Integrand size = 29, antiderivative size = 836 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x} \, dx=\frac {68}{27} b^2 d^2 \sqrt {d-c^2 d x^2}-\frac {2}{27} b^2 c^2 d^2 x^2 \sqrt {d-c^2 d x^2}-\frac {2 a b c d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{75 (1-c x) (1+c x)}+\frac {8 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{225 (1-c x) (1+c x)}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{125 (1-c x) (1+c x)}-\frac {2 b^2 c d^2 x \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {22 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{45 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 \sqrt {-1+c x} \sqrt {1+c x}}+d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {1}{3} d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 i b d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 i b d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 i b^2 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 i b^2 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \]
1/3*d*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2+1/5*(-c^2*d*x^2+d)^(5/2)*( a+b*arccosh(c*x))^2+68/27*b^2*d^2*(-c^2*d*x^2+d)^(1/2)-2/27*b^2*c^2*d^2*x^ 2*(-c^2*d*x^2+d)^(1/2)+16/75*b^2*d^2*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/(-c *x+1)/(c*x+1)+8/225*b^2*d^2*(-c^2*x^2+1)^2*(-c^2*d*x^2+d)^(1/2)/(-c*x+1)/( c*x+1)+2/125*b^2*d^2*(-c^2*x^2+1)^3*(-c^2*d*x^2+d)^(1/2)/(-c*x+1)/(c*x+1)+ d^2*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)-2*a*b*c*d^2*x*(-c^2*d*x^2+d) ^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2*b^2*c*d^2*x*arccosh(c*x)*(-c^2*d*x^2+ d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-16/15*b*c*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)+22/45*b*c^3*d^2*x^3*(a+b*ar ccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2/25*b*c^5*d^ 2*x^5*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)- 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)+2*I*b^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 )+2*I*b*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)-2*I*b^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)-2*I*b*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)
Time = 5.84 (sec) , antiderivative size = 963, normalized size of antiderivative = 1.15 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x} \, dx=\frac {1}{15} a^2 d^2 \sqrt {d-c^2 d x^2} \left (23-11 c^2 x^2+3 c^4 x^4\right )-\frac {1}{27} b^2 d^2 \sqrt {d-c^2 d x^2} \left (2 (-13+\cosh (2 \text {arccosh}(c x)))+9 \text {arccosh}(c x)^2 (-1+\cosh (2 \text {arccosh}(c x)))+\frac {3 \sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x) (9 c x-\cosh (3 \text {arccosh}(c x)))}{-1+c x}\right )-\frac {a b d^2 \sqrt {d-c^2 d x^2} \left (9 c x+12 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \text {arccosh}(c x)-\cosh (3 \text {arccosh}(c x))\right )}{9 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+a^2 d^{5/2} \log (c x)-a^2 d^{5/2} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {2 a b d^2 \sqrt {d-c^2 d x^2} \left (-c x+\sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x)+c x \sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x)+i \text {arccosh}(c x) \log \left (1-i e^{-\text {arccosh}(c x)}\right )-i \text {arccosh}(c x) \log \left (1+i e^{-\text {arccosh}(c x)}\right )+i \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )-i \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+b^2 d^2 \sqrt {d-c^2 d x^2} \left (2+\frac {2 c x \sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x)}{1-c x}+\text {arccosh}(c x)^2+\frac {i \left (\text {arccosh}(c x)^2 \log \left (1-i e^{-\text {arccosh}(c x)}\right )-\text {arccosh}(c x)^2 \log \left (1+i e^{-\text {arccosh}(c x)}\right )+2 \text {arccosh}(c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )-2 \text {arccosh}(c x) \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )+2 \operatorname {PolyLog}\left (3,-i e^{-\text {arccosh}(c x)}\right )-2 \operatorname {PolyLog}\left (3,i e^{-\text {arccosh}(c x)}\right )\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}\right )-\frac {a b d^2 \sqrt {d-c^2 d x^2} \left (25 \cosh (3 \text {arccosh}(c x))+9 (-50 c x+\cosh (5 \text {arccosh}(c x)))+15 \text {arccosh}(c x) \left (30 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)-5 \sinh (3 \text {arccosh}(c x))-3 \sinh (5 \text {arccosh}(c x))\right )\right )}{1800 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}-\frac {b^2 d^2 \sqrt {d-c^2 d x^2} \left (13500 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)+30 \text {arccosh}(c x) (25 \cosh (3 \text {arccosh}(c x))+9 (-50 c x+\cosh (5 \text {arccosh}(c x))))-250 \sinh (3 \text {arccosh}(c x))+225 \text {arccosh}(c x)^2 \left (30 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)-5 \sinh (3 \text {arccosh}(c x))-3 \sinh (5 \text {arccosh}(c x))\right )-54 \sinh (5 \text {arccosh}(c x))\right )}{54000 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \]
(a^2*d^2*Sqrt[d - c^2*d*x^2]*(23 - 11*c^2*x^2 + 3*c^4*x^4))/15 - (b^2*d^2* Sqrt[d - c^2*d*x^2]*(2*(-13 + Cosh[2*ArcCosh[c*x]]) + 9*ArcCosh[c*x]^2*(-1 + Cosh[2*ArcCosh[c*x]]) + (3*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]*(9*c *x - Cosh[3*ArcCosh[c*x]]))/(-1 + c*x)))/27 - (a*b*d^2*Sqrt[d - c^2*d*x^2] *(9*c*x + 12*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*ArcCosh[c*x] - Cosh[ 3*ArcCosh[c*x]]))/(9*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + a^2*d^(5/2)*L og[c*x] - a^2*d^(5/2)*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (2*a*b*d^2*Sq rt[d - c^2*d*x^2]*(-(c*x) + Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + c*x* Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + I*ArcCosh[c*x]*Log[1 - I/E^ArcCo sh[c*x]] - I*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + I*PolyLog[2, (-I)/E^ ArcCosh[c*x]] - I*PolyLog[2, I/E^ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x )]*(1 + c*x)) + b^2*d^2*Sqrt[d - c^2*d*x^2]*(2 + (2*c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x])/(1 - c*x) + ArcCosh[c*x]^2 + (I*(ArcCosh[c*x]^2*Log [1 - I/E^ArcCosh[c*x]] - ArcCosh[c*x]^2*Log[1 + I/E^ArcCosh[c*x]] + 2*ArcC osh[c*x]*PolyLog[2, (-I)/E^ArcCosh[c*x]] - 2*ArcCosh[c*x]*PolyLog[2, I/E^A rcCosh[c*x]] + 2*PolyLog[3, (-I)/E^ArcCosh[c*x]] - 2*PolyLog[3, I/E^ArcCos h[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))) - (a*b*d^2*Sqrt[d - c^2* d*x^2]*(25*Cosh[3*ArcCosh[c*x]] + 9*(-50*c*x + Cosh[5*ArcCosh[c*x]]) + 15* ArcCosh[c*x]*(30*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) - 5*Sinh[3*ArcCosh[c *x]] - 3*Sinh[5*ArcCosh[c*x]])))/(1800*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + ...
Time = 4.46 (sec) , antiderivative size = 613, normalized size of antiderivative = 0.73, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.793, Rules used = {6345, 6304, 6309, 27, 1905, 1576, 1140, 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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x} \, dx\) |
| \(\Big \downarrow \) 6345 |
| \(\displaystyle -\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int (1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))dx}{5 \sqrt {c x-1} \sqrt {c x+1}}+d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}dx+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6304 |
| \(\displaystyle -\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))dx}{5 \sqrt {c x-1} \sqrt {c x+1}}+d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}dx+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6309 |
| \(\displaystyle d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}dx-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-b c \int \frac {x \left (3 c^4 x^4-10 c^2 x^2+15\right )}{15 \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}dx-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {1}{15} b c \int \frac {x \left (3 c^4 x^4-10 c^2 x^2+15\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 1905 |
| \(\displaystyle d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}dx-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \int \frac {x \left (3 c^4 x^4-10 c^2 x^2+15\right )}{\sqrt {c^2 x^2-1}}dx}{15 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 1576 |
| \(\displaystyle d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}dx-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \int \frac {3 c^4 x^4-10 c^2 x^2+15}{\sqrt {c^2 x^2-1}}dx^2}{30 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}dx-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \int \left (3 \left (c^2 x^2-1\right )^{3/2}-4 \sqrt {c^2 x^2-1}+\frac {8}{\sqrt {c^2 x^2-1}}\right )dx^2}{30 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}dx+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^2}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^2}+\frac {16 \sqrt {c^2 x^2-1}}{c^2}\right )}{30 \sqrt {c x-1} \sqrt {c x+1}}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6345 |
| \(\displaystyle d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \int -((1-c x) (c x+1) (a+b \text {arccosh}(c x)))dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}dx+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^2}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^2}+\frac {16 \sqrt {c^2 x^2-1}}{c^2}\right )}{30 \sqrt {c x-1} \sqrt {c x+1}}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle d \left (-\frac {2 b c d \sqrt {d-c^2 d x^2} \int (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}dx+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^2}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^2}+\frac {16 \sqrt {c^2 x^2-1}}{c^2}\right )}{30 \sqrt {c x-1} \sqrt {c x+1}}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6304 |
| \(\displaystyle d \left (-\frac {2 b c d \sqrt {d-c^2 d x^2} \int \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}dx+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^2}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^2}+\frac {16 \sqrt {c^2 x^2-1}}{c^2}\right )}{30 \sqrt {c x-1} \sqrt {c x+1}}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6309 |
| \(\displaystyle d \left (d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-b c \int \frac {x \left (3-c^2 x^2\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^2}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^2}+\frac {16 \sqrt {c^2 x^2-1}}{c^2}\right )}{30 \sqrt {c x-1} \sqrt {c x+1}}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \left (d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} b c \int \frac {x \left (3-c^2 x^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^2}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^2}+\frac {16 \sqrt {c^2 x^2-1}}{c^2}\right )}{30 \sqrt {c x-1} \sqrt {c x+1}}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 960 |
| \(\displaystyle d \left (d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} b c \left (\frac {7}{3} \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^2}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^2}+\frac {16 \sqrt {c^2 x^2-1}}{c^2}\right )}{30 \sqrt {c x-1} \sqrt {c x+1}}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle d \left (d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x}dx+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {7 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^2}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^2}+\frac {16 \sqrt {c^2 x^2-1}}{c^2}\right )}{30 \sqrt {c x-1} \sqrt {c x+1}}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6341 |
| \(\displaystyle d \left (d \left (-\frac {2 b c \sqrt {d-c^2 d x^2} \int (a+b \text {arccosh}(c x))dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {7 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^2}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^2}+\frac {16 \sqrt {c^2 x^2-1}}{c^2}\right )}{30 \sqrt {c x-1} \sqrt {c x+1}}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {7 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^2}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^2}+\frac {16 \sqrt {c^2 x^2-1}}{c^2}\right )}{30 \sqrt {c x-1} \sqrt {c x+1}}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6362 |
| \(\displaystyle d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{c x}d\text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {7 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^2}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^2}+\frac {16 \sqrt {c^2 x^2-1}}{c^2}\right )}{30 \sqrt {c x-1} \sqrt {c x+1}}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \int (a+b \text {arccosh}(c x))^2 \csc \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {7 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^2}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^2}+\frac {16 \sqrt {c^2 x^2-1}}{c^2}\right )}{30 \sqrt {c x-1} \sqrt {c x+1}}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \left (-2 i b \int (a+b \text {arccosh}(c x)) \log \left (1-i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i b \int (a+b \text {arccosh}(c x)) \log \left (1+i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {7 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^2}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^2}+\frac {16 \sqrt {c^2 x^2-1}}{c^2}\right )}{30 \sqrt {c x-1} \sqrt {c x+1}}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \left (2 i b \left (b \int \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-2 i b \left (b \int \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {7 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^2}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^2}+\frac {16 \sqrt {c^2 x^2-1}}{c^2}\right )}{30 \sqrt {c x-1} \sqrt {c x+1}}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \left (2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {7 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^2}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^2}+\frac {16 \sqrt {c^2 x^2-1}}{c^2}\right )}{30 \sqrt {c x-1} \sqrt {c x+1}}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \left (2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2+2 i b \left (b \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c x)}\right )-\operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-2 i b \left (b \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c x)}\right )-\operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {7 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^2}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^2}+\frac {16 \sqrt {c^2 x^2-1}}{c^2}\right )}{30 \sqrt {c x-1} \sqrt {c x+1}}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\) |
((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/5 - (2*b*c*d^2*Sqrt[d - c^2 *d*x^2]*(-1/30*(b*c*Sqrt[-1 + c^2*x^2]*((16*Sqrt[-1 + c^2*x^2])/c^2 - (8*( -1 + c^2*x^2)^(3/2))/(3*c^2) + (6*(-1 + c^2*x^2)^(5/2))/(5*c^2)))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + x*(a + b*ArcCosh[c*x]) - (2*c^2*x^3*(a + b*ArcCos h[c*x]))/3 + (c^4*x^5*(a + b*ArcCosh[c*x]))/5))/(5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + d*(((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/3 - (2*b*c*d*Sq rt[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*Sq rt[-1 + c*x]*Sqrt[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])*Poly Log[2, I*E^ArcCosh[c*x]]) + b*PolyLog[3, I*E^ArcCosh[c*x]])))/(Sqrt[-1 + c *x]*Sqrt[1 + c*x])))
3.2.90.3.1 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_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[1/2 Subst[Int[(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[((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_)*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 + 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]
\[\int \frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}{x}d x\]
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
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, x)
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x} \, dx=\text {Timed out} \]
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
-1/15*(15*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) - 5*(-c^2*d*x^2 + d)^(3/2)*d - 15*sqrt(-c^2*d*x ^2 + d)*d^2)*a^2 + integrate((-c^2*d*x^2 + d)^(5/2)*b^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/x + 2*(-c^2*d*x^2 + d)^(5/2)*a*b*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/x, x)
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x} \,d x \]