Integrand size = 29, antiderivative size = 556 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d \sqrt {d-c^2 d x^2}}+\frac {94 b^2 (1-c x) (1+c x)}{27 c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 x^2 (1-c x) (1+c x)}{27 c^4 d \sqrt {d-c^2 d x^2}}+\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{3 c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {2 b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{9 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^6 d^2}+\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^4 d^2}+\frac {4 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}} \]
94/27*b^2*(-c*x+1)*(c*x+1)/c^6/d/(-c^2*d*x^2+d)^(1/2)+2/27*b^2*x^2*(-c*x+1 )*(c*x+1)/c^4/d/(-c^2*d*x^2+d)^(1/2)+x^4*(a+b*arccosh(c*x))^2/c^2/d/(-c^2* d*x^2+d)^(1/2)+16/3*a*b*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5/d/(-c^2*d*x^2+d) ^(1/2)+16/3*b^2*x*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5/d/(-c^2*d*x ^2+d)^(1/2)-2*b*x*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5/d/(-c ^2*d*x^2+d)^(1/2)+2/9*b*x^3*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2) /c^3/d/(-c^2*d*x^2+d)^(1/2)+4*b*(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)/c^6/d/(-c^2*d*x^2+d)^(1/2)+2 *b^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)/c^6/d/(-c^2*d*x^2+d)^(1/2)-2*b^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)/c^6/d/(-c^2*d*x^2+d)^(1/2)+8/3*(a+b*arcco sh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^6/d^2+4/3*x^2*(a+b*arccosh(c*x))^2*(-c^2 *d*x^2+d)^(1/2)/c^4/d^2
Time = 3.00 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.71 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-36 a^2 \left (-8+4 c^2 x^2+c^4 x^4\right )+3 a b \left (135 \text {arccosh}(c x)-60 \text {arccosh}(c x) \cosh (2 \text {arccosh}(c x))-3 \text {arccosh}(c x) \cosh (4 \text {arccosh}(c x))+72 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-72 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )+62 \sinh (2 \text {arccosh}(c x))+\sinh (4 \text {arccosh}(c x))\right )-b^2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (378 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)-378 c x \text {arccosh}(c x)+189 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)^2-6 \text {arccosh}(c x) \cosh (3 \text {arccosh}(c x))-54 \text {arccosh}(c x)^2 \coth \left (\frac {1}{2} \text {arccosh}(c x)\right )+216 \text {arccosh}(c x) \log \left (1-e^{-\text {arccosh}(c x)}\right )-216 \text {arccosh}(c x) \log \left (1+e^{-\text {arccosh}(c x)}\right )+216 \operatorname {PolyLog}\left (2,-e^{-\text {arccosh}(c x)}\right )-216 \operatorname {PolyLog}\left (2,e^{-\text {arccosh}(c x)}\right )+2 \sinh (3 \text {arccosh}(c x))+9 \text {arccosh}(c x)^2 \sinh (3 \text {arccosh}(c x))+54 \text {arccosh}(c x)^2 \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{108 c^6 d \sqrt {d-c^2 d x^2}} \]
(-36*a^2*(-8 + 4*c^2*x^2 + c^4*x^4) + 3*a*b*(135*ArcCosh[c*x] - 60*ArcCosh [c*x]*Cosh[2*ArcCosh[c*x]] - 3*ArcCosh[c*x]*Cosh[4*ArcCosh[c*x]] + 72*Sqrt [(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Cosh[ArcCosh[c*x]/2]] - 72*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Sinh[ArcCosh[c*x]/2]] + 62*Sinh[2*ArcCosh[c *x]] + Sinh[4*ArcCosh[c*x]]) - b^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(3 78*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) - 378*c*x*ArcCosh[c*x] + 189*Sqrt[ (-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]^2 - 6*ArcCosh[c*x]*Cosh[3*Arc Cosh[c*x]] - 54*ArcCosh[c*x]^2*Coth[ArcCosh[c*x]/2] + 216*ArcCosh[c*x]*Log [1 - E^(-ArcCosh[c*x])] - 216*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])] + 21 6*PolyLog[2, -E^(-ArcCosh[c*x])] - 216*PolyLog[2, E^(-ArcCosh[c*x])] + 2*S inh[3*ArcCosh[c*x]] + 9*ArcCosh[c*x]^2*Sinh[3*ArcCosh[c*x]] + 54*ArcCosh[c *x]^2*Tanh[ArcCosh[c*x]/2]))/(108*c^6*d*Sqrt[d - c^2*d*x^2])
Result contains complex when optimal does not.
Time = 3.09 (sec) , antiderivative size = 535, normalized size of antiderivative = 0.96, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.724, Rules used = {6349, 25, 6327, 6353, 111, 27, 83, 6298, 111, 27, 83, 6329, 2009, 6353, 83, 6318, 3042, 26, 4670, 2715, 2838}
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.
| \(\displaystyle \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6349 |
| \(\displaystyle -\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int -\frac {x^4 (a+b \text {arccosh}(c x))}{(1-c x) (c x+1)}dx}{c d \sqrt {d-c^2 d x^2}}-\frac {4 \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^4 (a+b \text {arccosh}(c x))}{(1-c x) (c x+1)}dx}{c d \sqrt {d-c^2 d x^2}}-\frac {4 \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^4 (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}-\frac {4 \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6353 |
| \(\displaystyle -\frac {4 \left (-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int x^2 (a+b \text {arccosh}(c x))dx}{3 c \sqrt {d-c^2 d x^2}}+\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}\right )}{c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\int \frac {x^2 (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c^2}+\frac {b \int \frac {x^3}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c}-\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle -\frac {4 \left (-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int x^2 (a+b \text {arccosh}(c x))dx}{3 c \sqrt {d-c^2 d x^2}}+\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}\right )}{c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\int \frac {x^2 (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c^2}+\frac {b \left (\frac {\int \frac {2 x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{3 c}-\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {4 \left (-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int x^2 (a+b \text {arccosh}(c x))dx}{3 c \sqrt {d-c^2 d x^2}}+\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}\right )}{c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\int \frac {x^2 (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c^2}+\frac {b \left (\frac {2 \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{3 c}-\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle -\frac {4 \left (-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int x^2 (a+b \text {arccosh}(c x))dx}{3 c \sqrt {d-c^2 d x^2}}+\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}\right )}{c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\int \frac {x^2 (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{3 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle -\frac {4 \left (\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \int \frac {x^3}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )}{3 c \sqrt {d-c^2 d x^2}}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}\right )}{c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\int \frac {x^2 (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{3 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle -\frac {4 \left (\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {\int \frac {2 x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}\right )}{c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\int \frac {x^2 (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{3 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {4 \left (\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {2 \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}\right )}{c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\int \frac {x^2 (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{3 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\int \frac {x^2 (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{3 c}\right )}{c d \sqrt {d-c^2 d x^2}}-\frac {4 \left (\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6329 |
| \(\displaystyle -\frac {4 \left (\frac {2 \left (-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int (a+b \text {arccosh}(c x))dx}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}\right )}{3 c^2}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\int \frac {x^2 (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{3 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\int \frac {x^2 (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{3 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {4 \left (-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}+\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{3 c^2}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 6353 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\frac {\int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx}{c^2}+\frac {b \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{c}-\frac {x (a+b \text {arccosh}(c x))}{c^2}}{c^2}-\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{3 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {4 \left (-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}+\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{3 c^2}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\frac {\int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx}{c^2}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}}{c^2}-\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{3 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {4 \left (-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}+\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{3 c^2}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 6318 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {-\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{c^3}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}}{c^2}-\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{3 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {4 \left (-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}+\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{3 c^2}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {-\frac {\int i (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{c^3}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}}{c^2}-\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{3 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {4 \left (-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}+\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{3 c^2}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {-\frac {i \int (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{c^3}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}}{c^2}-\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{3 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {4 \left (-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}+\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{3 c^2}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {-\frac {i \left (i b \int \log \left (1-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-i b \int \log \left (1+e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{c^3}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}}{c^2}-\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{3 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {4 \left (-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}+\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{3 c^2}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {-\frac {i \left (i b \int e^{-\text {arccosh}(c x)} \log \left (1-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-i b \int e^{-\text {arccosh}(c x)} \log \left (1+e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{c^3}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}}{c^2}-\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{3 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {4 \left (-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}+\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{3 c^2}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{c^3}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}}{c^2}-\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{3 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {4 \left (-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}+\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{3 c^2}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
(x^4*(a + b*ArcCosh[c*x])^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) - (4*(-1/3*(x^2*S qrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(c^2*d) - (2*b*Sqrt[-1 + c*x]*S qrt[1 + c*x]*(-1/3*(b*c*((2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^4) + (x^2*S qrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^2))) + (x^3*(a + b*ArcCosh[c*x]))/3))/(3 *c*Sqrt[d - c^2*d*x^2]) + (2*(-((Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^ 2)/(c^2*d)) - (2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a*x - (b*Sqrt[-1 + c*x]*S qrt[1 + c*x])/c + b*x*ArcCosh[c*x]))/(c*Sqrt[d - c^2*d*x^2])))/(3*c^2)))/( c^2*d) + (2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((b*((2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^4) + (x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^2)))/(3*c) - (x^3 *(a + b*ArcCosh[c*x]))/(3*c^2) + ((b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c^3 - ( x*(a + b*ArcCosh[c*x]))/c^2 - (I*((2*I)*(a + b*ArcCosh[c*x])*ArcTanh[E^Arc Cosh[c*x]] + I*b*PolyLog[2, -E^ArcCosh[c*x]] - I*b*PolyLog[2, E^ArcCosh[c* x]]))/c^3)/c^2))/(c*d*Sqrt[d - c^2*d*x^2])
3.3.4.3.1 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[((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 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
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_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & NeQ[m, -1]
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_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[(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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - S imp[b*f*(n/(2*c*(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}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IGtQ[m, 1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2 *p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*f*(n/(c*(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*Ar cCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2* d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]
Time = 1.46 (sec) , antiderivative size = 781, normalized size of antiderivative = 1.40
| method | result | size |
| default | \(a^{2} \left (-\frac {x^{4}}{3 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {-\frac {4 x^{2}}{3 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {8}{3 d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}}{c^{2}}\right )+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right )^{2} x^{4} c^{4}-6 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}+2 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+36 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}-84 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+92 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-54 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+54 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-54 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+54 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-72 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}+90 c x \,\operatorname {arccosh}\left (c x \right )-94 \sqrt {c x -1}\, \sqrt {c x +1}+54 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-54 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+54 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-54 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{27 \left (c^{2} x^{2}-1\right )^{2} d^{2} c^{6}}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (3 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-c^{5} x^{5}+12 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-14 c^{3} x^{3}-9 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+9 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) x^{2} c^{2}-24 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+15 c x +9 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-9 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )\right )}{9 \left (c^{2} x^{2}-1\right )^{2} d^{2} c^{6}}\) | \(781\) |
| parts | \(a^{2} \left (-\frac {x^{4}}{3 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {-\frac {4 x^{2}}{3 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {8}{3 d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}}{c^{2}}\right )+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right )^{2} x^{4} c^{4}-6 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}+2 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+36 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}-84 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+92 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-54 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+54 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-54 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+54 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-72 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}+90 c x \,\operatorname {arccosh}\left (c x \right )-94 \sqrt {c x -1}\, \sqrt {c x +1}+54 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-54 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+54 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-54 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{27 \left (c^{2} x^{2}-1\right )^{2} d^{2} c^{6}}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (3 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-c^{5} x^{5}+12 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-14 c^{3} x^{3}-9 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+9 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) x^{2} c^{2}-24 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+15 c x +9 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-9 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )\right )}{9 \left (c^{2} x^{2}-1\right )^{2} d^{2} c^{6}}\) | \(781\) |
a^2*(-1/3*x^4/c^2/d/(-c^2*d*x^2+d)^(1/2)+4/3/c^2*(-x^2/c^2/d/(-c^2*d*x^2+d )^(1/2)+2/d/c^4/(-c^2*d*x^2+d)^(1/2)))+1/27*b^2*(-d*(c^2*x^2-1))^(1/2)*(c* x-1)^(1/2)*(c*x+1)^(1/2)*(9*(c*x+1)^(1/2)*(c*x-1)^(1/2)*arccosh(c*x)^2*x^4 *c^4-6*arccosh(c*x)*c^5*x^5+2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+36*arcco sh(c*x)^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*c^2-84*c^3*x^3*arccosh(c*x)+92*( c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-54*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)* (c*x+1)^(1/2))*x^2*c^2+54*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2 ))*x^2*c^2-54*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))*x^2*c^2+54*polyl og(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*x^2*c^2-72*(c*x-1)^(1/2)*(c*x+1)^(1/ 2)*arccosh(c*x)^2+90*c*x*arccosh(c*x)-94*(c*x-1)^(1/2)*(c*x+1)^(1/2)+54*ar ccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-54*arccosh(c*x)*ln(1-c*x- (c*x-1)^(1/2)*(c*x+1)^(1/2))+54*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2) )-54*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))/(c^2*x^2-1)^2/d^2/c^6+2/9 *a*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(3*(c*x-1)^(1/2)*( c*x+1)^(1/2)*arccosh(c*x)*c^4*x^4-c^5*x^5+12*(c*x+1)^(1/2)*arccosh(c*x)*(c *x-1)^(1/2)*c^2*x^2-14*c^3*x^3-9*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*x^2 *c^2+9*ln((c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1)*x^2*c^2-24*arccosh(c*x)*(c*x- 1)^(1/2)*(c*x+1)^(1/2)+15*c*x+9*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-9*ln ((c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1))/(c^2*x^2-1)^2/d^2/c^6
\[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{5}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
integral((b^2*x^5*arccosh(c*x)^2 + 2*a*b*x^5*arccosh(c*x) + a^2*x^5)*sqrt( -c^2*d*x^2 + d)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)
\[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^{5} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{5}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
-1/3*a^2*(x^4/(sqrt(-c^2*d*x^2 + d)*c^2*d) + 4*x^2/(sqrt(-c^2*d*x^2 + d)*c ^4*d) - 8/(sqrt(-c^2*d*x^2 + d)*c^6*d)) + 1/3*(b^2*c^4*sqrt(d)*x^4 + 4*b^2 *c^2*sqrt(d)*x^2 - 8*b^2*sqrt(d))*sqrt(c*x + 1)*sqrt(-c*x + 1)*log(c*x + s qrt(c*x + 1)*sqrt(c*x - 1))^2/(c^8*d^2*x^2 - c^6*d^2) + integrate(-2/3*((4 *b^2*c^3*x^3 - (3*a*b*c^5 - b^2*c^5)*x^5 - 8*b^2*c*x)*(c*x + 1)*sqrt(c*x - 1) + (3*b^2*c^4*x^4 - (3*a*b*c^6 - b^2*c^6)*x^6 - 12*b^2*c^2*x^2 + 8*b^2) *sqrt(c*x + 1))*sqrt(-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(c^1 0*d^(3/2)*x^5 - 2*c^8*d^(3/2)*x^3 + c^6*d^(3/2)*x + (c^9*d^(3/2)*x^4 - 2*c ^7*d^(3/2)*x^2 + c^5*d^(3/2))*sqrt(c*x + 1)*sqrt(c*x - 1)), x)
Exception generated. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^5\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]