\(\int \frac {x (a+b \text {arccosh}(c x))}{(d-c^2 d x^2)^3} \, dx\) [49]

Optimal result

Integrand size = 23, antiderivative size = 91 \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {b x}{12 c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {b x}{6 c d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \text {arccosh}(c x)}{4 c^2 d^3 \left (1-c^2 x^2\right )^2} \] Output:

1/12*b*x/c/d^3/(c*x-1)^(3/2)/(c*x+1)^(3/2)-1/6*b*x/c/d^3/(c*x-1)^(1/2)/(c* 
x+1)^(1/2)+1/4*(a+b*arccosh(c*x))/c^2/d^3/(-c^2*x^2+1)^2
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70 \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {3 a+b c x \sqrt {-1+c x} \sqrt {1+c x} \left (3-2 c^2 x^2\right )+3 b \text {arccosh}(c x)}{12 c^2 d^3 \left (-1+c^2 x^2\right )^2} \] Input:

Integrate[(x*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^3,x]
 

Output:

(3*a + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(3 - 2*c^2*x^2) + 3*b*ArcCosh[c* 
x])/(12*c^2*d^3*(-1 + c^2*x^2)^2)
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6329, 42, 41}

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.

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

\(\Big \downarrow \) 6329

\(\displaystyle \frac {a+b \text {arccosh}(c x)}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \int \frac {1}{(c x-1)^{5/2} (c x+1)^{5/2}}dx}{4 c d^3}\)

\(\Big \downarrow \) 42

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

\(\Big \downarrow \) 41

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

Input:

Int[(x*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^3,x]
 

Output:

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

Defintions of rubi rules used

rule 41

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> S 
imp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && Eq 
Q[b*c + a*d, 0]
 

rule 42

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(- 
x)*(a + b*x)^(m + 1)*((c + d*x)^(m + 1)/(2*a*c*(m + 1))), x] + Simp[(2*m + 
3)/(2*a*c*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(m + 1), x], x] /; Fre 
eQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && ILtQ[m + 3/2, 0]
 

rule 6329

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

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.95

method	result	size
derivativedivides	\(\frac {\frac {a}{4 d^{3} \left (c^{2} x^{2}-1\right )^{2}}-\frac {b \left (-\frac {\operatorname {arccosh}\left (c x \right )}{4 \left (c^{2} x^{2}-1\right )^{2}}+\frac {c x \left (2 c^{2} x^{2}-3\right )}{12 \sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} x^{2}-1\right )}\right )}{d^{3}}}{c^{2}}\)	\(86\)
default	\(\frac {\frac {a}{4 d^{3} \left (c^{2} x^{2}-1\right )^{2}}-\frac {b \left (-\frac {\operatorname {arccosh}\left (c x \right )}{4 \left (c^{2} x^{2}-1\right )^{2}}+\frac {c x \left (2 c^{2} x^{2}-3\right )}{12 \sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} x^{2}-1\right )}\right )}{d^{3}}}{c^{2}}\)	\(86\)
parts	\(\frac {a}{4 d^{3} c^{2} \left (c^{2} x^{2}-1\right )^{2}}-\frac {b \left (-\frac {\operatorname {arccosh}\left (c x \right )}{4 \left (c^{2} x^{2}-1\right )^{2}}+\frac {c x \left (2 c^{2} x^{2}-3\right )}{12 \sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} x^{2}-1\right )}\right )}{d^{3} c^{2}}\)	\(88\)
orering	\(\frac {\left (c x -1\right ) \left (c x +1\right ) \left (10 c^{4} x^{4}-13 c^{2} x^{2}-6\right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{12 c^{2} \left (-c^{2} d \,x^{2}+d \right )^{3}}+\frac {\left (2 c^{2} x^{2}-3\right ) \left (c x -1\right )^{2} \left (c x +1\right )^{2} \left (\frac {a +b \,\operatorname {arccosh}\left (c x \right )}{\left (-c^{2} d \,x^{2}+d \right )^{3}}+\frac {x b c}{\sqrt {c x -1}\, \sqrt {c x +1}\, \left (-c^{2} d \,x^{2}+d \right )^{3}}+\frac {6 x^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c^{2} d}{\left (-c^{2} d \,x^{2}+d \right )^{4}}\right )}{12 c^{2}}\)	\(169\)

Input:

int(x*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^3,x,method=_RETURNVERBOSE)
 

Output:

1/c^2*(1/4*a/d^3/(c^2*x^2-1)^2-b/d^3*(-1/4/(c^2*x^2-1)^2*arccosh(c*x)+1/12 
/(c*x-1)^(1/2)/(c*x+1)^(1/2)*c*x*(2*c^2*x^2-3)/(c^2*x^2-1)))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.08 \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=-\frac {3 \, a c^{4} x^{4} - 6 \, a c^{2} x^{2} - 3 \, b \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (2 \, b c^{3} x^{3} - 3 \, b c x\right )} \sqrt {c^{2} x^{2} - 1}}{12 \, {\left (c^{6} d^{3} x^{4} - 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\right )}} \] Input:

integrate(x*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^3,x, algorithm="fricas")
 

Output:

-1/12*(3*a*c^4*x^4 - 6*a*c^2*x^2 - 3*b*log(c*x + sqrt(c^2*x^2 - 1)) + (2*b 
*c^3*x^3 - 3*b*c*x)*sqrt(c^2*x^2 - 1))/(c^6*d^3*x^4 - 2*c^4*d^3*x^2 + c^2* 
d^3)
 

Sympy [F]

\[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=- \frac {\int \frac {a x}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {b x \operatorname {acosh}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx}{d^{3}} \] Input:

integrate(x*(a+b*acosh(c*x))/(-c**2*d*x**2+d)**3,x)
 

Output:

-(Integral(a*x/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x) + Integral( 
b*x*acosh(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x))/d**3
 

Maxima [F]

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

integrate(x*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^3,x, algorithm="maxima")
 

Output:

1/16*b*((4*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + 1)/(c^6*d^3*x^4 - 2*c^ 
4*d^3*x^2 + c^2*d^3) + 16*integrate(1/4/(c^8*d^3*x^7 - 3*c^6*d^3*x^5 + 3*c 
^4*d^3*x^3 - c^2*d^3*x + (c^7*d^3*x^6 - 3*c^5*d^3*x^4 + 3*c^3*d^3*x^2 - c* 
d^3)*e^(1/2*log(c*x + 1) + 1/2*log(c*x - 1))), x)) + 1/4*a/(c^6*d^3*x^4 - 
2*c^4*d^3*x^2 + c^2*d^3)
 

Giac [F]

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

integrate(x*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^3,x, algorithm="giac")
 

Output:

integrate(-(b*arccosh(c*x) + a)*x/(c^2*d*x^2 - d)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\int \frac {x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {-4 \left (\int \frac {\mathit {acosh} \left (c x \right ) x}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b \,c^{6} x^{4}+8 \left (\int \frac {\mathit {acosh} \left (c x \right ) x}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b \,c^{4} x^{2}-4 \left (\int \frac {\mathit {acosh} \left (c x \right ) x}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b \,c^{2}+a}{4 c^{2} d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )} \] Input:

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

Output:

( - 4*int((acosh(c*x)*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x)*b* 
c**6*x**4 + 8*int((acosh(c*x)*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 
1),x)*b*c**4*x**2 - 4*int((acosh(c*x)*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2 
*x**2 - 1),x)*b*c**2 + a)/(4*c**2*d**3*(c**4*x**4 - 2*c**2*x**2 + 1))