Integrand size = 31, antiderivative size = 31 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\text {Int}\left ((f x)^m \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2,x\right ) \] Output:
Defer(Int)((f*x)^m*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x)
Not integrable
Time = 2.13 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\int (f x)^m \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx \] Input:
Integrate[(f*x)^m*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2,x]
Output:
Integrate[(f*x)^m*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2, x]
Not integrable
Time = 5.43 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \left (d-c^2 d x^2\right )^{5/2} (f x)^m (a+b \text {arccosh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6345 |
| \(\displaystyle -\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int (f x)^{m+1} (1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))dx}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 d \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx}{m+6}+\frac {\left (d-c^2 d x^2\right )^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+6)}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle -\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int (f x)^{m+1} \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))dx}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 d \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx}{m+6}+\frac {\left (d-c^2 d x^2\right )^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+6)}\) |
| \(\Big \downarrow \) 6336 |
| \(\displaystyle \frac {5 d \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx}{m+6}-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-b c \int \frac {(f x)^{m+2} \left (\frac {c^4 x^4}{m+6}-\frac {2 c^2 x^2}{m+4}+\frac {1}{m+2}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {c^4 (f x)^{m+6} (a+b \text {arccosh}(c x))}{f^5 (m+6)}-\frac {2 c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+6)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 d \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx}{m+6}-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \int \frac {(f x)^{m+2} \left (\frac {c^4 x^4}{m+6}-\frac {2 c^2 x^2}{m+4}+\frac {1}{m+2}\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx}{f}+\frac {c^4 (f x)^{m+6} (a+b \text {arccosh}(c x))}{f^5 (m+6)}-\frac {2 c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+6)}\) |
| \(\Big \downarrow \) 1905 |
| \(\displaystyle \frac {5 d \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx}{m+6}-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \int \frac {(f x)^{m+2} \left (\frac {c^4 x^4}{m+6}-\frac {2 c^2 x^2}{m+4}+\frac {1}{m+2}\right )}{\sqrt {c^2 x^2-1}}dx}{f \sqrt {c x-1} \sqrt {c x+1}}+\frac {c^4 (f x)^{m+6} (a+b \text {arccosh}(c x))}{f^5 (m+6)}-\frac {2 c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+6)}\) |
| \(\Big \downarrow \) 1590 |
| \(\displaystyle \frac {5 d \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx}{m+6}-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\int \frac {c^2 (f x)^{m+2} \left (\frac {m+6}{m+2}-\frac {c^2 \left (m^2+15 m+52\right ) x^2}{(m+4) (m+6)}\right )}{\sqrt {c^2 x^2-1}}dx}{c^2 (m+6)}+\frac {c^2 \sqrt {c^2 x^2-1} (f x)^{m+5}}{f^3 (m+6)^2}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}+\frac {c^4 (f x)^{m+6} (a+b \text {arccosh}(c x))}{f^5 (m+6)}-\frac {2 c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+6)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 d \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx}{m+6}-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\int \frac {(f x)^{m+2} \left (\frac {m+6}{m+2}-\frac {c^2 \left (m^2+15 m+52\right ) x^2}{(m+4) (m+6)}\right )}{\sqrt {c^2 x^2-1}}dx}{m+6}+\frac {c^2 \sqrt {c^2 x^2-1} (f x)^{m+5}}{f^3 (m+6)^2}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}+\frac {c^4 (f x)^{m+6} (a+b \text {arccosh}(c x))}{f^5 (m+6)}-\frac {2 c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+6)}\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {5 d \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx}{m+6}-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\left (15 m^2+130 m+264\right ) \int \frac {(f x)^{m+2}}{\sqrt {c^2 x^2-1}}dx}{(m+2) (m+4)^2 (m+6)}-\frac {\left (m^2+15 m+52\right ) \sqrt {c^2 x^2-1} (f x)^{m+3}}{f (m+4)^2 (m+6)}}{m+6}+\frac {c^2 \sqrt {c^2 x^2-1} (f x)^{m+5}}{f^3 (m+6)^2}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}+\frac {c^4 (f x)^{m+6} (a+b \text {arccosh}(c x))}{f^5 (m+6)}-\frac {2 c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+6)}\) |
| \(\Big \downarrow \) 279 |
| \(\displaystyle \frac {5 d \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx}{m+6}-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\left (15 m^2+130 m+264\right ) \sqrt {1-c^2 x^2} \int \frac {(f x)^{m+2}}{\sqrt {1-c^2 x^2}}dx}{(m+2) (m+4)^2 (m+6) \sqrt {c^2 x^2-1}}-\frac {\left (m^2+15 m+52\right ) \sqrt {c^2 x^2-1} (f x)^{m+3}}{f (m+4)^2 (m+6)}}{m+6}+\frac {c^2 \sqrt {c^2 x^2-1} (f x)^{m+5}}{f^3 (m+6)^2}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}+\frac {c^4 (f x)^{m+6} (a+b \text {arccosh}(c x))}{f^5 (m+6)}-\frac {2 c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+6)}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle \frac {5 d \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx}{m+6}+\frac {\left (d-c^2 d x^2\right )^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+6)}-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {c^4 (f x)^{m+6} (a+b \text {arccosh}(c x))}{f^5 (m+6)}-\frac {2 c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {c^2 \sqrt {c^2 x^2-1} (f x)^{m+5}}{f^3 (m+6)^2}+\frac {\frac {\left (15 m^2+130 m+264\right ) \sqrt {1-c^2 x^2} (f x)^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 (m+6) \sqrt {c^2 x^2-1}}-\frac {\left (m^2+15 m+52\right ) \sqrt {c^2 x^2-1} (f x)^{m+3}}{f (m+4)^2 (m+6)}}{m+6}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6345 |
| \(\displaystyle \frac {5 d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \int -(f x)^{m+1} (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 d \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx}{m+4}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+4)}\right )}{m+6}+\frac {\left (d-c^2 d x^2\right )^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+6)}-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {c^4 (f x)^{m+6} (a+b \text {arccosh}(c x))}{f^5 (m+6)}-\frac {2 c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {c^2 \sqrt {c^2 x^2-1} (f x)^{m+5}}{f^3 (m+6)^2}+\frac {\frac {\left (15 m^2+130 m+264\right ) \sqrt {1-c^2 x^2} (f x)^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 (m+6) \sqrt {c^2 x^2-1}}-\frac {\left (m^2+15 m+52\right ) \sqrt {c^2 x^2-1} (f x)^{m+3}}{f (m+4)^2 (m+6)}}{m+6}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5 d \left (-\frac {2 b c d \sqrt {d-c^2 d x^2} \int (f x)^{m+1} (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 d \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx}{m+4}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+4)}\right )}{m+6}+\frac {\left (d-c^2 d x^2\right )^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+6)}-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {c^4 (f x)^{m+6} (a+b \text {arccosh}(c x))}{f^5 (m+6)}-\frac {2 c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {c^2 \sqrt {c^2 x^2-1} (f x)^{m+5}}{f^3 (m+6)^2}+\frac {\frac {\left (15 m^2+130 m+264\right ) \sqrt {1-c^2 x^2} (f x)^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 (m+6) \sqrt {c^2 x^2-1}}-\frac {\left (m^2+15 m+52\right ) \sqrt {c^2 x^2-1} (f x)^{m+3}}{f (m+4)^2 (m+6)}}{m+6}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle \frac {5 d \left (-\frac {2 b c d \sqrt {d-c^2 d x^2} \int (f x)^{m+1} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))dx}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 d \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx}{m+4}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+4)}\right )}{m+6}+\frac {\left (d-c^2 d x^2\right )^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+6)}-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {c^4 (f x)^{m+6} (a+b \text {arccosh}(c x))}{f^5 (m+6)}-\frac {2 c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {c^2 \sqrt {c^2 x^2-1} (f x)^{m+5}}{f^3 (m+6)^2}+\frac {\frac {\left (15 m^2+130 m+264\right ) \sqrt {1-c^2 x^2} (f x)^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 (m+6) \sqrt {c^2 x^2-1}}-\frac {\left (m^2+15 m+52\right ) \sqrt {c^2 x^2-1} (f x)^{m+3}}{f (m+4)^2 (m+6)}}{m+6}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6336 |
| \(\displaystyle \frac {5 d \left (-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-b c \int \frac {(f x)^{m+2} \left (\frac {1}{m+2}-\frac {c^2 x^2}{m+4}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 d \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx}{m+4}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+4)}\right )}{m+6}+\frac {\left (d-c^2 d x^2\right )^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+6)}-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {c^4 (f x)^{m+6} (a+b \text {arccosh}(c x))}{f^5 (m+6)}-\frac {2 c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {c^2 \sqrt {c^2 x^2-1} (f x)^{m+5}}{f^3 (m+6)^2}+\frac {\frac {\left (15 m^2+130 m+264\right ) \sqrt {1-c^2 x^2} (f x)^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 (m+6) \sqrt {c^2 x^2-1}}-\frac {\left (m^2+15 m+52\right ) \sqrt {c^2 x^2-1} (f x)^{m+3}}{f (m+4)^2 (m+6)}}{m+6}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 d \left (-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {b c \int \frac {(f x)^{m+2} \left (\frac {1}{m+2}-\frac {c^2 x^2}{m+4}\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx}{f}-\frac {c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 d \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx}{m+4}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+4)}\right )}{m+6}+\frac {\left (d-c^2 d x^2\right )^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+6)}-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {c^4 (f x)^{m+6} (a+b \text {arccosh}(c x))}{f^5 (m+6)}-\frac {2 c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {c^2 \sqrt {c^2 x^2-1} (f x)^{m+5}}{f^3 (m+6)^2}+\frac {\frac {\left (15 m^2+130 m+264\right ) \sqrt {1-c^2 x^2} (f x)^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 (m+6) \sqrt {c^2 x^2-1}}-\frac {\left (m^2+15 m+52\right ) \sqrt {c^2 x^2-1} (f x)^{m+3}}{f (m+4)^2 (m+6)}}{m+6}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 960 |
| \(\displaystyle \frac {5 d \left (-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {b c \left (\frac {(3 m+10) \int \frac {(f x)^{m+2}}{\sqrt {c x-1} \sqrt {c x+1}}dx}{(m+2) (m+4)^2}-\frac {\sqrt {c x-1} \sqrt {c x+1} (f x)^{m+3}}{f (m+4)^2}\right )}{f}-\frac {c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 d \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx}{m+4}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+4)}\right )}{m+6}+\frac {\left (d-c^2 d x^2\right )^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+6)}-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {c^4 (f x)^{m+6} (a+b \text {arccosh}(c x))}{f^5 (m+6)}-\frac {2 c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {c^2 \sqrt {c^2 x^2-1} (f x)^{m+5}}{f^3 (m+6)^2}+\frac {\frac {\left (15 m^2+130 m+264\right ) \sqrt {1-c^2 x^2} (f x)^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 (m+6) \sqrt {c^2 x^2-1}}-\frac {\left (m^2+15 m+52\right ) \sqrt {c^2 x^2-1} (f x)^{m+3}}{f (m+4)^2 (m+6)}}{m+6}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 136 |
| \(\displaystyle \frac {5 d \left (-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {b c \left (\frac {(3 m+10) \sqrt {c^2 x^2-1} \int \frac {(f x)^{m+2}}{\sqrt {c^2 x^2-1}}dx}{(m+2) (m+4)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} (f x)^{m+3}}{f (m+4)^2}\right )}{f}-\frac {c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 d \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx}{m+4}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+4)}\right )}{m+6}+\frac {\left (d-c^2 d x^2\right )^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+6)}-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {c^4 (f x)^{m+6} (a+b \text {arccosh}(c x))}{f^5 (m+6)}-\frac {2 c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {c^2 \sqrt {c^2 x^2-1} (f x)^{m+5}}{f^3 (m+6)^2}+\frac {\frac {\left (15 m^2+130 m+264\right ) \sqrt {1-c^2 x^2} (f x)^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 (m+6) \sqrt {c^2 x^2-1}}-\frac {\left (m^2+15 m+52\right ) \sqrt {c^2 x^2-1} (f x)^{m+3}}{f (m+4)^2 (m+6)}}{m+6}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 279 |
| \(\displaystyle \frac {5 d \left (-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {b c \left (\frac {(3 m+10) \sqrt {1-c^2 x^2} \int \frac {(f x)^{m+2}}{\sqrt {1-c^2 x^2}}dx}{(m+2) (m+4)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} (f x)^{m+3}}{f (m+4)^2}\right )}{f}-\frac {c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 d \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx}{m+4}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+4)}\right )}{m+6}+\frac {\left (d-c^2 d x^2\right )^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+6)}-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {c^4 (f x)^{m+6} (a+b \text {arccosh}(c x))}{f^5 (m+6)}-\frac {2 c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {c^2 \sqrt {c^2 x^2-1} (f x)^{m+5}}{f^3 (m+6)^2}+\frac {\frac {\left (15 m^2+130 m+264\right ) \sqrt {1-c^2 x^2} (f x)^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 (m+6) \sqrt {c^2 x^2-1}}-\frac {\left (m^2+15 m+52\right ) \sqrt {c^2 x^2-1} (f x)^{m+3}}{f (m+4)^2 (m+6)}}{m+6}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle \frac {5 d \left (\frac {3 d \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx}{m+4}-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \left (\frac {(3 m+10) \sqrt {1-c^2 x^2} (f x)^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} (f x)^{m+3}}{f (m+4)^2}\right )}{f}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+4)}\right )}{m+6}+\frac {\left (d-c^2 d x^2\right )^{5/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+6)}-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {c^4 (f x)^{m+6} (a+b \text {arccosh}(c x))}{f^5 (m+6)}-\frac {2 c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {c^2 \sqrt {c^2 x^2-1} (f x)^{m+5}}{f^3 (m+6)^2}+\frac {\frac {\left (15 m^2+130 m+264\right ) \sqrt {1-c^2 x^2} (f x)^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 (m+6) \sqrt {c^2 x^2-1}}-\frac {\left (m^2+15 m+52\right ) \sqrt {c^2 x^2-1} (f x)^{m+3}}{f (m+4)^2 (m+6)}}{m+6}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6345 |
| \(\displaystyle \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 (f x)^{m+1}}{f (m+6)}-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {(a+b \text {arccosh}(c x)) (f x)^{m+2}}{f (m+2)}-\frac {2 c^2 (a+b \text {arccosh}(c x)) (f x)^{m+4}}{f^3 (m+4)}+\frac {c^4 (a+b \text {arccosh}(c x)) (f x)^{m+6}}{f^5 (m+6)}-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {c^2 \sqrt {c^2 x^2-1} (f x)^{m+5}}{f^3 (m+6)^2}+\frac {\frac {\left (15 m^2+130 m+264\right ) (f x)^{m+3} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 (m+6) \sqrt {c^2 x^2-1}}-\frac {\left (m^2+15 m+52\right ) (f x)^{m+3} \sqrt {c^2 x^2-1}}{f (m+4)^2 (m+6)}}{m+6}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 d \left (\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 (f x)^{m+1}}{f (m+4)}-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (\frac {(a+b \text {arccosh}(c x)) (f x)^{m+2}}{f (m+2)}-\frac {c^2 (a+b \text {arccosh}(c x)) (f x)^{m+4}}{f^3 (m+4)}-\frac {b c \left (\frac {(3 m+10) (f x)^{m+3} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {(f x)^{m+3} \sqrt {c x-1} \sqrt {c x+1}}{f (m+4)^2}\right )}{f}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 (f x)^{m+1}}{f (m+2)}-\frac {2 b c \sqrt {d-c^2 d x^2} \int (f x)^{m+1} (a+b \text {arccosh}(c x))dx}{f (m+2) \sqrt {c x-1} \sqrt {c x+1}}+\frac {d \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{m+2}\right )}{m+4}\right )}{m+6}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 (f x)^{m+1}}{f (m+6)}-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {(a+b \text {arccosh}(c x)) (f x)^{m+2}}{f (m+2)}-\frac {2 c^2 (a+b \text {arccosh}(c x)) (f x)^{m+4}}{f^3 (m+4)}+\frac {c^4 (a+b \text {arccosh}(c x)) (f x)^{m+6}}{f^5 (m+6)}-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {c^2 \sqrt {c^2 x^2-1} (f x)^{m+5}}{f^3 (m+6)^2}+\frac {\frac {\left (15 m^2+130 m+264\right ) (f x)^{m+3} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 (m+6) \sqrt {c^2 x^2-1}}-\frac {\left (m^2+15 m+52\right ) (f x)^{m+3} \sqrt {c^2 x^2-1}}{f (m+4)^2 (m+6)}}{m+6}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 d \left (\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 (f x)^{m+1}}{f (m+4)}-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (\frac {(a+b \text {arccosh}(c x)) (f x)^{m+2}}{f (m+2)}-\frac {c^2 (a+b \text {arccosh}(c x)) (f x)^{m+4}}{f^3 (m+4)}-\frac {b c \left (\frac {(3 m+10) (f x)^{m+3} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {(f x)^{m+3} \sqrt {c x-1} \sqrt {c x+1}}{f (m+4)^2}\right )}{f}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 (f x)^{m+1}}{f (m+2)}-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \int \frac {(f x)^{m+2}}{\sqrt {c x-1} \sqrt {c x+1}}dx}{f (m+2)}\right )}{f (m+2) \sqrt {c x-1} \sqrt {c x+1}}+\frac {d \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{m+2}\right )}{m+4}\right )}{m+6}\) |
| \(\Big \downarrow \) 136 |
| \(\displaystyle \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 (f x)^{m+1}}{f (m+6)}-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {(a+b \text {arccosh}(c x)) (f x)^{m+2}}{f (m+2)}-\frac {2 c^2 (a+b \text {arccosh}(c x)) (f x)^{m+4}}{f^3 (m+4)}+\frac {c^4 (a+b \text {arccosh}(c x)) (f x)^{m+6}}{f^5 (m+6)}-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {c^2 \sqrt {c^2 x^2-1} (f x)^{m+5}}{f^3 (m+6)^2}+\frac {\frac {\left (15 m^2+130 m+264\right ) (f x)^{m+3} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 (m+6) \sqrt {c^2 x^2-1}}-\frac {\left (m^2+15 m+52\right ) (f x)^{m+3} \sqrt {c^2 x^2-1}}{f (m+4)^2 (m+6)}}{m+6}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 d \left (\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 (f x)^{m+1}}{f (m+4)}-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (\frac {(a+b \text {arccosh}(c x)) (f x)^{m+2}}{f (m+2)}-\frac {c^2 (a+b \text {arccosh}(c x)) (f x)^{m+4}}{f^3 (m+4)}-\frac {b c \left (\frac {(3 m+10) (f x)^{m+3} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {(f x)^{m+3} \sqrt {c x-1} \sqrt {c x+1}}{f (m+4)^2}\right )}{f}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 (f x)^{m+1}}{f (m+2)}-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \sqrt {c^2 x^2-1} \int \frac {(f x)^{m+2}}{\sqrt {c^2 x^2-1}}dx}{f (m+2) \sqrt {c x-1} \sqrt {c x+1}}\right )}{f (m+2) \sqrt {c x-1} \sqrt {c x+1}}+\frac {d \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{m+2}\right )}{m+4}\right )}{m+6}\) |
| \(\Big \downarrow \) 279 |
| \(\displaystyle \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 (f x)^{m+1}}{f (m+6)}-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {(a+b \text {arccosh}(c x)) (f x)^{m+2}}{f (m+2)}-\frac {2 c^2 (a+b \text {arccosh}(c x)) (f x)^{m+4}}{f^3 (m+4)}+\frac {c^4 (a+b \text {arccosh}(c x)) (f x)^{m+6}}{f^5 (m+6)}-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {c^2 \sqrt {c^2 x^2-1} (f x)^{m+5}}{f^3 (m+6)^2}+\frac {\frac {\left (15 m^2+130 m+264\right ) (f x)^{m+3} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 (m+6) \sqrt {c^2 x^2-1}}-\frac {\left (m^2+15 m+52\right ) (f x)^{m+3} \sqrt {c^2 x^2-1}}{f (m+4)^2 (m+6)}}{m+6}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 d \left (\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 (f x)^{m+1}}{f (m+4)}-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (\frac {(a+b \text {arccosh}(c x)) (f x)^{m+2}}{f (m+2)}-\frac {c^2 (a+b \text {arccosh}(c x)) (f x)^{m+4}}{f^3 (m+4)}-\frac {b c \left (\frac {(3 m+10) (f x)^{m+3} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {(f x)^{m+3} \sqrt {c x-1} \sqrt {c x+1}}{f (m+4)^2}\right )}{f}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 (f x)^{m+1}}{f (m+2)}-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \sqrt {1-c^2 x^2} \int \frac {(f x)^{m+2}}{\sqrt {1-c^2 x^2}}dx}{f (m+2) \sqrt {c x-1} \sqrt {c x+1}}\right )}{f (m+2) \sqrt {c x-1} \sqrt {c x+1}}+\frac {d \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{m+2}\right )}{m+4}\right )}{m+6}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 (f x)^{m+1}}{f (m+6)}-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {(a+b \text {arccosh}(c x)) (f x)^{m+2}}{f (m+2)}-\frac {2 c^2 (a+b \text {arccosh}(c x)) (f x)^{m+4}}{f^3 (m+4)}+\frac {c^4 (a+b \text {arccosh}(c x)) (f x)^{m+6}}{f^5 (m+6)}-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {c^2 \sqrt {c^2 x^2-1} (f x)^{m+5}}{f^3 (m+6)^2}+\frac {\frac {\left (15 m^2+130 m+264\right ) (f x)^{m+3} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 (m+6) \sqrt {c^2 x^2-1}}-\frac {\left (m^2+15 m+52\right ) (f x)^{m+3} \sqrt {c^2 x^2-1}}{f (m+4)^2 (m+6)}}{m+6}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 d \left (\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 (f x)^{m+1}}{f (m+4)}-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (\frac {(a+b \text {arccosh}(c x)) (f x)^{m+2}}{f (m+2)}-\frac {c^2 (a+b \text {arccosh}(c x)) (f x)^{m+4}}{f^3 (m+4)}-\frac {b c \left (\frac {(3 m+10) (f x)^{m+3} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {(f x)^{m+3} \sqrt {c x-1} \sqrt {c x+1}}{f (m+4)^2}\right )}{f}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 (f x)^{m+1}}{f (m+2)}-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c (f x)^{m+3} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f^2 (m+2) (m+3) \sqrt {c x-1} \sqrt {c x+1}}\right )}{f (m+2) \sqrt {c x-1} \sqrt {c x+1}}+\frac {d \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{m+2}\right )}{m+4}\right )}{m+6}\) |
| \(\Big \downarrow \) 6375 |
| \(\displaystyle \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 (f x)^{m+1}}{f (m+6)}-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {(a+b \text {arccosh}(c x)) (f x)^{m+2}}{f (m+2)}-\frac {2 c^2 (a+b \text {arccosh}(c x)) (f x)^{m+4}}{f^3 (m+4)}+\frac {c^4 (a+b \text {arccosh}(c x)) (f x)^{m+6}}{f^5 (m+6)}-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {c^2 \sqrt {c^2 x^2-1} (f x)^{m+5}}{f^3 (m+6)^2}+\frac {\frac {\left (15 m^2+130 m+264\right ) (f x)^{m+3} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 (m+6) \sqrt {c^2 x^2-1}}-\frac {\left (m^2+15 m+52\right ) (f x)^{m+3} \sqrt {c^2 x^2-1}}{f (m+4)^2 (m+6)}}{m+6}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}\right )}{f (m+6) \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 d \left (\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 (f x)^{m+1}}{f (m+4)}-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (\frac {(a+b \text {arccosh}(c x)) (f x)^{m+2}}{f (m+2)}-\frac {c^2 (a+b \text {arccosh}(c x)) (f x)^{m+4}}{f^3 (m+4)}-\frac {b c \left (\frac {(3 m+10) (f x)^{m+3} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {(f x)^{m+3} \sqrt {c x-1} \sqrt {c x+1}}{f (m+4)^2}\right )}{f}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 (f x)^{m+1}}{f (m+2)}-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c (f x)^{m+3} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f^2 (m+2) (m+3) \sqrt {c x-1} \sqrt {c x+1}}\right )}{f (m+2) \sqrt {c x-1} \sqrt {c x+1}}+\frac {d \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{m+2}\right )}{m+4}\right )}{m+6}\) |
Input:
Int[(f*x)^m*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2,x]
Output:
$Aborted
Not integrable
Time = 1.86 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94
\[\int \left (f x \right )^{m} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}d x\]
Input:
int((f*x)^m*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x)
Output:
int((f*x)^m*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x)
Not integrable
Time = 0.11 (sec) , antiderivative size = 136, normalized size of antiderivative = 4.39 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \left (f x\right )^{m} \,d x } \] Input:
integrate((f*x)^m*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x, algorithm=" fricas")
Output:
integral((a^2*c^4*d^2*x^4 - 2*a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^4*d^2*x^4 - 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arccosh(c*x)^2 + 2*(a*b*c^4*d^2*x^4 - 2*a* b*c^2*d^2*x^2 + a*b*d^2)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d)*(f*x)^m, x)
Timed out. \[ \int (f x)^m \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\text {Timed out} \] Input:
integrate((f*x)**m*(-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x))**2,x)
Output:
Timed out
Not integrable
Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \left (f x\right )^{m} \,d x } \] Input:
integrate((f*x)^m*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x, algorithm=" maxima")
Output:
integrate((-c^2*d*x^2 + d)^(5/2)*(b*arccosh(c*x) + a)^2*(f*x)^m, x)
Exception generated. \[ \int (f x)^m \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: TypeError} \] Input:
integrate((f*x)^m*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x, algorithm=" giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Not integrable
Time = 3.56 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}\,{\left (f\,x\right )}^m \,d x \] Input:
int((a + b*acosh(c*x))^2*(d - c^2*d*x^2)^(5/2)*(f*x)^m,x)
Output:
int((a + b*acosh(c*x))^2*(d - c^2*d*x^2)^(5/2)*(f*x)^m, x)
Not integrable
Time = 0.64 (sec) , antiderivative size = 267, normalized size of antiderivative = 8.61 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=f^{m} \sqrt {d}\, d^{2} \left (2 \left (\int x^{m} \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{4}d x \right ) a b \,c^{4}-4 \left (\int x^{m} \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{2}d x \right ) a b \,c^{2}+2 \left (\int x^{m} \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )d x \right ) a b +\left (\int x^{m} \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{4}-2 \left (\int x^{m} \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{2}+\left (\int x^{m} \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )^{2}d x \right ) b^{2}+\left (\int x^{m} \sqrt {-c^{2} x^{2}+1}\, x^{4}d x \right ) a^{2} c^{4}-2 \left (\int x^{m} \sqrt {-c^{2} x^{2}+1}\, x^{2}d x \right ) a^{2} c^{2}+\left (\int x^{m} \sqrt {-c^{2} x^{2}+1}d x \right ) a^{2}\right ) \] Input:
int((f*x)^m*(-c^2*d*x^2+d)^(5/2)*(a+b*acosh(c*x))^2,x)
Output:
f**m*sqrt(d)*d**2*(2*int(x**m*sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**4,x)*a* b*c**4 - 4*int(x**m*sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**2,x)*a*b*c**2 + 2 *int(x**m*sqrt( - c**2*x**2 + 1)*acosh(c*x),x)*a*b + int(x**m*sqrt( - c**2 *x**2 + 1)*acosh(c*x)**2*x**4,x)*b**2*c**4 - 2*int(x**m*sqrt( - c**2*x**2 + 1)*acosh(c*x)**2*x**2,x)*b**2*c**2 + int(x**m*sqrt( - c**2*x**2 + 1)*aco sh(c*x)**2,x)*b**2 + int(x**m*sqrt( - c**2*x**2 + 1)*x**4,x)*a**2*c**4 - 2 *int(x**m*sqrt( - c**2*x**2 + 1)*x**2,x)*a**2*c**2 + int(x**m*sqrt( - c**2 *x**2 + 1),x)*a**2)