\(\int \frac {a+b \text {arccosh}(c x)}{x^4 (d-c^2 d x^2)^{3/2}} \, dx\) [118]

Optimal result

Integrand size = 27, antiderivative size = 246 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 d^2 x^3}-\frac {8 c^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 d^2 x}-\frac {5 b c^3 \sqrt {-1+c x} \sqrt {1+c x} \log (x)}{3 d \sqrt {d-c^2 d x^2}}-\frac {b c^3 \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{2 d \sqrt {d-c^2 d x^2}} \] Output:

1/6*b*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/x^2/(-c^2*d*x^2+d)^(1/2)+(a+b*arccos 
h(c*x))/d/x^3/(-c^2*d*x^2+d)^(1/2)-4/3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c 
*x))/d^2/x^3-8/3*c^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/d^2/x-5/3*b*c 
^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln(x)/d/(-c^2*d*x^2+d)^(1/2)-1/2*b*c^3*(c*x 
-1)^(1/2)*(c*x+1)^(1/2)*ln(-c^2*x^2+1)/d/(-c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.65 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-2 a-8 a c^2 x^2+16 a c^4 x^4+b c x \sqrt {-1+c x} \sqrt {1+c x}+2 b \left (-1-4 c^2 x^2+8 c^4 x^4\right ) \text {arccosh}(c x)-10 b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x} \log (x)-3 b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{6 d x^3 \sqrt {d-c^2 d x^2}} \] Input:

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

Output:

(-2*a - 8*a*c^2*x^2 + 16*a*c^4*x^4 + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 
2*b*(-1 - 4*c^2*x^2 + 8*c^4*x^4)*ArcCosh[c*x] - 10*b*c^3*x^3*Sqrt[-1 + c*x 
]*Sqrt[1 + c*x]*Log[x] - 3*b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 - 
c^2*x^2])/(6*d*x^3*Sqrt[d - c^2*d*x^2])
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.72, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6337, 27, 1578, 1195, 2009}

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.

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

\(\Big \downarrow \) 6337

\(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int -\frac {-8 c^4 x^4+4 c^2 x^2+1}{3 d^2 x^3 \left (1-c^2 x^2\right )}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {4 c^2 (a+b \text {arccosh}(c x))}{3 d x \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))}{3 d \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b c \sqrt {d-c^2 d x^2} \int \frac {-8 c^4 x^4+4 c^2 x^2+1}{x^3 \left (1-c^2 x^2\right )}dx}{3 d^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {4 c^2 (a+b \text {arccosh}(c x))}{3 d x \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))}{3 d \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 1578

\(\displaystyle \frac {b c \sqrt {d-c^2 d x^2} \int \frac {-8 c^4 x^4+4 c^2 x^2+1}{x^4 \left (1-c^2 x^2\right )}dx^2}{6 d^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {4 c^2 (a+b \text {arccosh}(c x))}{3 d x \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))}{3 d \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 1195

\(\displaystyle \frac {b c \sqrt {d-c^2 d x^2} \int \left (\frac {3 c^4}{c^2 x^2-1}+\frac {5 c^2}{x^2}+\frac {1}{x^4}\right )dx^2}{6 d^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {4 c^2 (a+b \text {arccosh}(c x))}{3 d x \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))}{3 d \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {4 c^2 (a+b \text {arccosh}(c x))}{3 d x \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))}{3 d \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \left (5 c^2 \log \left (x^2\right )+3 c^2 \log \left (1-c^2 x^2\right )-\frac {1}{x^2}\right )}{6 d^2 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

-1/3*(a + b*ArcCosh[c*x])/(d*x^3*Sqrt[d - c^2*d*x^2]) - (4*c^2*(a + b*ArcC 
osh[c*x]))/(3*d*x*Sqrt[d - c^2*d*x^2]) + (8*c^4*x*(a + b*ArcCosh[c*x]))/(3 
*d*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[d - c^2*d*x^2]*(-x^(-2) + 5*c^2*Log[x^ 
2] + 3*c^2*Log[1 - c^2*x^2]))/(6*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
 

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 1195

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && IGtQ[p, 0]
 

rule 1578

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ 
)^4)^(p_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a 
+ b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int 
egerQ[(m - 1)/2]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6337

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_ 
), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcCo 
sh[c*x])   u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c 
*x])]   Int[SimplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b 
, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] 
 && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
 

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.49

method	result	size
default	\(a \left (-\frac {1}{3 d \,x^{3} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {4 c^{2} \left (-\frac {1}{d x \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 c^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}\right )}{3}\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (16 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) x^{4} c^{4}+16 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}-6 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{5} c^{5}-10 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{5} c^{5}-8 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}-16 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+6 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{3} c^{3}+10 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}+c^{3} x^{3}-2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-c x \right )}{6 d^{2} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) x^{3}}\)	\(367\)
parts	\(a \left (-\frac {1}{3 d \,x^{3} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {4 c^{2} \left (-\frac {1}{d x \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 c^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}\right )}{3}\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (16 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) x^{4} c^{4}+16 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}-6 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{5} c^{5}-10 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{5} c^{5}-8 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}-16 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+6 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{3} c^{3}+10 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}+c^{3} x^{3}-2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-c x \right )}{6 d^{2} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) x^{3}}\)	\(367\)

Input:

int((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

a*(-1/3/d/x^3/(-c^2*d*x^2+d)^(1/2)+4/3*c^2*(-1/d/x/(-c^2*d*x^2+d)^(1/2)+2* 
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)*(16*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)*x^4*c^4+16*arcc 
osh(c*x)*c^5*x^5-6*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^5*c^5-10*ln 
(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*x^5*c^5-8*arccosh(c*x)*(c*x+1)^(1/ 
2)*(c*x-1)^(1/2)*c^2*x^2-16*c^3*x^3*arccosh(c*x)+6*ln((c*x+(c*x-1)^(1/2)*( 
c*x+1)^(1/2))^2-1)*x^3*c^3+10*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*x^ 
3*c^3+c^3*x^3-2*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)-c*x)/d^2/(c^4*x^4 
-2*c^2*x^2+1)/x^3
 

Fricas [F]

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

integrate((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^(3/2),x, algorithm="fricas 
")
 

Output:

integral(sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)/(c^4*d^2*x^8 - 2*c^2*d^ 
2*x^6 + d^2*x^4), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

integrate((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^(3/2),x, algorithm="maxima 
")
 

Output:

1/3*(8*c^4*x/(sqrt(-c^2*d*x^2 + d)*d) - 4*c^2/(sqrt(-c^2*d*x^2 + d)*d*x) - 
 1/(sqrt(-c^2*d*x^2 + d)*d*x^3))*a + b*integrate(log(c*x + sqrt(c*x + 1)*s 
qrt(c*x - 1))/((-c^2*d*x^2 + d)^(3/2)*x^4), x)
 

Giac [F]

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

integrate((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^(3/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^4\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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