\(\int (d-c^2 d x^2)^{3/2} (a+b \text {arccosh}(c x))^2 \, dx\) [59]

Optimal result

Integrand size = 26, antiderivative size = 324 \[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {15}{64} b^2 d x \sqrt {d-c^2 d x^2}+\frac {1}{32} b^2 d x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}+\frac {9 b^2 d \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{64 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c d x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d (-1+c x)^{3/2} (1+c x)^{3/2} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c}+\frac {3}{8} d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{8 b c \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

15/64*b^2*d*x*(-c^2*d*x^2+d)^(1/2)+1/32*b^2*d*x*(-c*x+1)*(c*x+1)*(-c^2*d*x 
^2+d)^(1/2)+9/64*b^2*d*(-c^2*d*x^2+d)^(1/2)*arccosh(c*x)/c/(c*x-1)^(1/2)/( 
c*x+1)^(1/2)-3/8*b*c*d*x^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/(c*x-1) 
^(1/2)/(c*x+1)^(1/2)+1/8*b*d*(c*x-1)^(3/2)*(c*x+1)^(3/2)*(-c^2*d*x^2+d)^(1 
/2)*(a+b*arccosh(c*x))/c+3/8*d*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2 
+1/4*x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2-1/8*d*(-c^2*d*x^2+d)^(1/2 
)*(a+b*arccosh(c*x))^3/b/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 3.20 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.15 \[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {-96 a^2 c d x \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (-5+2 c^2 x^2\right ) \sqrt {d-c^2 d x^2}-288 a^2 d^{3/2} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-192 a b d \sqrt {d-c^2 d x^2} (\cosh (2 \text {arccosh}(c x))+2 \text {arccosh}(c x) (\text {arccosh}(c x)-\sinh (2 \text {arccosh}(c x))))-32 b^2 d \sqrt {d-c^2 d x^2} \left (4 \text {arccosh}(c x)^3+6 \text {arccosh}(c x) \cosh (2 \text {arccosh}(c x))-3 \left (1+2 \text {arccosh}(c x)^2\right ) \sinh (2 \text {arccosh}(c x))\right )+12 a b d \sqrt {d-c^2 d x^2} \left (8 \text {arccosh}(c x)^2+\cosh (4 \text {arccosh}(c x))-4 \text {arccosh}(c x) \sinh (4 \text {arccosh}(c x))\right )+b^2 d \sqrt {d-c^2 d x^2} \left (32 \text {arccosh}(c x)^3+12 \text {arccosh}(c x) \cosh (4 \text {arccosh}(c x))-3 \left (1+8 \text {arccosh}(c x)^2\right ) \sinh (4 \text {arccosh}(c x))\right )}{768 c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \] Input:

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

Output:

(-96*a^2*c*d*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(-5 + 2*c^2*x^2)*Sqrt[ 
d - c^2*d*x^2] - 288*a^2*d^(3/2)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcT 
an[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 192*a*b*d*Sqrt[d 
- c^2*d*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c*x] - Sinh[2 
*ArcCosh[c*x]])) - 32*b^2*d*Sqrt[d - c^2*d*x^2]*(4*ArcCosh[c*x]^3 + 6*ArcC 
osh[c*x]*Cosh[2*ArcCosh[c*x]] - 3*(1 + 2*ArcCosh[c*x]^2)*Sinh[2*ArcCosh[c* 
x]]) + 12*a*b*d*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^2*d*Sqrt[d - c^2*d*x^2]*(32* 
ArcCosh[c*x]^3 + 12*ArcCosh[c*x]*Cosh[4*ArcCosh[c*x]] - 3*(1 + 8*ArcCosh[c 
*x]^2)*Sinh[4*ArcCosh[c*x]]))/(768*c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))
 

Rubi [A] (verified)

Time = 1.73 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.10, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {6312, 25, 6310, 6298, 101, 43, 6308, 6327, 6329, 40, 40, 43}

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.

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

\(\Big \downarrow \) 6312

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 6310

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 101

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

\(\Big \downarrow \) 43

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

\(\Big \downarrow \) 6308

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

\(\Big \downarrow \) 6327

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

\(\Big \downarrow \) 6329

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

\(\Big \downarrow \) 40

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

\(\Big \downarrow \) 40

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

\(\Big \downarrow \) 43

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

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

rule 40

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

rule 43

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

rule 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
 

rule 6298

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
 :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* 
(n/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + 
 c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& NeQ[m, -1]
 

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]
 

rule 6327

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + ( 
e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1 
*d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2 
, e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1060\) vs. \(2(276)=552\).

Time = 0.21 (sec) , antiderivative size = 1061, normalized size of antiderivative = 3.27

method	result	size
default	\(\text {Expression too large to display}\)	\(1061\)
parts	\(\text {Expression too large to display}\)	\(1061\)

Input:

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

Output:

1/4*a^2*x*(-c^2*d*x^2+d)^(3/2)+3/8*a^2*d*x*(-c^2*d*x^2+d)^(1/2)+3/8*a^2*d^ 
2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b^2*(-1/8*(-d 
*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c*arccosh(c*x)^3*d-1/512*( 
-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))*(8*arccosh(c*x)^2-4*arccosh(c*x)+1)*d/(c*x-1)/(c*x+1)/c+1/16*(-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))*(2*arccosh(c*x)^2-2*arccosh(c*x)+1)*d/(c*x-1)/( 
c*x+1)/c+1/16*(-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)*(2*arccosh(c*x)^2+2*arccos 
h(c*x)+1)*d/(c*x-1)/(c*x+1)/c-1/512*(-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^2-1 
2*c^3*x^3-(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c*x)*(8*arccosh(c*x)^2+4*arccosh(c 
*x)+1)*d/(c*x-1)/(c*x+1)/c)+2*a*b*(-3/16*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1 
/2)/(c*x+1)^(1/2)/c*arccosh(c*x)^2*d-1/256*(-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/( 
c*x-1)/(c*x+1)/c+1/16*(-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/(c*x-1)/(c*x+1)/c+1/16*(-d*(c^2*x^2-1))^(1/2)*(-2*(c*x-1)^(1/2)*(c*x...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: TypeError} \] Input:

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

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

Output:

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

Reduce [F]

\[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {\sqrt {d}\, d \left (3 \mathit {asin} \left (c x \right ) a^{2}-2 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{3} x^{3}+5 \sqrt {-c^{2} x^{2}+1}\, a^{2} c x -16 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{2}d x \right ) a b \,c^{3}+16 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )d x \right ) a b c -8 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}+8 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )^{2}d x \right ) b^{2} c \right )}{8 c} \] Input:

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

Output:

(sqrt(d)*d*(3*asin(c*x)*a**2 - 2*sqrt( - c**2*x**2 + 1)*a**2*c**3*x**3 + 5 
*sqrt( - c**2*x**2 + 1)*a**2*c*x - 16*int(sqrt( - c**2*x**2 + 1)*acosh(c*x 
)*x**2,x)*a*b*c**3 + 16*int(sqrt( - c**2*x**2 + 1)*acosh(c*x),x)*a*b*c - 8 
*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)**2*x**2,x)*b**2*c**3 + 8*int(sqrt( 
- c**2*x**2 + 1)*acosh(c*x)**2,x)*b**2*c))/(8*c)