x^3 - b*c*x)*sqrt(c^2*x^2 - 1) - 2*(2 
*a*c^4*x^4 - a*c^2*x^2 - a)*sqrt(-c^2*d*x^2 + d))/(c^2*d*x^5 - d*x^3)]
 

Sympy [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x^{4} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \] Input:

integrate((a+b*acosh(c*x))/x**4/(-c**2*d*x**2+d)**(1/2),x)
 

Output:

Integral((a + b*acosh(c*x))/(x**4*sqrt(-d*(c*x - 1)*(c*x + 1))), x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.83 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\frac {1}{6} \, {\left (\frac {4 \, c^{2} \sqrt {-d} \log \left (x\right )}{d} - \frac {\sqrt {-d}}{d x^{2}}\right )} b c - \frac {1}{3} \, b {\left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} c^{2}}{d x} + \frac {\sqrt {-c^{2} d x^{2} + d}}{d x^{3}}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {1}{3} \, a {\left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} c^{2}}{d x} + \frac {\sqrt {-c^{2} d x^{2} + d}}{d x^{3}}\right )} \] Input:

integrate((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima 
")
 

Output:

1/6*(4*c^2*sqrt(-d)*log(x)/d - sqrt(-d)/(d*x^2))*b*c - 1/3*b*(2*sqrt(-c^2* 
d*x^2 + d)*c^2/(d*x) + sqrt(-c^2*d*x^2 + d)/(d*x^3))*arccosh(c*x) - 1/3*a* 
(2*sqrt(-c^2*d*x^2 + d)*c^2/(d*x) + sqrt(-c^2*d*x^2 + d)/(d*x^3))
 

Giac [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{\sqrt {-c^{2} d x^{2} + d} x^{4}} \,d x } \] Input:

integrate((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")
 

Output:

integrate((b*arccosh(c*x) + a)/(sqrt(-c^2*d*x^2 + d)*x^4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^4\,\sqrt {d-c^2\,d\,x^2}} \,d x \] Input:

int((a + b*acosh(c*x))/(x^4*(d - c^2*d*x^2)^(1/2)),x)
 

Output:

int((a + b*acosh(c*x))/(x^4*(d - c^2*d*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\frac {-2 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}\, a +3 \left (\int \frac {\mathit {acosh} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, x^{4}}d x \right ) b \,x^{3}}{3 \sqrt {d}\, x^{3}} \] Input:

int((a+b*acosh(c*x))/x^4/(-c^2*d*x^2+d)^(1/2),x)
                                                                                    
                                                                                    
 

Output:

( - 2*sqrt( - c**2*x**2 + 1)*a*c**2*x**2 - sqrt( - c**2*x**2 + 1)*a + 3*in 
t(acosh(c*x)/(sqrt( - c**2*x**2 + 1)*x**4),x)*b*x**3)/(3*sqrt(d)*x**3)