Integrand size = 29, antiderivative size = 650 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \text {arccosh}(c x))^2}{2 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2 \arctan \left (e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {4 b c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 i b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}} \]
3/2*c^2*(a+b*arccosh(c*x))^2/d/(-c^2*d*x^2+d)^(1/2)-1/2*(a+b*arccosh(c*x)) ^2/d/x^2/(-c^2*d*x^2+d)^(1/2)+b*c*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1) ^(1/2)/d/x/(-c^2*d*x^2+d)^(1/2)+3*c^2*(a+b*arccosh(c*x))^2*arctan(c*x+(c*x -1)^(1/2)*(c*x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2 )-b^2*c^2*arctan((c*x-1)^(1/2)*(c*x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/ d/(-c^2*d*x^2+d)^(1/2)+4*b*c^2*(a+b*arccosh(c*x))*arctanh(c*x+(c*x-1)^(1/2 )*(c*x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2)+2*b^2* c^2*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2 )/d/(-c^2*d*x^2+d)^(1/2)-3*I*b*c^2*(a+b*arccosh(c*x))*polylog(2,-I*(c*x+(c *x-1)^(1/2)*(c*x+1)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(-c^2*d*x^2+d)^( 1/2)+3*I*b*c^2*(a+b*arccosh(c*x))*polylog(2,I*(c*x+(c*x-1)^(1/2)*(c*x+1)^( 1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2)-2*b^2*c^2*polylo g(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(-c^2*d *x^2+d)^(1/2)+3*I*b^2*c^2*polylog(3,-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))* (c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2)-3*I*b^2*c^2*polylog(3,I *(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(-c^2*d* x^2+d)^(1/2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(5362\) vs. \(2(650)=1300\).
Time = 63.67 (sec) , antiderivative size = 5362, normalized size of antiderivative = 8.25 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Result too large to show} \]
Time = 4.98 (sec) , antiderivative size = 433, normalized size of antiderivative = 0.67, number of steps used = 28, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.931, Rules used = {6347, 25, 6327, 6347, 103, 218, 6318, 3042, 26, 4670, 2715, 2838, 6351, 25, 6304, 6318, 3042, 26, 4670, 2715, 2838, 6361, 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.
| \(\displaystyle \int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6347 |
| \(\displaystyle \frac {3}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \int -\frac {a+b \text {arccosh}(c x)}{x^2 (1-c x) (c x+1)}dx}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x^2 (1-c x) (c x+1)}dx}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle \frac {3}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (1-c^2 x^2\right )}dx}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6347 |
| \(\displaystyle \frac {3}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (c^2 \int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx+b c \int \frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {a+b \text {arccosh}(c x)}{x}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle \frac {3}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (c^2 \int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx+b c^2 \int \frac {1}{(c x-1) (c x+1) c+c}d\left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {a+b \text {arccosh}(c x)}{x}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (c^2 \int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx-\frac {a+b \text {arccosh}(c x)}{x}+b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6318 |
| \(\displaystyle -\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (-c \int \frac {a+b \text {arccosh}(c x)}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)-\frac {a+b \text {arccosh}(c x)}{x}+b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (-c \int i (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)-\frac {a+b \text {arccosh}(c x)}{x}+b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (-i c \int (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)-\frac {a+b \text {arccosh}(c x)}{x}+b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (-i c \left (i b \int \log \left (1-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-i b \int \log \left (1+e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-\frac {a+b \text {arccosh}(c x)}{x}+b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (-i c \left (i b \int e^{-\text {arccosh}(c x)} \log \left (1-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-i b \int e^{-\text {arccosh}(c x)} \log \left (1+e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-\frac {a+b \text {arccosh}(c x)}{x}+b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {3}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (-i c \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )-\frac {a+b \text {arccosh}(c x)}{x}+b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6351 |
| \(\displaystyle \frac {3}{2} c^2 \left (-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int -\frac {a+b \text {arccosh}(c x)}{(1-c x) (c x+1)}dx}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (-i c \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )-\frac {a+b \text {arccosh}(c x)}{x}+b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{2} c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{(1-c x) (c x+1)}dx}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (-i c \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )-\frac {a+b \text {arccosh}(c x)}{x}+b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6304 |
| \(\displaystyle \frac {3}{2} c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (-i c \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )-\frac {a+b \text {arccosh}(c x)}{x}+b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6318 |
| \(\displaystyle \frac {3}{2} c^2 \left (-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (-i c \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )-\frac {a+b \text {arccosh}(c x)}{x}+b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{2} c^2 \left (\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int i (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (-i c \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )-\frac {a+b \text {arccosh}(c x)}{x}+b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {3}{2} c^2 \left (\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \int (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (-i c \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )-\frac {a+b \text {arccosh}(c x)}{x}+b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {3}{2} c^2 \left (-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (i b \int \log \left (1-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-i b \int \log \left (1+e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (-i c \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )-\frac {a+b \text {arccosh}(c x)}{x}+b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {3}{2} c^2 \left (-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (i b \int e^{-\text {arccosh}(c x)} \log \left (1-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-i b \int e^{-\text {arccosh}(c x)} \log \left (1+e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (-i c \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )-\frac {a+b \text {arccosh}(c x)}{x}+b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {3}{2} c^2 \left (\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (-i c \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )-\frac {a+b \text {arccosh}(c x)}{x}+b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6361 |
| \(\displaystyle \frac {3}{2} c^2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} \int \frac {(a+b \text {arccosh}(c x))^2}{c x}d\text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (-i c \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )-\frac {a+b \text {arccosh}(c x)}{x}+b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{2} c^2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} \int (a+b \text {arccosh}(c x))^2 \csc \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (-i c \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )-\frac {a+b \text {arccosh}(c x)}{x}+b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle \frac {3}{2} c^2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} \left (-2 i b \int (a+b \text {arccosh}(c x)) \log \left (1-i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i b \int (a+b \text {arccosh}(c x)) \log \left (1+i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (-i c \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )-\frac {a+b \text {arccosh}(c x)}{x}+b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {3}{2} c^2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} \left (2 i b \left (b \int \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-2 i b \left (b \int \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (-i c \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )-\frac {a+b \text {arccosh}(c x)}{x}+b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {3}{2} c^2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} \left (2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (-i c \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )-\frac {a+b \text {arccosh}(c x)}{x}+b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {3}{2} c^2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} \left (2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2+2 i b \left (b \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c x)}\right )-\operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-2 i b \left (b \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c x)}\right )-\operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (-i c \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )-\frac {a+b \text {arccosh}(c x)}{x}+b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}\) |
-1/2*(a + b*ArcCosh[c*x])^2/(d*x^2*Sqrt[d - c^2*d*x^2]) - (b*c*Sqrt[-1 + c *x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/x) + b*c*ArcTan[Sqrt[-1 + c*x]*S qrt[1 + c*x]] - I*c*((2*I)*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]] + I*b*PolyLog[2, -E^ArcCosh[c*x]] - I*b*PolyLog[2, E^ArcCosh[c*x]])))/(d*Sqr t[d - c^2*d*x^2]) + (3*c^2*((a + b*ArcCosh[c*x])^2/(d*Sqrt[d - c^2*d*x^2]) - ((2*I)*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((2*I)*(a + b*ArcCosh[c*x])*ArcTa nh[E^ArcCosh[c*x]] + I*b*PolyLog[2, -E^ArcCosh[c*x]] - I*b*PolyLog[2, E^Ar cCosh[c*x]]))/(d*Sqrt[d - c^2*d*x^2]) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(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^ArcCosh[c*x]]) + b*PolyL og[3, I*E^ArcCosh[c*x]])))/(d*Sqrt[d - c^2*d*x^2])))/2
3.3.12.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ ))), x_] :> Simp[b*f Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d *e - f*(b*c + a*d), 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 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[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
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[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && 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_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[-(c*d)^(-1) Subst[Int[(a + b*x)^n*Csch[x], x], x, ArcCosh[c*x ]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 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_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1 ))) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[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 + 1)*((a + b*ArcCosh[c*x])^n/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1 )) Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - Simp[ b*c*(n/(2*f*(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] && LtQ[p, -1] && !GtQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.) *(x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x ]/Sqrt[d + e*x^2])] Subst[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]] , x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && Int egerQ[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]
\[\int \frac {\left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}{x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}d x\]
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{3}} \,d x } \]
integral(sqrt(-c^2*d*x^2 + d)*(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a ^2)/(c^4*d^2*x^7 - 2*c^2*d^2*x^5 + d^2*x^3), x)
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{x^{3} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{3}} \,d x } \]
-1/2*(3*c^2*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x))/d^(3/2 ) - 3*c^2/(sqrt(-c^2*d*x^2 + d)*d) + 1/(sqrt(-c^2*d*x^2 + d)*d*x^2))*a^2 + integrate(b^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/((-c^2*d*x^2 + d)^ (3/2)*x^3) + 2*a*b*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/((-c^2*d*x^2 + d )^(3/2)*x^3), x)
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{3}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{x^3\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]