Integrand size = 31, antiderivative size = 31 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {2 b^2 c^2 d (f x)^{3+m} \sqrt {d-c^2 d x^2}}{f^3 (4+m)^3}-\frac {6 b c d (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m)^2 (4+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m) (4+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d (f x)^{4+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^4 (4+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 d (f x)^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{f \left (8+6 m+m^2\right )}+\frac {(f x)^{1+m} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{f (4+m)}-\frac {6 b^2 c^2 d (f x)^{3+m} \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{f^3 (2+m)^2 (3+m) (4+m) (1-c x) (1+c x)}-\frac {2 b^2 c^2 d (10+3 m) (f x)^{3+m} \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{f^3 (2+m) (3+m) (4+m)^3 (1-c x) (1+c x)}+\frac {3 d^2 \text {Int}\left (\frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}},x\right )}{8+6 m+m^2} \]
(f*x)^(1+m)*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2/f/(4+m)-2*b^2*c^2*d* (f*x)^(3+m)*(-c^2*d*x^2+d)^(1/2)/f^3/(4+m)^3+3*d*(f*x)^(1+m)*(a+b*arccosh( c*x))^2*(-c^2*d*x^2+d)^(1/2)/f/(m^2+6*m+8)-6*b*c*d*(f*x)^(2+m)*(a+b*arccos h(c*x))*(-c^2*d*x^2+d)^(1/2)/f^2/(2+m)^2/(4+m)/(c*x-1)^(1/2)/(c*x+1)^(1/2) -2*b*c*d*(f*x)^(2+m)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/f^2/(2+m)/(4+ m)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2*b*c^3*d*(f*x)^(4+m)*(a+b*arccosh(c*x))*(- c^2*d*x^2+d)^(1/2)/f^4/(4+m)^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-6*b^2*c^2*d*(f* x)^(3+m)*hypergeom([1/2, 3/2+1/2*m],[5/2+1/2*m],c^2*x^2)*(-c^2*x^2+1)^(1/2 )*(-c^2*d*x^2+d)^(1/2)/f^3/(2+m)^2/(3+m)/(4+m)/(-c*x+1)/(c*x+1)-2*b^2*c^2* d*(10+3*m)*(f*x)^(3+m)*hypergeom([1/2, 3/2+1/2*m],[5/2+1/2*m],c^2*x^2)*(-c ^2*x^2+1)^(1/2)*(-c^2*d*x^2+d)^(1/2)/f^3/(4+m)^3/(m^2+5*m+6)/(-c*x+1)/(c*x +1)+3*d^2*Unintegrable((f*x)^m*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(1/2),x )/(m^2+6*m+8)
Not integrable
Time = 1.10 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx \]
Not integrable
Time = 2.41 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {6345, 25, 6327, 6336, 27, 960, 136, 279, 278, 6345, 6298, 136, 279, 278, 6375}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \left (d-c^2 d x^2\right )^{3/2} (f x)^m (a+b \text {arccosh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6345 |
| \(\displaystyle \frac {2 b c d \sqrt {d-c^2 d x^2} \int -(f x)^{m+1} (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 d \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx}{m+4}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+4)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 b c d \sqrt {d-c^2 d x^2} \int (f x)^{m+1} (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 d \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx}{m+4}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+4)}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle -\frac {2 b c d \sqrt {d-c^2 d x^2} \int (f x)^{m+1} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))dx}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 d \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx}{m+4}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+4)}\) |
| \(\Big \downarrow \) 6336 |
| \(\displaystyle -\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-b c \int \frac {(f x)^{m+2} \left (\frac {1}{m+2}-\frac {c^2 x^2}{m+4}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 d \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx}{m+4}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+4)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {b c \int \frac {(f x)^{m+2} \left (\frac {1}{m+2}-\frac {c^2 x^2}{m+4}\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx}{f}-\frac {c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 d \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx}{m+4}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+4)}\) |
| \(\Big \downarrow \) 960 |
| \(\displaystyle -\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {b c \left (\frac {(3 m+10) \int \frac {(f x)^{m+2}}{\sqrt {c x-1} \sqrt {c x+1}}dx}{(m+2) (m+4)^2}-\frac {\sqrt {c x-1} \sqrt {c x+1} (f x)^{m+3}}{f (m+4)^2}\right )}{f}-\frac {c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 d \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx}{m+4}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+4)}\) |
| \(\Big \downarrow \) 136 |
| \(\displaystyle -\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {b c \left (\frac {(3 m+10) \sqrt {c^2 x^2-1} \int \frac {(f x)^{m+2}}{\sqrt {c^2 x^2-1}}dx}{(m+2) (m+4)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} (f x)^{m+3}}{f (m+4)^2}\right )}{f}-\frac {c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 d \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx}{m+4}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+4)}\) |
| \(\Big \downarrow \) 279 |
| \(\displaystyle -\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {b c \left (\frac {(3 m+10) \sqrt {1-c^2 x^2} \int \frac {(f x)^{m+2}}{\sqrt {1-c^2 x^2}}dx}{(m+2) (m+4)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} (f x)^{m+3}}{f (m+4)^2}\right )}{f}-\frac {c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 d \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx}{m+4}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+4)}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle \frac {3 d \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx}{m+4}-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \left (\frac {(3 m+10) \sqrt {1-c^2 x^2} (f x)^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} (f x)^{m+3}}{f (m+4)^2}\right )}{f}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+4)}\) |
| \(\Big \downarrow \) 6345 |
| \(\displaystyle \frac {3 d \left (-\frac {2 b c \sqrt {d-c^2 d x^2} \int (f x)^{m+1} (a+b \text {arccosh}(c x))dx}{f (m+2) \sqrt {c x-1} \sqrt {c x+1}}+\frac {d \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{m+2}+\frac {\sqrt {d-c^2 d x^2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+2)}\right )}{m+4}-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \left (\frac {(3 m+10) \sqrt {1-c^2 x^2} (f x)^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} (f x)^{m+3}}{f (m+4)^2}\right )}{f}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+4)}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {3 d \left (-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \int \frac {(f x)^{m+2}}{\sqrt {c x-1} \sqrt {c x+1}}dx}{f (m+2)}\right )}{f (m+2) \sqrt {c x-1} \sqrt {c x+1}}+\frac {d \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{m+2}+\frac {\sqrt {d-c^2 d x^2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+2)}\right )}{m+4}-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \left (\frac {(3 m+10) \sqrt {1-c^2 x^2} (f x)^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} (f x)^{m+3}}{f (m+4)^2}\right )}{f}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+4)}\) |
| \(\Big \downarrow \) 136 |
| \(\displaystyle \frac {3 d \left (-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \sqrt {c^2 x^2-1} \int \frac {(f x)^{m+2}}{\sqrt {c^2 x^2-1}}dx}{f (m+2) \sqrt {c x-1} \sqrt {c x+1}}\right )}{f (m+2) \sqrt {c x-1} \sqrt {c x+1}}+\frac {d \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{m+2}+\frac {\sqrt {d-c^2 d x^2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+2)}\right )}{m+4}-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \left (\frac {(3 m+10) \sqrt {1-c^2 x^2} (f x)^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} (f x)^{m+3}}{f (m+4)^2}\right )}{f}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+4)}\) |
| \(\Big \downarrow \) 279 |
| \(\displaystyle \frac {3 d \left (-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \sqrt {1-c^2 x^2} \int \frac {(f x)^{m+2}}{\sqrt {1-c^2 x^2}}dx}{f (m+2) \sqrt {c x-1} \sqrt {c x+1}}\right )}{f (m+2) \sqrt {c x-1} \sqrt {c x+1}}+\frac {d \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{m+2}+\frac {\sqrt {d-c^2 d x^2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+2)}\right )}{m+4}-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \left (\frac {(3 m+10) \sqrt {1-c^2 x^2} (f x)^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} (f x)^{m+3}}{f (m+4)^2}\right )}{f}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+4)}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle \frac {3 d \left (\frac {d \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{m+2}-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \sqrt {1-c^2 x^2} (f x)^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f^2 (m+2) (m+3) \sqrt {c x-1} \sqrt {c x+1}}\right )}{f (m+2) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\sqrt {d-c^2 d x^2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+2)}\right )}{m+4}-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \left (\frac {(3 m+10) \sqrt {1-c^2 x^2} (f x)^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} (f x)^{m+3}}{f (m+4)^2}\right )}{f}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+4)}\) |
| \(\Big \downarrow \) 6375 |
| \(\displaystyle \frac {3 d \left (\frac {d \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{m+2}-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \sqrt {1-c^2 x^2} (f x)^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f^2 (m+2) (m+3) \sqrt {c x-1} \sqrt {c x+1}}\right )}{f (m+2) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\sqrt {d-c^2 d x^2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+2)}\right )}{m+4}-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {c^2 (f x)^{m+4} (a+b \text {arccosh}(c x))}{f^3 (m+4)}+\frac {(f x)^{m+2} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \left (\frac {(3 m+10) \sqrt {1-c^2 x^2} (f x)^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{f (m+2) (m+3) (m+4)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} (f x)^{m+3}}{f (m+4)^2}\right )}{f}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))^2}{f (m+4)}\) |
3.3.34.3.1 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[((f_.)*(x_))^(p_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_] :> Simp[(a + b*x)^FracPart[m]*((c + d*x)^FracPart[m]/(a*c + b*d*x^2)^Fr acPart[m]) Int[(a*c + b*d*x^2)^m*(f*x)^p, x], x] /; FreeQ[{a, b, c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p]) Int[(c*x)^m* (1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && !(ILtQ[p, 0] || GtQ[a, 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[((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_)*((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_.)*((d_) + (e _.)*(x_)^2)^(p_.), x_Symbol] :> Unintegrable[(f*x)^m*(d + e*x^2)^p*(a + b*A rcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
Not integrable
Time = 2.77 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94
\[\int \left (f x \right )^{m} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}d x\]
Not integrable
Time = 0.29 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.81 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \left (f x\right )^{m} \,d x } \]
integral(-(a^2*c^2*d*x^2 - a^2*d + (b^2*c^2*d*x^2 - b^2*d)*arccosh(c*x)^2 + 2*(a*b*c^2*d*x^2 - a*b*d)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d)*(f*x)^m, x)
Timed out. \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\text {Timed out} \]
Not integrable
Time = 0.38 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \left (f x\right )^{m} \,d x } \]
Exception generated. \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]
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 = 3.44 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2}\,{\left (f\,x\right )}^m \,d x \]