\(\int x^2 (d-c^2 d x^2)^{3/2} (a+b \text {arccosh}(c x))^2 \, dx\) [168]

Optimal result

Integrand size = 29, antiderivative size = 441 \[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {7 b^2 d x \sqrt {d-c^2 d x^2}}{1152 c^2}+\frac {43 b^2 d x^3 \sqrt {d-c^2 d x^2}}{1728}-\frac {1}{108} b^2 c^2 d x^5 \sqrt {d-c^2 d x^2}+\frac {7 b^2 d \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{1152 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{16 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {7 b c d x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{48 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^6 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{18 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{48 b c^3 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

7/1152*b^2*d*x*(-c^2*d*x^2+d)^(1/2)/c^2+43/1728*b^2*d*x^3*(-c^2*d*x^2+d)^( 
1/2)-1/108*b^2*c^2*d*x^5*(-c^2*d*x^2+d)^(1/2)+7/1152*b^2*d*(-c^2*d*x^2+d)^ 
(1/2)*arccosh(c*x)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/16*b*d*x^2*(-c^2*d*x^ 
2+d)^(1/2)*(a+b*arccosh(c*x))/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-7/48*b*c*d*x^4 
*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/18* 
b*c^3*d*x^6*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/(c*x-1)^(1/2)/(c*x+1)^ 
(1/2)-1/16*d*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2/c^2+1/8*d*x^3*(-c 
^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2+1/6*x^3*(-c^2*d*x^2+d)^(3/2)*(a+b*a 
rccosh(c*x))^2-1/48*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^3/b/c^3/(c*x 
-1)^(1/2)/(c*x+1)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 4.68 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.10 \[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {-288 a^2 c d x \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2} \left (3-14 c^2 x^2+8 c^4 x^4\right )-864 a^2 d^{3/2} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-216 a b d \sqrt {d-c^2 d x^2} \left (8 \text {arccosh}(c x)^2+\cosh (4 \text {arccosh}(c x))-4 \text {arccosh}(c x) \sinh (4 \text {arccosh}(c x))\right )-18 b^2 d \sqrt {d-c^2 d x^2} \left (32 \text {arccosh}(c x)^3+12 \text {arccosh}(c x) \cosh (4 \text {arccosh}(c x))-3 \left (1+8 \text {arccosh}(c x)^2\right ) \sinh (4 \text {arccosh}(c x))\right )-12 a b d \sqrt {d-c^2 d x^2} \left (-72 \text {arccosh}(c x)^2+18 \cosh (2 \text {arccosh}(c x))-9 \cosh (4 \text {arccosh}(c x))-2 \cosh (6 \text {arccosh}(c x))+12 \text {arccosh}(c x) (-3 \sinh (2 \text {arccosh}(c x))+3 \sinh (4 \text {arccosh}(c x))+\sinh (6 \text {arccosh}(c x)))\right )+b^2 d \sqrt {d-c^2 d x^2} \left (288 \text {arccosh}(c x)^3+12 \text {arccosh}(c x) (-18 \cosh (2 \text {arccosh}(c x))+9 \cosh (4 \text {arccosh}(c x))+2 \cosh (6 \text {arccosh}(c x)))+108 \sinh (2 \text {arccosh}(c x))-27 \sinh (4 \text {arccosh}(c x))-4 \sinh (6 \text {arccosh}(c x))-72 \text {arccosh}(c x)^2 (-3 \sinh (2 \text {arccosh}(c x))+3 \sinh (4 \text {arccosh}(c x))+\sinh (6 \text {arccosh}(c x)))\right )}{13824 c^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \] Input:

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

Output:

(-288*a^2*c*d*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*( 
3 - 14*c^2*x^2 + 8*c^4*x^4) - 864*a^2*d^(3/2)*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))] - 216* 
a*b*d*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*Arc 
Cosh[c*x]*Sinh[4*ArcCosh[c*x]]) - 18*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]]) - 12*a*b*d*Sqrt[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*ArcCosh[c*x]] + 3*Sinh[4*ArcCosh[c*x]] + Sin 
h[6*ArcCosh[c*x]])) + b^2*d*Sqrt[d - c^2*d*x^2]*(288*ArcCosh[c*x]^3 + 12*A 
rcCosh[c*x]*(-18*Cosh[2*ArcCosh[c*x]] + 9*Cosh[4*ArcCosh[c*x]] + 2*Cosh[6* 
ArcCosh[c*x]]) + 108*Sinh[2*ArcCosh[c*x]] - 27*Sinh[4*ArcCosh[c*x]] - 4*Si 
nh[6*ArcCosh[c*x]] - 72*ArcCosh[c*x]^2*(-3*Sinh[2*ArcCosh[c*x]] + 3*Sinh[4 
*ArcCosh[c*x]] + Sinh[6*ArcCosh[c*x]])))/(13824*c^3*Sqrt[(-1 + c*x)/(1 + c 
*x)]*(1 + c*x))
 

Rubi [A] (verified)

Time = 4.77 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.25, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.724, Rules used = {6345, 25, 6327, 6336, 27, 960, 111, 27, 101, 43, 6341, 6298, 111, 27, 101, 43, 6354, 6298, 101, 43, 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.

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

\(\Big \downarrow \) 6345

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 6327

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

\(\Big \downarrow \) 6336

\(\displaystyle \frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-b c \int \frac {x^4 \left (3-2 c^2 x^2\right )}{12 \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{12} b c \int \frac {x^4 \left (3-2 c^2 x^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\)

\(\Big \downarrow \) 960

\(\displaystyle \frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{12} b c \left (\frac {4}{3} \int \frac {x^4}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\)

\(\Big \downarrow \) 111

\(\displaystyle \frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {\int \frac {3 x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \int \frac {x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\)

\(\Big \downarrow \) 101

\(\displaystyle \frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\)

\(\Big \downarrow \) 43

\(\displaystyle \frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 6341

\(\displaystyle \frac {1}{2} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \int x^3 (a+b \text {arccosh}(c x))dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 6298

\(\displaystyle \frac {1}{2} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \int \frac {x^4}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 111

\(\displaystyle \frac {1}{2} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {\int \frac {3 x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \int \frac {x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 101

\(\displaystyle \frac {1}{2} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 43

\(\displaystyle \frac {1}{2} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 6354

\(\displaystyle \frac {1}{2} d \left (-\frac {\sqrt {d-c^2 d x^2} \left (\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}-\frac {b \int x (a+b \text {arccosh}(c x))dx}{c}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{2 c^2}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 6298

\(\displaystyle \frac {1}{2} d \left (-\frac {\sqrt {d-c^2 d x^2} \left (\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}-\frac {b \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )}{c}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{2 c^2}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 101

\(\displaystyle \frac {1}{2} d \left (-\frac {\sqrt {d-c^2 d x^2} \left (-\frac {b \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{c}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{2 c^2}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 43

\(\displaystyle \frac {1}{2} d \left (-\frac {\sqrt {d-c^2 d x^2} \left (\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{2 c^2}-\frac {b \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{c}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 6308

\(\displaystyle \frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{2} d \left (\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\sqrt {d-c^2 d x^2} \left (\frac {(a+b \text {arccosh}(c x))^3}{6 b c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{2 c^2}-\frac {b \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{c}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

(x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/6 - (b*c*d*Sqrt[d - c^2 
*d*x^2]*((x^4*(a + b*ArcCosh[c*x]))/4 - (c^2*x^6*(a + b*ArcCosh[c*x]))/6 - 
 (b*c*(-1/3*(x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*((x^3*Sqrt[-1 + c*x]*S 
qrt[1 + c*x])/(4*c^2) + (3*((x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*c^2) + Arc 
Cosh[c*x]/(2*c^3)))/(4*c^2)))/3))/12))/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + 
(d*((x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/4 - (b*c*Sqrt[d - c^2 
*d*x^2]*((x^4*(a + b*ArcCosh[c*x]))/4 - (b*c*((x^3*Sqrt[-1 + c*x]*Sqrt[1 + 
 c*x])/(4*c^2) + (3*((x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*c^2) + ArcCosh[c* 
x]/(2*c^3)))/(4*c^2)))/4))/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (Sqrt[d - c^ 
2*d*x^2]*((x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(2*c^2) 
+ (a + b*ArcCosh[c*x])^3/(6*b*c^3) - (b*((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))/ 
c))/(4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/2
 

Defintions of rubi rules used

rule 25

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

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 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 111

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

rule 960

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.) 
*(x_)^(non2_.))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^( 
m + 1)*(a1 + b1*x^(n/2))^(p + 1)*((a2 + b2*x^(n/2))^(p + 1)/(b1*b2*e*(m + n 
*(p + 1) + 1))), x] - Simp[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/ 
(b1*b2*(m + n*(p + 1) + 1))   Int[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n 
/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] && EqQ[non2, 
 n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 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 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 6336

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

rule 6341

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + 
 (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc 
Cosh[c*x])^n/(f*(m + 2))), x] + (-Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/(Sq 
rt[1 + c*x]*Sqrt[-1 + c*x])]   Int[(f*x)^m*((a + b*ArcCosh[c*x])^n/(Sqrt[1 
+ c*x]*Sqrt[-1 + c*x])), x], x] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e* 
x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])]   Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x 
])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] 
 && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
 

rule 6345

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*Arc 
Cosh[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1))   Int[(f* 
x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*c*(n/(f*(m 
 + 2*p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]   Int[(f*x)^(m 
 + 1)*(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, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 
 0] && GtQ[p, 0] &&  !LtQ[m, -1]
 

rule 6354

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 
1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 
1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1*e2*( 
m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1)))   Int[(f*x)^(m 
 - 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*f 
*(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/( 
-1 + c*x)^p]   Int[(f*x)^(m - 1)*(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, d1, e1, d2, e2, f, 
p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IGtQ[m, 1] && N 
eQ[m + 2*p + 1, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1720\) vs. \(2(377)=754\).

Time = 0.50 (sec) , antiderivative size = 1721, normalized size of antiderivative = 3.90

method	result	size
default	\(\text {Expression too large to display}\)	\(1721\)
parts	\(\text {Expression too large to display}\)	\(1721\)

Input:

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

Output:

-1/6*a^2*x*(-c^2*d*x^2+d)^(5/2)/c^2/d+1/24*a^2/c^2*x*(-c^2*d*x^2+d)^(3/2)+ 
1/16*a^2/c^2*d*x*(-c^2*d*x^2+d)^(1/2)+1/16*a^2/c^2*d^2/(c^2*d)^(1/2)*arcta 
n((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b^2*(-1/48*(-d*(c^2*x^2-1))^(1/2)/ 
(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^3*arccosh(c*x)^3*d-1/6912*(-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))*(18*arccosh(c*x)^2-6*arccosh(c* 
x)+1)*d/(c*x+1)/c^3/(c*x-1)+1/1024*(-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^3/(c*x-1)+1/256*(-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))*(2*arcco 
sh(c*x)^2-2*arccosh(c*x)+1)*d/(c*x+1)/c^3/(c*x-1)+1/256*(-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)*(2*arccosh(c*x)^2+2*arccosh(c*x)+1)*d/(c*x+1)/c^3/(c*x-1)+ 
1/1024*(-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)*(8*arccosh(c*x)^2+4*arccosh(c*x)+1)*d/(c*x+1)/c^3/(c*x-1)- 
1/6912*(-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...
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {\sqrt {d}\, d \left (3 \mathit {asin} \left (c x \right ) a^{2}-8 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{5} x^{5}+14 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{3} x^{3}-3 \sqrt {-c^{2} x^{2}+1}\, a^{2} c x -96 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{4}d x \right ) a b \,c^{5}+96 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{2}d x \right ) a b \,c^{3}-48 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{5}+48 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}\right )}{48 c^{3}} \] Input:

int(x^2*(-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 - 8*sqrt( - c**2*x**2 + 1)*a**2*c**5*x**5 + 1 
4*sqrt( - c**2*x**2 + 1)*a**2*c**3*x**3 - 3*sqrt( - c**2*x**2 + 1)*a**2*c* 
x - 96*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**4,x)*a*b*c**5 + 96*int(sqr 
t( - c**2*x**2 + 1)*acosh(c*x)*x**2,x)*a*b*c**3 - 48*int(sqrt( - c**2*x**2 
 + 1)*acosh(c*x)**2*x**4,x)*b**2*c**5 + 48*int(sqrt( - c**2*x**2 + 1)*acos 
h(c*x)**2*x**2,x)*b**2*c**3))/(48*c**3)