\(\int (d-c^2 d x^2)^{5/2} (a+b \text {arccosh}(c x)) \, dx\) [86]

Optimal result

Integrand size = 24, antiderivative size = 278 \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {5 b c d^2 x^2 \sqrt {d-c^2 d x^2}}{32 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 (-1+c x)^{3/2} (1+c x)^{3/2} \sqrt {d-c^2 d x^2}}{96 c}-\frac {b d^2 (-1+c x)^{5/2} (1+c x)^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{32 b c \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-5/32*b*c*d^2*x^2*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+5/96*b* 
d^2*(c*x-1)^(3/2)*(c*x+1)^(3/2)*(-c^2*d*x^2+d)^(1/2)/c-1/36*b*d^2*(c*x-1)^ 
(5/2)*(c*x+1)^(5/2)*(-c^2*d*x^2+d)^(1/2)/c+5/16*d^2*x*(-c^2*d*x^2+d)^(1/2) 
*(a+b*arccosh(c*x))+5/24*d*x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))+1/6*x 
*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))-5/32*d^2*(-c^2*d*x^2+d)^(1/2)*(a+ 
b*arccosh(c*x))^2/b/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 1.69 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.25 \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {48 a c d^2 x \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2} \left (33-26 c^2 x^2+8 c^4 x^4\right )-720 a d^{5/2} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-288 b d^2 \sqrt {d-c^2 d x^2} (\cosh (2 \text {arccosh}(c x))+2 \text {arccosh}(c x) (\text {arccosh}(c x)-\sinh (2 \text {arccosh}(c x))))+36 b d^2 \sqrt {d-c^2 d x^2} \left (8 \text {arccosh}(c x)^2+\cosh (4 \text {arccosh}(c x))-4 \text {arccosh}(c x) \sinh (4 \text {arccosh}(c x))\right )+b d^2 \sqrt {d-c^2 d x^2} \left (-72 \text {arccosh}(c x)^2+18 \cosh (2 \text {arccosh}(c x))-9 \cosh (4 \text {arccosh}(c x))-2 \cosh (6 \text {arccosh}(c x))+12 \text {arccosh}(c x) (-3 \sinh (2 \text {arccosh}(c x))+3 \sinh (4 \text {arccosh}(c x))+\sinh (6 \text {arccosh}(c x)))\right )}{2304 c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \] Input:

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

Output:

(48*a*c*d^2*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(33 
 - 26*c^2*x^2 + 8*c^4*x^4) - 720*a*d^(5/2)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + 
 c*x)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 288*b*d 
^2*Sqrt[d - c^2*d*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c*x 
] - Sinh[2*ArcCosh[c*x]])) + 36*b*d^2*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^ 
2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]) + b*d^2*Sq 
rt[d - c^2*d*x^2]*(-72*ArcCosh[c*x]^2 + 18*Cosh[2*ArcCosh[c*x]] - 9*Cosh[4 
*ArcCosh[c*x]] - 2*Cosh[6*ArcCosh[c*x]] + 12*ArcCosh[c*x]*(-3*Sinh[2*ArcCo 
sh[c*x]] + 3*Sinh[4*ArcCosh[c*x]] + Sinh[6*ArcCosh[c*x]])))/(2304*c*Sqrt[( 
-1 + c*x)/(1 + c*x)]*(1 + c*x))
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6312, 82, 241, 6312, 25, 82, 244, 2009, 6310, 15, 6308}

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.

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

\(\Big \downarrow \) 6312

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

\(\Big \downarrow \) 82

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

\(\Big \downarrow \) 241

\(\displaystyle \frac {5}{6} d \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))dx+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 6312

\(\displaystyle \frac {5}{6} d \left (\frac {3}{4} d \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx+\frac {b c d \sqrt {d-c^2 d x^2} \int -x (1-c x) (c x+1)dx}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {5}{6} d \left (\frac {3}{4} d \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx-\frac {b c d \sqrt {d-c^2 d x^2} \int x (1-c x) (c x+1)dx}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 82

\(\displaystyle \frac {5}{6} d \left (\frac {3}{4} d \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx-\frac {b c d \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right )dx}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 244

\(\displaystyle \frac {5}{6} d \left (\frac {3}{4} d \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx-\frac {b c d \sqrt {d-c^2 d x^2} \int \left (x-c^2 x^3\right )dx}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5}{6} d \left (\frac {3}{4} d \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right ) \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 6310

\(\displaystyle \frac {5}{6} d \left (\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {a+b \text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \int xdx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right ) \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {5}{6} d \left (\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {a+b \text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right ) \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 6308

\(\displaystyle \frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))+\frac {5}{6} d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))+\frac {3}{4} d \left (\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b c \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {b c d \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right ) \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

(b*d^2*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(36*c*Sqrt[-1 + c*x]*Sqrt[1 + 
c*x]) + (x*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/6 + (5*d*(-1/4*(b*c 
*d*Sqrt[d - c^2*d*x^2]*(x^2/2 - (c^2*x^4)/4))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x 
]) + (x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/4 + (3*d*(-1/4*(b*c*x^ 
2*Sqrt[d - c^2*d*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x*Sqrt[d - c^2*d* 
x^2]*(a + b*ArcCosh[c*x]))/2 - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2 
)/(4*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/4))/6
 

Defintions of rubi rules used

rule 15

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

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

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

rule 6308

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq 
rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + 
 c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ 
c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 
] && EqQ[e2, (-c)*d2] && NeQ[n, -1]
 

rule 6310

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcCosh[c*x])^n/2), x] + (-Simp[( 
1/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])]   Int[(a + b*ArcC 
osh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Simp[b*c*(n/2)*Simp[Sq 
rt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])]   Int[x*(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]
 

rule 6312

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(884\) vs. \(2(234)=468\).

Time = 0.00 (sec) , antiderivative size = 885, normalized size of antiderivative = 3.18

method	result	size
default	\(\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{6}+\frac {5 a d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{24}+\frac {5 a \,d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{16}+\frac {5 a \,d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{16 \sqrt {c^{2} d}}+b \left (-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2} d^{2}}{32 \sqrt {c x -1}\, \sqrt {c x +1}\, c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (32 c^{7} x^{7}-64 c^{5} x^{5}+32 c^{6} x^{6} \sqrt {c x -1}\, \sqrt {c x +1}+38 c^{3} x^{3}-48 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}-6 c x +18 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+6 \,\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{2304 \left (c x -1\right ) \left (c x +1\right ) c}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}+4 c x -8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+4 \,\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{512 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {15 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{256 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {15 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{256 \left (c x -1\right ) \left (c x +1\right ) c}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}+8 c^{5} x^{5}+8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) \left (1+4 \,\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{512 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-32 c^{6} x^{6} \sqrt {c x -1}\, \sqrt {c x +1}+32 c^{7} x^{7}+48 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}-64 c^{5} x^{5}-18 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+38 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-6 c x \right ) \left (1+6 \,\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{2304 \left (c x -1\right ) \left (c x +1\right ) c}\right )\)	\(885\)
parts	\(\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{6}+\frac {5 a d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{24}+\frac {5 a \,d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{16}+\frac {5 a \,d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{16 \sqrt {c^{2} d}}+b \left (-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2} d^{2}}{32 \sqrt {c x -1}\, \sqrt {c x +1}\, c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (32 c^{7} x^{7}-64 c^{5} x^{5}+32 c^{6} x^{6} \sqrt {c x -1}\, \sqrt {c x +1}+38 c^{3} x^{3}-48 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}-6 c x +18 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+6 \,\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{2304 \left (c x -1\right ) \left (c x +1\right ) c}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}+4 c x -8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+4 \,\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{512 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {15 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{256 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {15 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{256 \left (c x -1\right ) \left (c x +1\right ) c}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}+8 c^{5} x^{5}+8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) \left (1+4 \,\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{512 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-32 c^{6} x^{6} \sqrt {c x -1}\, \sqrt {c x +1}+32 c^{7} x^{7}+48 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}-64 c^{5} x^{5}-18 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+38 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-6 c x \right ) \left (1+6 \,\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{2304 \left (c x -1\right ) \left (c x +1\right ) c}\right )\)	\(885\)

Input:

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

Output:

1/6*a*x*(-c^2*d*x^2+d)^(5/2)+5/24*a*d*x*(-c^2*d*x^2+d)^(3/2)+5/16*a*d^2*x* 
(-c^2*d*x^2+d)^(1/2)+5/16*a*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2 
*d*x^2+d)^(1/2))+b*(-5/32*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/ 
2)/c*arccosh(c*x)^2*d^2+1/2304*(-d*(c^2*x^2-1))^(1/2)*(32*c^7*x^7-64*c^5*x 
^5+32*c^6*x^6*(c*x-1)^(1/2)*(c*x+1)^(1/2)+38*c^3*x^3-48*c^4*x^4*(c*x-1)^(1 
/2)*(c*x+1)^(1/2)-6*c*x+18*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/ 
2)*(c*x+1)^(1/2))*(-1+6*arccosh(c*x))*d^2/(c*x-1)/(c*x+1)/c-3/512*(-d*(c^2 
*x^2-1))^(1/2)*(8*c^5*x^5-12*c^3*x^3+8*c^4*x^4*(c*x-1)^(1/2)*(c*x+1)^(1/2) 
+4*c*x-8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+(c*x-1)^(1/2)*(c*x+1)^(1/2))* 
(-1+4*arccosh(c*x))*d^2/(c*x-1)/(c*x+1)/c+15/256*(-d*(c^2*x^2-1))^(1/2)*(2 
*c^3*x^3-2*c*x+2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1) 
^(1/2))*(-1+2*arccosh(c*x))*d^2/(c*x-1)/(c*x+1)/c+15/256*(-d*(c^2*x^2-1))^ 
(1/2)*(-2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+2*c^3*x^3+(c*x-1)^(1/2)*(c*x 
+1)^(1/2)-2*c*x)*(1+2*arccosh(c*x))*d^2/(c*x-1)/(c*x+1)/c-3/512*(-d*(c^2*x 
^2-1))^(1/2)*(-8*c^4*x^4*(c*x-1)^(1/2)*(c*x+1)^(1/2)+8*c^5*x^5+8*(c*x-1)^( 
1/2)*(c*x+1)^(1/2)*c^2*x^2-12*c^3*x^3-(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c*x)*( 
1+4*arccosh(c*x))*d^2/(c*x-1)/(c*x+1)/c+1/2304*(-d*(c^2*x^2-1))^(1/2)*(-32 
*c^6*x^6*(c*x-1)^(1/2)*(c*x+1)^(1/2)+32*c^7*x^7+48*c^4*x^4*(c*x-1)^(1/2)*( 
c*x+1)^(1/2)-64*c^5*x^5-18*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+38*c^3*x^3+ 
(c*x-1)^(1/2)*(c*x+1)^(1/2)-6*c*x)*(1+6*arccosh(c*x))*d^2/(c*x-1)/(c*x+...
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

1/48*(8*(-c^2*d*x^2 + d)^(5/2)*x + 10*(-c^2*d*x^2 + d)^(3/2)*d*x + 15*sqrt 
(-c^2*d*x^2 + d)*d^2*x + 15*d^(5/2)*arcsin(c*x)/c)*a + b*integrate((-c^2*d 
*x^2 + d)^(5/2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)), x)
 

Giac [F(-2)]

Exception generated. \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \] Input:

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

\[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {\sqrt {d}\, d^{2} \left (15 \mathit {asin} \left (c x \right ) a +8 \sqrt {-c^{2} x^{2}+1}\, a \,c^{5} x^{5}-26 \sqrt {-c^{2} x^{2}+1}\, a \,c^{3} x^{3}+33 \sqrt {-c^{2} x^{2}+1}\, a c x +48 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{4}d x \right ) b \,c^{5}-96 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{2}d x \right ) b \,c^{3}+48 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )d x \right ) b c \right )}{48 c} \] Input:

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

Output:

(sqrt(d)*d**2*(15*asin(c*x)*a + 8*sqrt( - c**2*x**2 + 1)*a*c**5*x**5 - 26* 
sqrt( - c**2*x**2 + 1)*a*c**3*x**3 + 33*sqrt( - c**2*x**2 + 1)*a*c*x + 48* 
int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**4,x)*b*c**5 - 96*int(sqrt( - c**2 
*x**2 + 1)*acosh(c*x)*x**2,x)*b*c**3 + 48*int(sqrt( - c**2*x**2 + 1)*acosh 
(c*x),x)*b*c))/(48*c)