Integrand size = 29, antiderivative size = 796 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b^2 c^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 b c^3 x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))^2}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2 \arctan \left (e^{\text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {26 b c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {13 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {13 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 i b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}} \]
5/6*c^2*(a+b*arccosh(c*x))^2/d/(-c^2*d*x^2+d)^(3/2)-1/2*(a+b*arccosh(c*x)) ^2/d/x^2/(-c^2*d*x^2+d)^(3/2)-1/3*b^2*c^2/d^2/(-c^2*d*x^2+d)^(1/2)+5/2*c^2 *(a+b*arccosh(c*x))^2/d^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^2/x/(-c^2*x^2+1)/(-c^2*d*x^2+d)^(1/2)-2/3*b*c^3* x*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/(-c^2*x^2+1)/(-c^2*d* x^2+d)^(1/2)+5*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^2/(-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^2/(-c^2*d*x^2+d )^(1/2)+26/3*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^2/(-c^2*d*x^2+d)^(1/2)+13/3*b^2*c^2*pol ylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/( -c^2*d*x^2+d)^(1/2)-5*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^2/(-c^2*d*x^2+d)^(1/2 )+5*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^2/(-c^2*d*x^2+d)^(1/2)-13/3*b^2*c^2*poly log(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/(-c ^2*d*x^2+d)^(1/2)+5*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^2/(-c^2*d*x^2+d)^(1/2)-5*I*b^2*c^2*polyl og(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^2/ (-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. \(5532\) vs. \(2(796)=1592\).
Time = 65.92 (sec) , antiderivative size = 5532, normalized size of antiderivative = 6.95 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Result too large to show} \]
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.
| \(\displaystyle \int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6347 |
| \(\displaystyle -\frac {b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x^2 (1-c x)^2 (c x+1)^2}dx}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle -\frac {b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6347 |
| \(\displaystyle -\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^2}dx-b c \int \frac {1}{x (c x-1)^{3/2} (c x+1)^{3/2}}dx-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle -\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^2}dx-b c \left (-\frac {\int \frac {c}{x \sqrt {c x-1} \sqrt {c x+1}}dx}{c}-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^2}dx-b c \left (-\int \frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle -\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^2}dx-b c \left (-c \int \frac {1}{(c x-1) (c x+1) c+c}d\left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^2}dx-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6316 |
| \(\displaystyle -\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \left (\frac {1}{2} \int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx+\frac {1}{2} b c \int \frac {x}{(c x-1)^{3/2} (c x+1)^{3/2}}dx+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle -\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \left (\frac {1}{2} \int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6318 |
| \(\displaystyle -\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \left (-\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \left (-\frac {\int i (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \left (-\frac {i \int (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \left (-\frac {i \left (i b \int \log \left (1-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-i b \int \log \left (1+e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \left (-\frac {i \left (i b \int e^{-\text {arccosh}(c x)} \log \left (1-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-i b \int e^{-\text {arccosh}(c x)} \log \left (1+e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {5}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6351 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{(1-c x)^2 (c x+1)^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6304 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6316 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{2} \int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx+\frac {1}{2} b c \int \frac {x}{(c x-1)^{3/2} (c x+1)^{3/2}}dx+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{2} \int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6318 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {\int i (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \int (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (i b \int \log \left (1-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-i b \int \log \left (1+e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (i b \int e^{-\text {arccosh}(c x)} \log \left (1-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-i b \int e^{-\text {arccosh}(c x)} \log \left (1+e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6351 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int -\frac {a+b \text {arccosh}(c x)}{(1-c x) (c x+1)}dx}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{(1-c x) (c x+1)}dx}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6304 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6318 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int i (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (3 c^2 \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}-b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
3.3.22.3.1 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[(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[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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 *n, 2*p]
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[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[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)^(p_), x _Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*d*(p + 1))), x] + (Simp[(2*p + 3)/(2*d*(p + 1)) Int[(d + e*x^2)^(p + 1)*(a + b* ArcCosh[c*x])^n, x], x] - Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[x*(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}, x] && EqQ[c^2* d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
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])
\[\int \frac {\left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}{x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
integral(-sqrt(-c^2*d*x^2 + d)*(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2)/(c^6*d^3*x^9 - 3*c^4*d^3*x^7 + 3*c^2*d^3*x^5 - d^3*x^3), x)
Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
-1/6*a^2*(15*c^2*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x))/d ^(5/2) - 15*c^2/(sqrt(-c^2*d*x^2 + d)*d^2) - 5*c^2/((-c^2*d*x^2 + d)^(3/2) *d) + 3/((-c^2*d*x^2 + d)^(3/2)*d*x^2)) + integrate(b^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/((-c^2*d*x^2 + d)^(5/2)*x^3) + 2*a*b*log(c*x + sqrt (c*x + 1)*sqrt(c*x - 1))/((-c^2*d*x^2 + d)^(5/2)*x^3), x)
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{x^3\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]