Integrand size = 29, antiderivative size = 841 \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {35 b^2 d^2 x \sqrt {d-c^2 d x^2}}{9216 c^2}+\frac {215 b^2 d^2 x^3 \sqrt {d-c^2 d x^2}}{13824}-\frac {5}{864} b^2 c^2 d^2 x^5 \sqrt {d-c^2 d x^2}+\frac {73 b^2 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{12288 c^2 (1-c x) (1+c x)}+\frac {73 b^2 d^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{18432 (1-c x) (1+c x)}-\frac {43 b^2 c^2 d^2 x^5 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{4608 (1-c x) (1+c x)}+\frac {b^2 c^4 d^2 x^7 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{256 (1-c x) (1+c x)}+\frac {35 b^2 d^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{9216 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{128 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{384 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{144 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{384 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {73 b^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2} \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{12288 c^3 (1-c x) (1+c x)} \]
5/48*d*x^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2+1/8*x^3*(-c^2*d*x^2+d )^(5/2)*(a+b*arccosh(c*x))^2+35/9216*b^2*d^2*x*(-c^2*d*x^2+d)^(1/2)/c^2+21 5/13824*b^2*d^2*x^3*(-c^2*d*x^2+d)^(1/2)-5/864*b^2*c^2*d^2*x^5*(-c^2*d*x^2 +d)^(1/2)+73/12288*b^2*d^2*x*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/c^2/(-c*x+1 )/(c*x+1)+73/18432*b^2*d^2*x^3*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/(-c*x+1)/ (c*x+1)-43/4608*b^2*c^2*d^2*x^5*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/(-c*x+1) /(c*x+1)+1/256*b^2*c^4*d^2*x^7*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/(-c*x+1)/ (c*x+1)-5/128*d^2*x*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^2+5/64*d^2 *x^3*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)+35/9216*b^2*d^2*arccosh(c*x )*(-c^2*d*x^2+d)^(1/2)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+5/128*b*d^2*x^2*(a+ b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-59/384* b*c*d^2*x^4*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^ (1/2)+17/144*b*c^3*d^2*x^6*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/(c*x-1) ^(1/2)/(c*x+1)^(1/2)-1/32*b*c^5*d^2*x^8*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^ (1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-5/384*d^2*(a+b*arccosh(c*x))^3*(-c^2*d*x ^2+d)^(1/2)/b/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)-73/12288*b^2*d^2*arctanh(c*x /(c^2*x^2-1)^(1/2))*(c^2*x^2-1)^(1/2)*(-c^2*d*x^2+d)^(1/2)/c^3/(-c*x+1)/(c *x+1)
Time = 5.92 (sec) , antiderivative size = 910, normalized size of antiderivative = 1.08 \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {d^2 \left (34560 a^2 c x \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}+34560 a^2 c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}-271872 a^2 c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}-271872 a^2 c^4 x^4 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}+313344 a^2 c^5 x^5 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}+313344 a^2 c^6 x^6 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}-110592 a^2 c^7 x^7 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}-110592 a^2 c^8 x^8 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}+11520 b^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)^3+34560 a^2 \sqrt {d} \sqrt {\frac {-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 )+34560 a^2 c \sqrt {d} x \sqrt {\frac {-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 )+13824 a b \sqrt {d-c^2 d x^2} \cosh (2 \text {arccosh}(c x))+3456 a b \sqrt {d-c^2 d x^2} \cosh (4 \text {arccosh}(c x))-1536 a b \sqrt {d-c^2 d x^2} \cosh (6 \text {arccosh}(c x))+216 a b \sqrt {d-c^2 d x^2} \cosh (8 \text {arccosh}(c x))-6912 b^2 \sqrt {d-c^2 d x^2} \sinh (2 \text {arccosh}(c x))-864 b^2 \sqrt {d-c^2 d x^2} \sinh (4 \text {arccosh}(c x))+256 b^2 \sqrt {d-c^2 d x^2} \sinh (6 \text {arccosh}(c x))-27 b^2 \sqrt {d-c^2 d x^2} \sinh (8 \text {arccosh}(c x))+24 b \sqrt {d-c^2 d x^2} \text {arccosh}(c x) (576 b \cosh (2 \text {arccosh}(c x))+144 b \cosh (4 \text {arccosh}(c x))-64 b \cosh (6 \text {arccosh}(c x))+9 b \cosh (8 \text {arccosh}(c x))-1152 a \sinh (2 \text {arccosh}(c x))-576 a \sinh (4 \text {arccosh}(c x))+384 a \sinh (6 \text {arccosh}(c x))-72 a \sinh (8 \text {arccosh}(c x)))-288 b \sqrt {d-c^2 d x^2} \text {arccosh}(c x)^2 (-120 a+48 b \sinh (2 \text {arccosh}(c x))+24 b \sinh (4 \text {arccosh}(c x))-16 b \sinh (6 \text {arccosh}(c x))+3 b \sinh (8 \text {arccosh}(c x)))\right )}{884736 c^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \]
-1/884736*(d^2*(34560*a^2*c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^ 2] + 34560*a^2*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] - 27 1872*a^2*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] - 271872*a ^2*c^4*x^4*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] + 313344*a^2*c^5 *x^5*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] + 313344*a^2*c^6*x^6*S qrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] - 110592*a^2*c^7*x^7*Sqrt[(- 1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] - 110592*a^2*c^8*x^8*Sqrt[(-1 + c* x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] + 11520*b^2*Sqrt[d - c^2*d*x^2]*ArcCosh[ c*x]^3 + 34560*a^2*Sqrt[d]*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 34560*a^2*c*Sqrt[d]*x*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 13824*a*b*Sqrt[d - c^2*d*x^2]*Cosh[2*ArcCosh[c*x]] + 3456*a*b*Sqrt[d - c^2*d*x^2]*Cosh[4*ArcCosh[c*x]] - 1536*a*b*Sqrt[d - c^2*d*x^2]*Cosh[6*ArcC osh[c*x]] + 216*a*b*Sqrt[d - c^2*d*x^2]*Cosh[8*ArcCosh[c*x]] - 6912*b^2*Sq rt[d - c^2*d*x^2]*Sinh[2*ArcCosh[c*x]] - 864*b^2*Sqrt[d - c^2*d*x^2]*Sinh[ 4*ArcCosh[c*x]] + 256*b^2*Sqrt[d - c^2*d*x^2]*Sinh[6*ArcCosh[c*x]] - 27*b^ 2*Sqrt[d - c^2*d*x^2]*Sinh[8*ArcCosh[c*x]] + 24*b*Sqrt[d - c^2*d*x^2]*ArcC osh[c*x]*(576*b*Cosh[2*ArcCosh[c*x]] + 144*b*Cosh[4*ArcCosh[c*x]] - 64*b*C osh[6*ArcCosh[c*x]] + 9*b*Cosh[8*ArcCosh[c*x]] - 1152*a*Sinh[2*ArcCosh[c*x ]] - 576*a*Sinh[4*ArcCosh[c*x]] + 384*a*Sinh[6*ArcCosh[c*x]] - 72*a*Sin...
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.
| \(\displaystyle \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6345 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x^3 (1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))dx}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x^3 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))dx}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6336 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-b c \int \frac {x^4 \left (3 c^4 x^4-8 c^2 x^2+6\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {1}{24} b c \int \frac {x^4 \left (3 c^4 x^4-8 c^2 x^2+6\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 1905 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \int \frac {x^4 \left (3 c^4 x^4-8 c^2 x^2+6\right )}{\sqrt {c^2 x^2-1}}dx}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 1590 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\int \frac {c^2 x^4 \left (48-43 c^2 x^2\right )}{\sqrt {c^2 x^2-1}}dx}{8 c^2}+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \int \frac {x^4 \left (48-43 c^2 x^2\right )}{\sqrt {c^2 x^2-1}}dx+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \int \frac {x^4}{\sqrt {c^2 x^2-1}}dx-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \int \frac {x^2}{\sqrt {c^2 x^2-1}}dx}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {c^2 x^2-1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\int \frac {1}{1-\frac {c^2 x^2}{c^2 x^2-1}}d\frac {x}{\sqrt {c^2 x^2-1}}}{2 c^2}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6345 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx+\frac {b c d \sqrt {d-c^2 d x^2} \int -x^3 (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \int x^3 (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \int x^3 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6336 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-b c \int \frac {x^4 \left (3-2 c^2 x^2\right )}{12 \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{12} b c \int \frac {x^4 \left (3-2 c^2 x^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 960 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{12} b c \left (\frac {4}{3} \int \frac {x^4}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {\int \frac {3 x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \int \frac {x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6341 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \int x^3 (a+b \text {arccosh}(c x))dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \int \frac {x^4}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {\int \frac {3 x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \int \frac {x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6354 |
| \(\displaystyle \frac {1}{8} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 x^3-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 (a+b \text {arccosh}(c x)) x^8-\frac {1}{3} c^2 (a+b \text {arccosh}(c x)) x^6+\frac {1}{4} (a+b \text {arccosh}(c x)) x^4-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {3}{8} c^2 \sqrt {c^2 x^2-1} x^7+\frac {1}{8} \left (\frac {73}{6} \left (\frac {\sqrt {c^2 x^2-1} x^3}{4 c^2}+\frac {3 \left (\frac {\sqrt {c^2 x^2-1} x}{2 c^2}+\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}\right )}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{8} d \left (\frac {1}{6} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 x^3-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 (a+b \text {arccosh}(c x)) x^6+\frac {1}{4} (a+b \text {arccosh}(c x)) x^4-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^3}{4 c^2}+\frac {3 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x}{2 c^2}+\frac {\text {arccosh}(c x)}{2 c^3}\right )}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} d \left (\frac {1}{4} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 x^3-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^3}{4 c^2}+\frac {3 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x}{2 c^2}+\frac {\text {arccosh}(c x)}{2 c^3}\right )}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \left (\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{2 c^2}-\frac {b \int x (a+b \text {arccosh}(c x))dx}{c}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {1}{8} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 x^3-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 (a+b \text {arccosh}(c x)) x^8-\frac {1}{3} c^2 (a+b \text {arccosh}(c x)) x^6+\frac {1}{4} (a+b \text {arccosh}(c x)) x^4-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {3}{8} c^2 \sqrt {c^2 x^2-1} x^7+\frac {1}{8} \left (\frac {73}{6} \left (\frac {\sqrt {c^2 x^2-1} x^3}{4 c^2}+\frac {3 \left (\frac {\sqrt {c^2 x^2-1} x}{2 c^2}+\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}\right )}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{8} d \left (\frac {1}{6} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 x^3-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 (a+b \text {arccosh}(c x)) x^6+\frac {1}{4} (a+b \text {arccosh}(c x)) x^4-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^3}{4 c^2}+\frac {3 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x}{2 c^2}+\frac {\text {arccosh}(c x)}{2 c^3}\right )}{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}}+\frac {1}{2} d \left (\frac {1}{4} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 x^3-\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 {\sqrt {c x-1} \sqrt {c x+1} x^3}{4 c^2}+\frac {3 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x}{2 c^2}+\frac {\text {arccosh}(c x)}{2 c^3}\right )}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \left (\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 \int \frac {x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )}{c}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
3.2.87.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 0]
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]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[c^p*(f*x)^(m + 4*p - 1)*((d + e*x^2)^ (q + 1)/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1))), x] + Simp[1/(e*(m + 4*p + 2*q + 1)) Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + b*x^2 + c*x^4)^p - c^p*x^(4*p)) - d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]
Int[((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_)^(non2_.))^(q_.)*((d2_) + (e2_.) *(x_)^(non2_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x _Symbol] :> Simp[(d1 + e1*x^(n/2))^FracPart[q]*((d2 + e2*x^(n/2))^FracPart[ q]/(d1*d2 + e1*e2*x^n)^FracPart[q]) Int[(f*x)^m*(d1*d2 + e1*e2*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[non2, n/2] && EqQ[d2*e1 + d1*e2, 0]
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]
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]
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]
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])
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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(2527\) vs. \(2(741)=1482\).
Time = 0.88 (sec) , antiderivative size = 2528, normalized size of antiderivative = 3.01
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2528\) |
| parts | \(\text {Expression too large to display}\) | \(2528\) |
-1/8*a^2*x*(-c^2*d*x^2+d)^(7/2)/c^2/d+1/48*a^2/c^2*x*(-c^2*d*x^2+d)^(5/2)+ 5/192*a^2/c^2*d*x*(-c^2*d*x^2+d)^(3/2)+5/128*a^2/c^2*d^2*x*(-c^2*d*x^2+d)^ (1/2)+5/128*a^2/c^2*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d )^(1/2))+b^2*(-5/384*(-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^2+1/65536*(-d*(c^2*x^2-1))^(1/2)*(128*c^9*x^9-320*c^7*x ^7+128*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^8*x^8+272*c^5*x^5-256*(c*x+1)^(1/2)*( c*x-1)^(1/2)*c^6*x^6-88*c^3*x^3+160*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+8* c*x-32*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(3 2*arccosh(c*x)^2-8*arccosh(c*x)+1)*d^2/(c*x+1)/c^3/(c*x-1)-1/6912*(-d*(c^2 *x^2-1))^(1/2)*(32*c^7*x^7-64*c^5*x^5+32*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^6*x ^6+38*c^3*x^3-48*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4-6*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^2/(c*x+1)/c^3/(c*x-1)+1/2048*(-d*(c^2*x^2-1))^(1/2)*(8*c ^5*x^5-12*c^3*x^3+8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+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^2/(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*arccosh(c*x)^2-2*arccosh(c*x)+1)*d^2/(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^2...
\[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
integral((a^2*c^4*d^2*x^6 - 2*a^2*c^2*d^2*x^4 + a^2*d^2*x^2 + (b^2*c^4*d^2 *x^6 - 2*b^2*c^2*d^2*x^4 + b^2*d^2*x^2)*arccosh(c*x)^2 + 2*(a*b*c^4*d^2*x^ 6 - 2*a*b*c^2*d^2*x^4 + a*b*d^2*x^2)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d), x )
Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\text {Timed out} \]
\[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
1/384*(8*(-c^2*d*x^2 + d)^(5/2)*x/c^2 - 48*(-c^2*d*x^2 + d)^(7/2)*x/(c^2*d ) + 10*(-c^2*d*x^2 + d)^(3/2)*d*x/c^2 + 15*sqrt(-c^2*d*x^2 + d)*d^2*x/c^2 + 15*d^(5/2)*arcsin(c*x)/c^3)*a^2 + integrate((-c^2*d*x^2 + d)^(5/2)*b^2*x ^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2 + 2*(-c^2*d*x^2 + d)^(5/2)*a*b *x^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)), x)
\[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]