\(\int \frac {a+b \text {arccosh}(c x)}{(d-c^2 d x^2)^{5/2}} \, dx\) [57]

Optimal result

Integrand size = 24, antiderivative size = 162 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b}{6 c d^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {d-c^2 d x^2}}+\frac {x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{3 c d^2 \sqrt {d-c^2 d x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.81 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {-6 a c x+4 a c^3 x^3-b \sqrt {-1+c x} \sqrt {1+c x}+2 b c x \left (-3+2 c^2 x^2\right ) \text {arccosh}(c x)-2 b \sqrt {-1+c x} \sqrt {1+c x} \left (-1+c^2 x^2\right ) \log \left (1-c^2 x^2\right )}{6 c d^2 \left (-1+c^2 x^2\right ) \sqrt {d-c^2 d x^2}} \] Input:

Integrate[(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^(5/2),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.10, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6316, 82, 241, 6314, 240}

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.

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

\(\Big \downarrow \) 6316

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

\(\Big \downarrow \) 82

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 6314

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

\(\Big \downarrow \) 240

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

Input:

Int[(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^(5/2),x]
 

Output:

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

Defintions of rubi rules used

rule 82

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

rule 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
 

rule 241

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ 
(2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
 

rule 6314

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

rule 6316

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(476\) vs. \(2(138)=276\).

Time = 0.25 (sec) , antiderivative size = 477, normalized size of antiderivative = 2.94

method	result	size
default	\(a \left (\frac {x}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-3 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-2 \sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (8 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{5} c^{5}-8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{6} c^{6}-20 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{3} c^{3}+24 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{4} c^{4}+2 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}-2 c^{4} x^{4}+6 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+12 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x c -24 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{2} c^{2}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +4 c^{2} x^{2}-8 \,\operatorname {arccosh}\left (c x \right )+8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )-2\right )}{6 \left (3 c^{6} x^{6}-10 c^{4} x^{4}+11 c^{2} x^{2}-4\right ) c \,d^{3}}\)	\(477\)
parts	\(a \left (\frac {x}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-3 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-2 \sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (8 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{5} c^{5}-8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{6} c^{6}-20 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{3} c^{3}+24 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{4} c^{4}+2 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}-2 c^{4} x^{4}+6 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+12 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x c -24 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{2} c^{2}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +4 c^{2} x^{2}-8 \,\operatorname {arccosh}\left (c x \right )+8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )-2\right )}{6 \left (3 c^{6} x^{6}-10 c^{4} x^{4}+11 c^{2} x^{2}-4\right ) c \,d^{3}}\)	\(477\)

Input:

int((a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

a*(1/3*x/d/(-c^2*d*x^2+d)^(3/2)+2/3/d^2*x/(-c^2*d*x^2+d)^(1/2))-1/6*b*(-d* 
(c^2*x^2-1))^(1/2)*(2*c^3*x^3-3*c*x+2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2- 
2*(c*x-1)^(1/2)*(c*x+1)^(1/2))*(8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln((c*x+(c*x 
-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^5*c^5-8*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2 
))^2-1)*x^6*c^6-20*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln((c*x+(c*x-1)^(1/2)*(c*x+ 
1)^(1/2))^2-1)*x^3*c^3+24*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^4*c^ 
4+2*c^3*x^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)-2*c^4*x^4+6*c^2*x^2*arccosh(c*x)+1 
2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x* 
c-24*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^2*c^2-3*(c*x-1)^(1/2)*(c* 
x+1)^(1/2)*c*x+4*c^2*x^2-8*arccosh(c*x)+8*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1 
/2))^2-1)-2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/c/d^3
 

Fricas [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((a+b*acosh(c*x))/(-c**2*d*x**2+d)**(5/2),x)
 

Output:

Integral((a + b*acosh(c*x))/(-d*(c*x - 1)*(c*x + 1))**(5/2), x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.97 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {1}{6} \, b c {\left (\frac {\sqrt {-d}}{c^{4} d^{3} x^{2} - c^{2} d^{3}} + \frac {2 \, \sqrt {-d} \log \left (c x + 1\right )}{c^{2} d^{3}} + \frac {2 \, \sqrt {-d} \log \left (c x - 1\right )}{c^{2} d^{3}}\right )} + \frac {1}{3} \, b {\left (\frac {2 \, x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d}\right )} \operatorname {arcosh}\left (c x\right ) + \frac {1}{3} \, a {\left (\frac {2 \, x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d}\right )} \] Input:

integrate((a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")
 

Output:

integrate((b*arccosh(c*x) + a)/(-c^2*d*x^2 + d)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \] Input:

int((a + b*acosh(c*x))/(d - c^2*d*x^2)^(5/2),x)
 

Output:

int((a + b*acosh(c*x))/(d - c^2*d*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{2} x^{2}-3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b +2 a \,c^{2} x^{3}-3 a x}{3 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} \left (c^{2} x^{2}-1\right )} \] Input:

int((a+b*acosh(c*x))/(-c^2*d*x^2+d)^(5/2),x)
 

Output:

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