Integrand size = 29, antiderivative size = 890 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^3} \, dx=-\frac {170}{27} b^2 c^2 d^2 \sqrt {d-c^2 d x^2}+\frac {5}{27} b^2 c^4 d^2 x^2 \sqrt {d-c^2 d x^2}+\frac {5 a b c^3 d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b^2 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{3 (1-c x) (1+c x)}+\frac {b^2 c^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{9 (1-c x) (1+c x)}+\frac {5 b^2 c^3 d^2 x \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{2 x^2}+\frac {5 c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b^2 c^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )}{(1-c x) (1+c x)}-\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 i b^2 c^2 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 i b^2 c^2 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \]
-5/6*c^2*d*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2-1/2*(-c^2*d*x^2+d)^(5 /2)*(a+b*arccosh(c*x))^2/x^2-170/27*b^2*c^2*d^2*(-c^2*d*x^2+d)^(1/2)+5/27* b^2*c^4*d^2*x^2*(-c^2*d*x^2+d)^(1/2)+5/3*b^2*c^2*d^2*(-c^2*x^2+1)*(-c^2*d* x^2+d)^(1/2)/(-c*x+1)/(c*x+1)+1/9*b^2*c^2*d^2*(-c^2*x^2+1)^2*(-c^2*d*x^2+d )^(1/2)/(-c*x+1)/(c*x+1)-5/2*c^2*d^2*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^( 1/2)+5*a*b*c^3*d^2*x*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+5*b^ 2*c^3*d^2*x*arccosh(c*x)*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)- b*c*d^2*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/x/(c*x-1)^(1/2)/(c*x+1)^(1 /2)-1/3*b*c^3*d^2*x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/ (c*x+1)^(1/2)-2/9*b*c^5*d^2*x^3*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/(c *x-1)^(1/2)/(c*x+1)^(1/2)+5*c^2*d^2*(a+b*arccosh(c*x))^2*arctan(c*x+(c*x-1 )^(1/2)*(c*x+1)^(1/2))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+5* I*b^2*c^2*d^2*polylog(3,-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(-c^2*d*x^2+ d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-5*I*b*c^2*d^2*(a+b*arccosh(c*x))*poly log(2,-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^( 1/2)/(c*x+1)^(1/2)+5*I*b*c^2*d^2*(a+b*arccosh(c*x))*polylog(2,I*(c*x+(c*x- 1)^(1/2)*(c*x+1)^(1/2)))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)- 5*I*b^2*c^2*d^2*polylog(3,I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(-c^2*d*x^2 +d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-b^2*c^2*d^2*arctan((c^2*x^2-1)^(1/2) )*(c^2*x^2-1)^(1/2)*(-c^2*d*x^2+d)^(1/2)/(-c*x+1)/(c*x+1)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(5694\) vs. \(2(890)=1780\).
Time = 65.18 (sec) , antiderivative size = 5694, normalized size of antiderivative = 6.40 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^3} \, dx=\text {Result too large to show} \]
Time = 4.56 (sec) , antiderivative size = 609, normalized size of antiderivative = 0.68, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.828, Rules used = {6343, 6327, 6336, 27, 1905, 1578, 1192, 1467, 2009, 6345, 25, 6304, 6309, 27, 960, 83, 6341, 2009, 6362, 3042, 4668, 3011, 2720, 7143}
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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^3} \, dx\) |
| \(\Big \downarrow \) 6343 |
| \(\displaystyle \frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {(1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))}{x^2}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle \frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 6336 |
| \(\displaystyle -\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-b c \int -\frac {-c^4 x^4+6 c^2 x^2+3}{3 x \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{3} c^4 x^3 (a+b \text {arccosh}(c x))-2 c^2 x (a+b \text {arccosh}(c x))-\frac {a+b \text {arccosh}(c x)}{x}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} b c \int \frac {-c^4 x^4+6 c^2 x^2+3}{x \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{3} c^4 x^3 (a+b \text {arccosh}(c x))-2 c^2 x (a+b \text {arccosh}(c x))-\frac {a+b \text {arccosh}(c x)}{x}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 1905 |
| \(\displaystyle -\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {b c \sqrt {c^2 x^2-1} \int \frac {-c^4 x^4+6 c^2 x^2+3}{x \sqrt {c^2 x^2-1}}dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{3} c^4 x^3 (a+b \text {arccosh}(c x))-2 c^2 x (a+b \text {arccosh}(c x))-\frac {a+b \text {arccosh}(c x)}{x}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle -\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {b c \sqrt {c^2 x^2-1} \int \frac {-c^4 x^4+6 c^2 x^2+3}{x^2 \sqrt {c^2 x^2-1}}dx^2}{6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{3} c^4 x^3 (a+b \text {arccosh}(c x))-2 c^2 x (a+b \text {arccosh}(c x))-\frac {a+b \text {arccosh}(c x)}{x}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 1192 |
| \(\displaystyle -\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {b \sqrt {c^2 x^2-1} \int \frac {-c^4 x^8+4 c^4 x^4+8 c^4}{x^4+1}d\sqrt {c^2 x^2-1}}{3 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{3} c^4 x^3 (a+b \text {arccosh}(c x))-2 c^2 x (a+b \text {arccosh}(c x))-\frac {a+b \text {arccosh}(c x)}{x}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 1467 |
| \(\displaystyle -\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {b \sqrt {c^2 x^2-1} \int \left (-x^4 c^4+\frac {3 c^4}{x^4+1}+5 c^4\right )d\sqrt {c^2 x^2-1}}{3 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{3} c^4 x^3 (a+b \text {arccosh}(c x))-2 c^2 x (a+b \text {arccosh}(c x))-\frac {a+b \text {arccosh}(c x)}{x}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \text {arccosh}(c x))-2 c^2 x (a+b \text {arccosh}(c x))-\frac {a+b \text {arccosh}(c x)}{x}+\frac {b \sqrt {c^2 x^2-1} \left (3 c^4 \arctan \left (\sqrt {c^2 x^2-1}\right )-\frac {1}{3} c^4 x^6+5 c^4 \sqrt {c^2 x^2-1}\right )}{3 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 6345 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \int -((1-c x) (c x+1) (a+b \text {arccosh}(c x)))dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}dx+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \text {arccosh}(c x))-2 c^2 x (a+b \text {arccosh}(c x))-\frac {a+b \text {arccosh}(c x)}{x}+\frac {b \sqrt {c^2 x^2-1} \left (3 c^4 \arctan \left (\sqrt {c^2 x^2-1}\right )-\frac {1}{3} c^4 x^6+5 c^4 \sqrt {c^2 x^2-1}\right )}{3 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (-\frac {2 b c d \sqrt {d-c^2 d x^2} \int (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}dx+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \text {arccosh}(c x))-2 c^2 x (a+b \text {arccosh}(c x))-\frac {a+b \text {arccosh}(c x)}{x}+\frac {b \sqrt {c^2 x^2-1} \left (3 c^4 \arctan \left (\sqrt {c^2 x^2-1}\right )-\frac {1}{3} c^4 x^6+5 c^4 \sqrt {c^2 x^2-1}\right )}{3 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 6304 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (-\frac {2 b c d \sqrt {d-c^2 d x^2} \int \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}dx+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \text {arccosh}(c x))-2 c^2 x (a+b \text {arccosh}(c x))-\frac {a+b \text {arccosh}(c x)}{x}+\frac {b \sqrt {c^2 x^2-1} \left (3 c^4 \arctan \left (\sqrt {c^2 x^2-1}\right )-\frac {1}{3} c^4 x^6+5 c^4 \sqrt {c^2 x^2-1}\right )}{3 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 6309 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-b c \int \frac {x \left (3-c^2 x^2\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \text {arccosh}(c x))-2 c^2 x (a+b \text {arccosh}(c x))-\frac {a+b \text {arccosh}(c x)}{x}+\frac {b \sqrt {c^2 x^2-1} \left (3 c^4 \arctan \left (\sqrt {c^2 x^2-1}\right )-\frac {1}{3} c^4 x^6+5 c^4 \sqrt {c^2 x^2-1}\right )}{3 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} b c \int \frac {x \left (3-c^2 x^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \text {arccosh}(c x))-2 c^2 x (a+b \text {arccosh}(c x))-\frac {a+b \text {arccosh}(c x)}{x}+\frac {b \sqrt {c^2 x^2-1} \left (3 c^4 \arctan \left (\sqrt {c^2 x^2-1}\right )-\frac {1}{3} c^4 x^6+5 c^4 \sqrt {c^2 x^2-1}\right )}{3 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 960 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} b c \left (\frac {7}{3} \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \text {arccosh}(c x))-2 c^2 x (a+b \text {arccosh}(c x))-\frac {a+b \text {arccosh}(c x)}{x}+\frac {b \sqrt {c^2 x^2-1} \left (3 c^4 \arctan \left (\sqrt {c^2 x^2-1}\right )-\frac {1}{3} c^4 x^6+5 c^4 \sqrt {c^2 x^2-1}\right )}{3 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}dx+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {7 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \text {arccosh}(c x))-2 c^2 x (a+b \text {arccosh}(c x))-\frac {a+b \text {arccosh}(c x)}{x}+\frac {b \sqrt {c^2 x^2-1} \left (3 c^4 \arctan \left (\sqrt {c^2 x^2-1}\right )-\frac {1}{3} c^4 x^6+5 c^4 \sqrt {c^2 x^2-1}\right )}{3 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 6341 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \left (-\frac {2 b c \sqrt {d-c^2 d x^2} \int (a+b \text {arccosh}(c x))dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {7 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \text {arccosh}(c x))-2 c^2 x (a+b \text {arccosh}(c x))-\frac {a+b \text {arccosh}(c x)}{x}+\frac {b \sqrt {c^2 x^2-1} \left (3 c^4 \arctan \left (\sqrt {c^2 x^2-1}\right )-\frac {1}{3} c^4 x^6+5 c^4 \sqrt {c^2 x^2-1}\right )}{3 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {7 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \text {arccosh}(c x))-2 c^2 x (a+b \text {arccosh}(c x))-\frac {a+b \text {arccosh}(c x)}{x}+\frac {b \sqrt {c^2 x^2-1} \left (3 c^4 \arctan \left (\sqrt {c^2 x^2-1}\right )-\frac {1}{3} c^4 x^6+5 c^4 \sqrt {c^2 x^2-1}\right )}{3 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 6362 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{c x}d\text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {7 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \text {arccosh}(c x))-2 c^2 x (a+b \text {arccosh}(c x))-\frac {a+b \text {arccosh}(c x)}{x}+\frac {b \sqrt {c^2 x^2-1} \left (3 c^4 \arctan \left (\sqrt {c^2 x^2-1}\right )-\frac {1}{3} c^4 x^6+5 c^4 \sqrt {c^2 x^2-1}\right )}{3 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \int (a+b \text {arccosh}(c x))^2 \csc \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {7 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \text {arccosh}(c x))-2 c^2 x (a+b \text {arccosh}(c x))-\frac {a+b \text {arccosh}(c x)}{x}+\frac {b \sqrt {c^2 x^2-1} \left (3 c^4 \arctan \left (\sqrt {c^2 x^2-1}\right )-\frac {1}{3} c^4 x^6+5 c^4 \sqrt {c^2 x^2-1}\right )}{3 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \left (-2 i b \int (a+b \text {arccosh}(c x)) \log \left (1-i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i b \int (a+b \text {arccosh}(c x)) \log \left (1+i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {7 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \text {arccosh}(c x))-2 c^2 x (a+b \text {arccosh}(c x))-\frac {a+b \text {arccosh}(c x)}{x}+\frac {b \sqrt {c^2 x^2-1} \left (3 c^4 \arctan \left (\sqrt {c^2 x^2-1}\right )-\frac {1}{3} c^4 x^6+5 c^4 \sqrt {c^2 x^2-1}\right )}{3 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \left (2 i b \left (b \int \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-2 i b \left (b \int \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {7 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \text {arccosh}(c x))-2 c^2 x (a+b \text {arccosh}(c x))-\frac {a+b \text {arccosh}(c x)}{x}+\frac {b \sqrt {c^2 x^2-1} \left (3 c^4 \arctan \left (\sqrt {c^2 x^2-1}\right )-\frac {1}{3} c^4 x^6+5 c^4 \sqrt {c^2 x^2-1}\right )}{3 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \left (2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {7 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \text {arccosh}(c x))-2 c^2 x (a+b \text {arccosh}(c x))-\frac {a+b \text {arccosh}(c x)}{x}+\frac {b \sqrt {c^2 x^2-1} \left (3 c^4 \arctan \left (\sqrt {c^2 x^2-1}\right )-\frac {1}{3} c^4 x^6+5 c^4 \sqrt {c^2 x^2-1}\right )}{3 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \left (2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2+2 i b \left (b \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c x)}\right )-\operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-2 i b \left (b \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c x)}\right )-\operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {7 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \text {arccosh}(c x))-2 c^2 x (a+b \text {arccosh}(c x))-\frac {a+b \text {arccosh}(c x)}{x}+\frac {b \sqrt {c^2 x^2-1} \left (3 c^4 \arctan \left (\sqrt {c^2 x^2-1}\right )-\frac {1}{3} c^4 x^6+5 c^4 \sqrt {c^2 x^2-1}\right )}{3 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{2 x^2}\) |
-1/2*((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/x^2 + (b*c*d^2*Sqrt[d - c^2*d*x^2]*(-((a + b*ArcCosh[c*x])/x) - 2*c^2*x*(a + b*ArcCosh[c*x]) + ( c^4*x^3*(a + b*ArcCosh[c*x]))/3 + (b*Sqrt[-1 + c^2*x^2]*(-1/3*(c^4*x^6) + 5*c^4*Sqrt[-1 + c^2*x^2] + 3*c^4*ArcTan[Sqrt[-1 + c^2*x^2]]))/(3*c^3*Sqrt[ -1 + c*x]*Sqrt[1 + c*x])))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*c^2*d*(((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/3 - (2*b*c*d*Sqrt[d - c^2*d*x^2 ]*(-1/3*(b*c*((7*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^2) - (x^2*Sqrt[-1 + c* x]*Sqrt[1 + c*x])/3)) + x*(a + b*ArcCosh[c*x]) - (c^2*x^3*(a + b*ArcCosh[c *x]))/3))/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + d*(Sqrt[d - c^2*d*x^2]*(a + b *ArcCosh[c*x])^2 - (2*b*c*Sqrt[d - c^2*d*x^2]*(a*x - (b*Sqrt[-1 + c*x]*Sqr t[1 + c*x])/c + b*x*ArcCosh[c*x]))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (Sqrt[ d - c^2*d*x^2]*(2*(a + b*ArcCosh[c*x])^2*ArcTan[E^ArcCosh[c*x]] + (2*I)*b* (-((a + b*ArcCosh[c*x])*PolyLog[2, (-I)*E^ArcCosh[c*x]]) + b*PolyLog[3, (- I)*E^ArcCosh[c*x]]) - (2*I)*b*(-((a + b*ArcCosh[c*x])*PolyLog[2, I*E^ArcCo sh[c*x]]) + b*PolyLog[3, I*E^ArcCosh[c*x]])))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x ]))))/2
3.2.92.3.1 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[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f *(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2/e^(n + 2*p + 1) Subst[Int[x^( 2*m + 1)*(e*f - d*g + g*x^2)^n*(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4)^p, x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m + 1/2]
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/2]
Int[((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_)^(non2_.))^(q_.)*((d2_) + (e2_.) *(x_)^(non2_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x _Symbol] :> Simp[(d1 + e1*x^(n/2))^FracPart[q]*((d2 + e2*x^(n/2))^FracPart[ q]/(d1*d2 + e1*e2*x^n)^FracPart[q]) Int[(f*x)^m*(d1*d2 + e1*e2*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[non2, n/2] && EqQ[d2*e1 + d1*e2, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ ))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[ 1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c , d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*( (d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(d1*d2 + e1*e2*x^2)^p*(a + b*A rcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symb ol] :> With[{u = IntHide[(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}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + ( e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1 *d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2 , e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ )^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Simp [(a + b*ArcCosh[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && E qQ[c^2*d + e, 0] && IGtQ[p, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc Cosh[c*x])^n/(f*(m + 2))), x] + (-Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/(Sq rt[1 + c*x]*Sqrt[-1 + c*x])] Int[(f*x)^m*((a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e* x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])] Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x ])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*Arc Cosh[c*x])^n/(f*(m + 1))), x] + (-Simp[2*e*(p/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*c*(n/(f*( m + 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}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && G tQ[p, 0] && LtQ[m, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*Arc Cosh[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1)) Int[(f* x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*c*(n/(f*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[(f*x)^(m + 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1) , x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -1]
Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1 _.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/c^(m + 1))*Simp[ Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]] Subst [Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[n, 0] && Inte gerQ[m]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
\[\int \frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}{x^{3}}d x\]
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^3} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
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)/x^3, x)
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^3} \, dx=\text {Timed out} \]
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^3} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
1/6*(15*c^2*d^(5/2)*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x) ) - 3*(-c^2*d*x^2 + d)^(5/2)*c^2 - 5*(-c^2*d*x^2 + d)^(3/2)*c^2*d - 15*sqr t(-c^2*d*x^2 + d)*c^2*d^2 - 3*(-c^2*d*x^2 + d)^(7/2)/(d*x^2))*a^2 + integr ate((-c^2*d*x^2 + d)^(5/2)*b^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/x^ 3 + 2*(-c^2*d*x^2 + d)^(5/2)*a*b*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/x^ 3, x)
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^3} \, dx=\text {Exception raised: TypeError} \]
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
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^3} \,d x \]