Integrand size = 27, antiderivative size = 371 \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2}}{256 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2}}{768 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2}}{288 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{256 b c^3 \sqrt {-1+c x} \sqrt {1+c x}} \]
5/48*d*x^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))+1/8*x^3*(-c^2*d*x^2+d)^ (5/2)*(a+b*arccosh(c*x))-5/128*d^2*x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/ 2)/c^2+5/64*d^2*x^3*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)+5/256*b*d^2*x^ 2*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-59/768*b*c*d^2*x^4*(- c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+17/288*b*c^3*d^2*x^6*(-c^2* d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/64*b*c^5*d^2*x^8*(-c^2*d*x^2+ d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-5/256*d^2*(a+b*arccosh(c*x))^2*(-c^2* d*x^2+d)^(1/2)/b/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 3.70 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.12 \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {192 a c d^2 x \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2} \left (-15+118 c^2 x^2-136 c^4 x^4+48 c^6 x^6\right )-2880 a d^{5/2} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-576 b d^2 \sqrt {d-c^2 d x^2} \left (8 \text {arccosh}(c x)^2+\cosh (4 \text {arccosh}(c x))-4 \text {arccosh}(c x) \sinh (4 \text {arccosh}(c x))\right )-64 b d^2 \sqrt {d-c^2 d x^2} \left (-72 \text {arccosh}(c x)^2+18 \cosh (2 \text {arccosh}(c x))-9 \cosh (4 \text {arccosh}(c x))-2 \cosh (6 \text {arccosh}(c x))+12 \text {arccosh}(c x) (-3 \sinh (2 \text {arccosh}(c x))+3 \sinh (4 \text {arccosh}(c x))+\sinh (6 \text {arccosh}(c x)))\right )+b d^2 \sqrt {d-c^2 d x^2} \left (-1440 \text {arccosh}(c x)^2+576 \cosh (2 \text {arccosh}(c x))-144 \cosh (4 \text {arccosh}(c x))-64 \cosh (6 \text {arccosh}(c x))-9 \cosh (8 \text {arccosh}(c x))+24 \text {arccosh}(c x) (-48 \sinh (2 \text {arccosh}(c x))+24 \sinh (4 \text {arccosh}(c x))+16 \sinh (6 \text {arccosh}(c x))+3 \sinh (8 \text {arccosh}(c x)))\right )}{73728 c^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \]
(192*a*c*d^2*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(- 15 + 118*c^2*x^2 - 136*c^4*x^4 + 48*c^6*x^6) - 2880*a*d^(5/2)*Sqrt[(-1 + c *x)/(1 + c*x)]*(1 + c*x)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c ^2*x^2))] - 576*b*d^2*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCo sh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]) - 64*b*d^2*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]] + Sinh[6*ArcCosh[c*x]])) + b*d^2*Sqrt[d - c^2*d*x^2]* (-1440*ArcCosh[c*x]^2 + 576*Cosh[2*ArcCosh[c*x]] - 144*Cosh[4*ArcCosh[c*x] ] - 64*Cosh[6*ArcCosh[c*x]] - 9*Cosh[8*ArcCosh[c*x]] + 24*ArcCosh[c*x]*(-4 8*Sinh[2*ArcCosh[c*x]] + 24*Sinh[4*ArcCosh[c*x]] + 16*Sinh[6*ArcCosh[c*x]] + 3*Sinh[8*ArcCosh[c*x]])))/(73728*c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c* x))
Time = 1.57 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.01, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {6345, 82, 243, 49, 2009, 6345, 25, 82, 244, 2009, 6341, 15, 6354, 15, 6308}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6345 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x^3 (1-c x)^2 (c x+1)^2dx}{8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 82 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x^3 \left (1-c^2 x^2\right )^2dx}{8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x^2 \left (1-c^2 x^2\right )^2dx^2}{16 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \left (c^4 x^6-2 c^2 x^4+x^2\right )dx^2}{16 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))dx+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6345 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx+\frac {b c d \sqrt {d-c^2 d x^2} \int -x^3 (1-c x) (c x+1)dx}{6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx-\frac {b c d \sqrt {d-c^2 d x^2} \int x^3 (1-c x) (c x+1)dx}{6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 82 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx-\frac {b c d \sqrt {d-c^2 d x^2} \int x^3 \left (1-c^2 x^2\right )dx}{6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx-\frac {b c d \sqrt {d-c^2 d x^2} \int \left (x^3-c^2 x^5\right )dx}{6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (\frac {x^4}{4}-\frac {c^2 x^6}{6}\right ) \sqrt {d-c^2 d x^2}}{6 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6341 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \int x^3dx}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (\frac {x^4}{4}-\frac {c^2 x^6}{6}\right ) \sqrt {d-c^2 d x^2}}{6 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (\frac {x^4}{4}-\frac {c^2 x^6}{6}\right ) \sqrt {d-c^2 d x^2}}{6 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6354 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (-\frac {\sqrt {d-c^2 d x^2} \left (\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}-\frac {b \int xdx}{2 c}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 c^2}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (\frac {x^4}{4}-\frac {c^2 x^6}{6}\right ) \sqrt {d-c^2 d x^2}}{6 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (-\frac {\sqrt {d-c^2 d x^2} \left (\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (\frac {x^4}{4}-\frac {c^2 x^6}{6}\right ) \sqrt {d-c^2 d x^2}}{6 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6308 |
| \(\displaystyle \frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))+\frac {5}{8} d \left (\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))+\frac {1}{2} d \left (\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\sqrt {d-c^2 d x^2} \left (\frac {(a+b \text {arccosh}(c x))^2}{4 b c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {b c d \left (\frac {x^4}{4}-\frac {c^2 x^6}{6}\right ) \sqrt {d-c^2 d x^2}}{6 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}\) |
-1/16*(b*c*d^2*Sqrt[d - c^2*d*x^2]*(x^4/2 - (2*c^2*x^6)/3 + (c^4*x^8)/4))/ (Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh [c*x]))/8 + (5*d*(-1/6*(b*c*d*Sqrt[d - c^2*d*x^2]*(x^4/4 - (c^2*x^6)/6))/( Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[ c*x]))/6 + (d*(-1/16*(b*c*x^4*Sqrt[d - c^2*d*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/4 - (Sqrt[d - c^2 *d*x^2]*(-1/4*(b*x^2)/c + (x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c *x]))/(2*c^2) + (a + b*ArcCosh[c*x])^2/(4*b*c^3)))/(4*Sqrt[-1 + c*x]*Sqrt[ 1 + c*x])))/2))/8
3.1.88.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_) )^(p_.), x_] :> Int[(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && IntegerQ[m]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
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]
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. \(1288\) vs. \(2(315)=630\).
Time = 0.74 (sec) , antiderivative size = 1289, normalized size of antiderivative = 3.47
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1289\) |
| parts | \(\text {Expression too large to display}\) | \(1289\) |
-1/8*a*x*(-c^2*d*x^2+d)^(7/2)/c^2/d+1/48*a/c^2*x*(-c^2*d*x^2+d)^(5/2)+5/19 2*a/c^2*d*x*(-c^2*d*x^2+d)^(3/2)+5/128*a/c^2*d^2*x*(-c^2*d*x^2+d)^(1/2)+5/ 128*a/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 *(-5/256*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^3*arccosh(c* x)^2*d^2+1/16384*(-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))*(-1+8*arccosh( c*x))*d^2/(c*x+1)/c^3/(c*x-1)-1/2304*(-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))*(-1+6*arccosh(c*x))*d^2/(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*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))*(-1+4*arccosh(c*x))*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))*(-1+2*arccosh(c*x))*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)*(1+2*arccosh(c*x))*d^2/(c*x+1)/c^3/(c*x-1)+1/1 024*(-d*(c^2*x^2-1))^(1/2)*(-8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+8*c^...
\[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2} \,d x } \]
integral((a*c^4*d^2*x^6 - 2*a*c^2*d^2*x^4 + a*d^2*x^2 + (b*c^4*d^2*x^6 - 2 *b*c^2*d^2*x^4 + b*d^2*x^2)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d), x)
Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\text {Timed out} \]
\[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2} \,d x } \]
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 + b*integrate((-c^2*d*x^2 + d)^(5/2)*x^2*l og(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)), x)
\[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2} \,d x } \]
Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]