3.3.33 \(\int (f x)^m (d-c^2 d x^2)^{5/2} (a+b \text {arccosh}(c x))^2 \, dx\) [233]

3.3.33.1 Optimal result

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=-\frac {10 b^2 c^2 d^2 (f x)^{3+m} \sqrt {d-c^2 d x^2}}{f^3 (4+m)^3 (6+m)}-\frac {2 b^2 c^2 d^2 \left (52+15 m+m^2\right ) (f x)^{3+m} \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{f^3 (4+m)^2 (6+m)^3 (1-c x) (1+c x)}+\frac {2 b^2 c^4 d^2 (f x)^{5+m} \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{f^5 (6+m)^3 (1-c x) (1+c x)}-\frac {2 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {30 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m)^2 (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {10 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m) (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {10 b c^3 d^2 (f x)^{4+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^4 (4+m)^2 (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b c^3 d^2 (f x)^{4+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^4 (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 (f x)^{6+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^6 (6+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {15 d^2 (f x)^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{f (6+m) \left (8+6 m+m^2\right )}+\frac {5 d (f x)^{1+m} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{f (4+m) (6+m)}+\frac {(f x)^{1+m} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{f (6+m)}-\frac {30 b^2 c^2 d^2 (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) (6+m) (1-c x) (1+c x)}-\frac {10 b^2 c^2 d^2 (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 (6+m) (1-c x) (1+c x)}-\frac {2 b^2 c^2 d^2 \left (264+130 m+15 m^2\right ) (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)^2 (6+m)^3 (1-c x) (1+c x)}+\frac {15 d^3 \text {Int}\left (\frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}},x\right )}{(6+m) \left (8+6 m+m^2\right )} \]

output

5*d*(f*x)^(1+m)*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2/f/(4+m)/(6+m)+(f 
*x)^(1+m)*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2/f/(6+m)-10*b^2*c^2*d^2 
*(f*x)^(3+m)*(-c^2*d*x^2+d)^(1/2)/f^3/(4+m)^3/(6+m)-2*b^2*c^2*d^2*(m^2+15* 
m+52)*(f*x)^(3+m)*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/f^3/(4+m)^2/(6+m)^3/(- 
c*x+1)/(c*x+1)+2*b^2*c^4*d^2*(f*x)^(5+m)*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2) 
/f^5/(6+m)^3/(-c*x+1)/(c*x+1)+15*d^2*(f*x)^(1+m)*(a+b*arccosh(c*x))^2*(-c^ 
2*d*x^2+d)^(1/2)/f/(6+m)/(m^2+6*m+8)-2*b*c*d^2*(f*x)^(2+m)*(a+b*arccosh(c* 
x))*(-c^2*d*x^2+d)^(1/2)/f^2/(2+m)/(6+m)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-30*b* 
c*d^2*(f*x)^(2+m)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/f^2/(2+m)^2/(4+m 
)/(6+m)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-10*b*c*d^2*(f*x)^(2+m)*(a+b*arccosh(c* 
x))*(-c^2*d*x^2+d)^(1/2)/f^2/(6+m)/(m^2+6*m+8)/(c*x-1)^(1/2)/(c*x+1)^(1/2) 
+10*b*c^3*d^2*(f*x)^(4+m)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/f^4/(4+m 
)^2/(6+m)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+4*b*c^3*d^2*(f*x)^(4+m)*(a+b*arccosh 
(c*x))*(-c^2*d*x^2+d)^(1/2)/f^4/(4+m)/(6+m)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2* 
b*c^5*d^2*(f*x)^(6+m)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/f^6/(6+m)^2/ 
(c*x-1)^(1/2)/(c*x+1)^(1/2)-30*b^2*c^2*d^2*(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)/(6+m)/(-c*x+1)/(c*x+1)-10*b^2*c^2*d^2*(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/(6+m)/(m^2+5*m+6)/(-c*x+1)/(c*x+1)-2*b^2*...
 

3.3.33.2 Mathematica [N/A]

Not integrable

Time = 2.23 (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]
 

3.3.33.3 Rubi [N/A]

Not integrable

Time = 5.21 (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 = {6345, 6327, 6336, 27, 1905, 1590, 27, 363, 279, 278, 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.

\(\displaystyle \int \left (d-c^2 d x^2\right )^{5/2} (f x)^m (a+b \text {arccosh}(c x))^2 \, dx\)

\(\Big \downarrow \) 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
 

3.3.33.3.1 Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 136

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]
 

rule 278

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])
 

rule 279

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])
 

rule 363

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]
 

rule 960

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

rule 1590

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]
 

rule 1905

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]
 

rule 6298

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
 :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* 
(n/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + 
 c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& NeQ[m, -1]
 

rule 6327

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + ( 
e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1 
*d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2 
, e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
 

rule 6336

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

rule 6345

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

rule 6375

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]
 

3.3.33.4 Maple [N/A] (verified)

Not integrable

Time = 2.79 (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)
 

3.3.33.5 Fricas [N/A]

Not integrable

Time = 0.26 (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)
 

3.3.33.6 Sympy [F(-1)]

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
 

3.3.33.7 Maxima [N/A]

Not integrable

Time = 0.42 (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)
 

3.3.33.8 Giac [F(-2)]

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
 

3.3.33.9 Mupad [N/A]

Not integrable

Time = 3.22 (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)