Integrand size = 22, antiderivative size = 605 \[ \int \left (c-a^2 c x^2\right )^{5/2} \text {arccosh}(a x)^3 \, dx=-\frac {865 a c^2 x^2 \sqrt {c-a^2 c x^2}}{2304 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {65 a^3 c^2 x^4 \sqrt {c-a^2 c x^2}}{2304 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {c^2 \left (1-a^2 x^2\right )^3 \sqrt {c-a^2 c x^2}}{216 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {245}{384} c^2 x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)+\frac {65}{576} c^2 x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \text {arccosh}(a x)+\frac {1}{36} c^2 x (1-a x)^2 (1+a x)^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)+\frac {115 c^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^2}{768 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 a c^2 x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^2}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5 c^2 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^2}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {c^2 \left (1-a^2 x^2\right )^3 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^2}{12 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5}{16} c^2 x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^3+\frac {5}{24} c x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^3+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2} \text {arccosh}(a x)^3-\frac {5 c^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^4}{64 a \sqrt {-1+a x} \sqrt {1+a x}} \]
5/24*c*x*(-a^2*c*x^2+c)^(3/2)*arccosh(a*x)^3+1/6*x*(-a^2*c*x^2+c)^(5/2)*ar ccosh(a*x)^3+245/384*c^2*x*arccosh(a*x)*(-a^2*c*x^2+c)^(1/2)+65/576*c^2*x* (-a*x+1)*(a*x+1)*arccosh(a*x)*(-a^2*c*x^2+c)^(1/2)+1/36*c^2*x*(-a*x+1)^2*( a*x+1)^2*arccosh(a*x)*(-a^2*c*x^2+c)^(1/2)+5/16*c^2*x*arccosh(a*x)^3*(-a^2 *c*x^2+c)^(1/2)-865/2304*a*c^2*x^2*(-a^2*c*x^2+c)^(1/2)/(a*x-1)^(1/2)/(a*x +1)^(1/2)+65/2304*a^3*c^2*x^4*(-a^2*c*x^2+c)^(1/2)/(a*x-1)^(1/2)/(a*x+1)^( 1/2)+1/216*c^2*(-a^2*x^2+1)^3*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1) ^(1/2)+115/768*c^2*arccosh(a*x)^2*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a* x+1)^(1/2)-15/32*a*c^2*x^2*arccosh(a*x)^2*(-a^2*c*x^2+c)^(1/2)/(a*x-1)^(1/ 2)/(a*x+1)^(1/2)+5/32*c^2*(-a^2*x^2+1)^2*arccosh(a*x)^2*(-a^2*c*x^2+c)^(1/ 2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)+1/12*c^2*(-a^2*x^2+1)^3*arccosh(a*x)^2*(- a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)-5/64*c^2*arccosh(a*x)^4*( -a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)
Time = 0.92 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.31 \[ \int \left (c-a^2 c x^2\right )^{5/2} \text {arccosh}(a x)^3 \, dx=\frac {c^2 \sqrt {c-a^2 c x^2} \left (-4320 \text {arccosh}(a x)^4-9720 \cosh (2 \text {arccosh}(a x))+243 \cosh (4 \text {arccosh}(a x))-8 \cosh (6 \text {arccosh}(a x))-72 \text {arccosh}(a x)^2 (270 \cosh (2 \text {arccosh}(a x))-27 \cosh (4 \text {arccosh}(a x))+2 \cosh (6 \text {arccosh}(a x)))+288 \text {arccosh}(a x)^3 (45 \sinh (2 \text {arccosh}(a x))-9 \sinh (4 \text {arccosh}(a x))+\sinh (6 \text {arccosh}(a x)))+12 \text {arccosh}(a x) (1620 \sinh (2 \text {arccosh}(a x))-81 \sinh (4 \text {arccosh}(a x))+4 \sinh (6 \text {arccosh}(a x)))\right )}{55296 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \]
(c^2*Sqrt[c - a^2*c*x^2]*(-4320*ArcCosh[a*x]^4 - 9720*Cosh[2*ArcCosh[a*x]] + 243*Cosh[4*ArcCosh[a*x]] - 8*Cosh[6*ArcCosh[a*x]] - 72*ArcCosh[a*x]^2*( 270*Cosh[2*ArcCosh[a*x]] - 27*Cosh[4*ArcCosh[a*x]] + 2*Cosh[6*ArcCosh[a*x] ]) + 288*ArcCosh[a*x]^3*(45*Sinh[2*ArcCosh[a*x]] - 9*Sinh[4*ArcCosh[a*x]] + Sinh[6*ArcCosh[a*x]]) + 12*ArcCosh[a*x]*(1620*Sinh[2*ArcCosh[a*x]] - 81* Sinh[4*ArcCosh[a*x]] + 4*Sinh[6*ArcCosh[a*x]])))/(55296*a*Sqrt[(-1 + a*x)/ (1 + a*x)]*(1 + a*x))
Time = 6.94 (sec) , antiderivative size = 651, normalized size of antiderivative = 1.08, number of steps used = 28, number of rules used = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.273, Rules used = {6312, 6312, 25, 6310, 6298, 6308, 6327, 6329, 6313, 25, 82, 241, 244, 2009, 6311, 15, 6308, 6313, 25, 82, 244, 2009, 6311, 15, 6308, 6354, 15, 6308}
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.
| \(\displaystyle \int \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 6312 |
| \(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \int x (1-a x)^2 (a x+1)^2 \text {arccosh}(a x)^2dx}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{6} c \int \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^3dx+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 6312 |
| \(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \int x (1-a x)^2 (a x+1)^2 \text {arccosh}(a x)^2dx}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{6} c \left (\frac {3 a c \sqrt {c-a^2 c x^2} \int -x (1-a x) (a x+1) \text {arccosh}(a x)^2dx}{4 \sqrt {a x-1} \sqrt {a x+1}}+\frac {3}{4} c \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^3dx+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \int x (1-a x)^2 (a x+1)^2 \text {arccosh}(a x)^2dx}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{6} c \left (-\frac {3 a c \sqrt {c-a^2 c x^2} \int x (1-a x) (a x+1) \text {arccosh}(a x)^2dx}{4 \sqrt {a x-1} \sqrt {a x+1}}+\frac {3}{4} c \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^3dx+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 6310 |
| \(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \int x (1-a x)^2 (a x+1)^2 \text {arccosh}(a x)^2dx}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{6} c \left (-\frac {3 a c \sqrt {c-a^2 c x^2} \int x (1-a x) (a x+1) \text {arccosh}(a x)^2dx}{4 \sqrt {a x-1} \sqrt {a x+1}}+\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \int x \text {arccosh}(a x)^2dx}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {c-a^2 c x^2} \int \frac {\text {arccosh}(a x)^3}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \int x (1-a x)^2 (a x+1)^2 \text {arccosh}(a x)^2dx}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{6} c \left (-\frac {3 a c \sqrt {c-a^2 c x^2} \int x (1-a x) (a x+1) \text {arccosh}(a x)^2dx}{4 \sqrt {a x-1} \sqrt {a x+1}}+\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {c-a^2 c x^2} \int \frac {\text {arccosh}(a x)^3}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 6308 |
| \(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \int x (1-a x)^2 (a x+1)^2 \text {arccosh}(a x)^2dx}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\right )-\frac {3 a c \sqrt {c-a^2 c x^2} \int x (1-a x) (a x+1) \text {arccosh}(a x)^2dx}{4 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \int x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2dx}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\right )-\frac {3 a c \sqrt {c-a^2 c x^2} \int x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^2dx}{4 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 6329 |
| \(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (-\frac {\int (a x-1)^{5/2} (a x+1)^{5/2} \text {arccosh}(a x)dx}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2}{6 a^2}\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\right )-\frac {3 a c \sqrt {c-a^2 c x^2} \left (\frac {\int (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)dx}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2}{4 a^2}\right )}{4 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 6313 |
| \(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (-\frac {-\frac {5}{6} \int (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)dx-\frac {1}{6} a \int x (1-a x)^2 (a x+1)^2dx+\frac {1}{6} x (a x-1)^{5/2} (a x+1)^{5/2} \text {arccosh}(a x)}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2}{6 a^2}\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\right )-\frac {3 a c \sqrt {c-a^2 c x^2} \left (\frac {-\frac {3}{4} \int \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)dx-\frac {1}{4} a \int -x (1-a x) (a x+1)dx+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2}{4 a^2}\right )}{4 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (-\frac {-\frac {5}{6} \int (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)dx-\frac {1}{6} a \int x (1-a x)^2 (a x+1)^2dx+\frac {1}{6} x (a x-1)^{5/2} (a x+1)^{5/2} \text {arccosh}(a x)}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2}{6 a^2}\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\right )-\frac {3 a c \sqrt {c-a^2 c x^2} \left (\frac {-\frac {3}{4} \int \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)dx+\frac {1}{4} a \int x (1-a x) (a x+1)dx+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2}{4 a^2}\right )}{4 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 82 |
| \(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (-\frac {-\frac {1}{6} a \int x \left (1-a^2 x^2\right )^2dx-\frac {5}{6} \int (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)dx+\frac {1}{6} x (a x-1)^{5/2} (a x+1)^{5/2} \text {arccosh}(a x)}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2}{6 a^2}\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\right )-\frac {3 a c \sqrt {c-a^2 c x^2} \left (\frac {\frac {1}{4} a \int x \left (1-a^2 x^2\right )dx-\frac {3}{4} \int \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)dx+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2}{4 a^2}\right )}{4 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (-\frac {-\frac {5}{6} \int (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)dx+\frac {\left (1-a^2 x^2\right )^3}{36 a}+\frac {1}{6} x (a x-1)^{5/2} (a x+1)^{5/2} \text {arccosh}(a x)}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2}{6 a^2}\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\right )-\frac {3 a c \sqrt {c-a^2 c x^2} \left (\frac {\frac {1}{4} a \int x \left (1-a^2 x^2\right )dx-\frac {3}{4} \int \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)dx+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2}{4 a^2}\right )}{4 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (-\frac {-\frac {5}{6} \int (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)dx+\frac {\left (1-a^2 x^2\right )^3}{36 a}+\frac {1}{6} x (a x-1)^{5/2} (a x+1)^{5/2} \text {arccosh}(a x)}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2}{6 a^2}\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\right )-\frac {3 a c \sqrt {c-a^2 c x^2} \left (\frac {\frac {1}{4} a \int \left (x-a^2 x^3\right )dx-\frac {3}{4} \int \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)dx+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2}{4 a^2}\right )}{4 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (-\frac {-\frac {5}{6} \int (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)dx+\frac {\left (1-a^2 x^2\right )^3}{36 a}+\frac {1}{6} x (a x-1)^{5/2} (a x+1)^{5/2} \text {arccosh}(a x)}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2}{6 a^2}\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\right )-\frac {3 a c \sqrt {c-a^2 c x^2} \left (\frac {-\frac {3}{4} \int \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)dx+\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2}{4 a^2}\right )}{4 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 6311 |
| \(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (-\frac {-\frac {5}{6} \int (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)dx+\frac {\left (1-a^2 x^2\right )^3}{36 a}+\frac {1}{6} x (a x-1)^{5/2} (a x+1)^{5/2} \text {arccosh}(a x)}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2}{6 a^2}\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\right )-\frac {3 a c \sqrt {c-a^2 c x^2} \left (\frac {-\frac {3}{4} \left (-\frac {1}{2} \int \frac {\text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx-\frac {a \int xdx}{2}+\frac {1}{2} x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)\right )+\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2}{4 a^2}\right )}{4 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (-\frac {-\frac {5}{6} \int (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)dx+\frac {\left (1-a^2 x^2\right )^3}{36 a}+\frac {1}{6} x (a x-1)^{5/2} (a x+1)^{5/2} \text {arccosh}(a x)}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2}{6 a^2}\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\right )-\frac {3 a c \sqrt {c-a^2 c x^2} \left (\frac {-\frac {3}{4} \left (-\frac {1}{2} \int \frac {\text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx+\frac {1}{2} x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)-\frac {a x^2}{4}\right )+\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2}{4 a^2}\right )}{4 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 6308 |
| \(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (-\frac {-\frac {5}{6} \int (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)dx+\frac {\left (1-a^2 x^2\right )^3}{36 a}+\frac {1}{6} x (a x-1)^{5/2} (a x+1)^{5/2} \text {arccosh}(a x)}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2}{6 a^2}\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {3 a c \left (\frac {\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )-\frac {3}{4} \left (\frac {1}{2} x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)-\frac {\text {arccosh}(a x)^2}{4 a}-\frac {a x^2}{4}\right )+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2}{4 a^2}\right ) \sqrt {c-a^2 c x^2}}{4 \sqrt {a x-1} \sqrt {a x+1}}\right )+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 6313 |
| \(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (-\frac {-\frac {5}{6} \left (-\frac {3}{4} \int \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)dx-\frac {1}{4} a \int -x (1-a x) (a x+1)dx+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)\right )+\frac {\left (1-a^2 x^2\right )^3}{36 a}+\frac {1}{6} x (a x-1)^{5/2} (a x+1)^{5/2} \text {arccosh}(a x)}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2}{6 a^2}\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {3 a c \left (\frac {\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )-\frac {3}{4} \left (\frac {1}{2} x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)-\frac {\text {arccosh}(a x)^2}{4 a}-\frac {a x^2}{4}\right )+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2}{4 a^2}\right ) \sqrt {c-a^2 c x^2}}{4 \sqrt {a x-1} \sqrt {a x+1}}\right )+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (-\frac {-\frac {5}{6} \left (-\frac {3}{4} \int \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)dx+\frac {1}{4} a \int x (1-a x) (a x+1)dx+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)\right )+\frac {\left (1-a^2 x^2\right )^3}{36 a}+\frac {1}{6} x (a x-1)^{5/2} (a x+1)^{5/2} \text {arccosh}(a x)}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2}{6 a^2}\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {3 a c \left (\frac {\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )-\frac {3}{4} \left (\frac {1}{2} x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)-\frac {\text {arccosh}(a x)^2}{4 a}-\frac {a x^2}{4}\right )+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2}{4 a^2}\right ) \sqrt {c-a^2 c x^2}}{4 \sqrt {a x-1} \sqrt {a x+1}}\right )+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 82 |
| \(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (-\frac {-\frac {5}{6} \left (\frac {1}{4} a \int x \left (1-a^2 x^2\right )dx-\frac {3}{4} \int \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)dx+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)\right )+\frac {\left (1-a^2 x^2\right )^3}{36 a}+\frac {1}{6} x (a x-1)^{5/2} (a x+1)^{5/2} \text {arccosh}(a x)}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2}{6 a^2}\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {3 a c \left (\frac {\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )-\frac {3}{4} \left (\frac {1}{2} x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)-\frac {\text {arccosh}(a x)^2}{4 a}-\frac {a x^2}{4}\right )+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2}{4 a^2}\right ) \sqrt {c-a^2 c x^2}}{4 \sqrt {a x-1} \sqrt {a x+1}}\right )+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (-\frac {-\frac {5}{6} \left (\frac {1}{4} a \int \left (x-a^2 x^3\right )dx-\frac {3}{4} \int \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)dx+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)\right )+\frac {\left (1-a^2 x^2\right )^3}{36 a}+\frac {1}{6} x (a x-1)^{5/2} (a x+1)^{5/2} \text {arccosh}(a x)}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2}{6 a^2}\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {3 a c \left (\frac {\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )-\frac {3}{4} \left (\frac {1}{2} x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)-\frac {\text {arccosh}(a x)^2}{4 a}-\frac {a x^2}{4}\right )+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2}{4 a^2}\right ) \sqrt {c-a^2 c x^2}}{4 \sqrt {a x-1} \sqrt {a x+1}}\right )+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (-\frac {-\frac {5}{6} \left (-\frac {3}{4} \int \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)dx+\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)\right )+\frac {\left (1-a^2 x^2\right )^3}{36 a}+\frac {1}{6} x (a x-1)^{5/2} (a x+1)^{5/2} \text {arccosh}(a x)}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2}{6 a^2}\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {3 a c \left (\frac {\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )-\frac {3}{4} \left (\frac {1}{2} x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)-\frac {\text {arccosh}(a x)^2}{4 a}-\frac {a x^2}{4}\right )+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2}{4 a^2}\right ) \sqrt {c-a^2 c x^2}}{4 \sqrt {a x-1} \sqrt {a x+1}}\right )+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 6311 |
| \(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (-\frac {-\frac {5}{6} \left (-\frac {3}{4} \left (-\frac {1}{2} \int \frac {\text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx-\frac {a \int xdx}{2}+\frac {1}{2} x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)\right )+\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)\right )+\frac {\left (1-a^2 x^2\right )^3}{36 a}+\frac {1}{6} x (a x-1)^{5/2} (a x+1)^{5/2} \text {arccosh}(a x)}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2}{6 a^2}\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {3 a c \left (\frac {\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )-\frac {3}{4} \left (\frac {1}{2} x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)-\frac {\text {arccosh}(a x)^2}{4 a}-\frac {a x^2}{4}\right )+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2}{4 a^2}\right ) \sqrt {c-a^2 c x^2}}{4 \sqrt {a x-1} \sqrt {a x+1}}\right )+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (-\frac {-\frac {5}{6} \left (-\frac {3}{4} \left (-\frac {1}{2} \int \frac {\text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx+\frac {1}{2} x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)-\frac {a x^2}{4}\right )+\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)\right )+\frac {\left (1-a^2 x^2\right )^3}{36 a}+\frac {1}{6} x (a x-1)^{5/2} (a x+1)^{5/2} \text {arccosh}(a x)}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2}{6 a^2}\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {3 a c \left (\frac {\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )-\frac {3}{4} \left (\frac {1}{2} x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)-\frac {\text {arccosh}(a x)^2}{4 a}-\frac {a x^2}{4}\right )+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2}{4 a^2}\right ) \sqrt {c-a^2 c x^2}}{4 \sqrt {a x-1} \sqrt {a x+1}}\right )+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 6308 |
| \(\displaystyle \frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {3 a c \left (\frac {\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )-\frac {3}{4} \left (\frac {1}{2} x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)-\frac {\text {arccosh}(a x)^2}{4 a}-\frac {a x^2}{4}\right )+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2}{4 a^2}\right ) \sqrt {c-a^2 c x^2}}{4 \sqrt {a x-1} \sqrt {a x+1}}\right )-\frac {a c^2 \left (-\frac {\left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2}{6 a^2}-\frac {-\frac {5}{6} \left (\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )-\frac {3}{4} \left (\frac {1}{2} x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)-\frac {\text {arccosh}(a x)^2}{4 a}-\frac {a x^2}{4}\right )+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)\right )+\frac {\left (1-a^2 x^2\right )^3}{36 a}+\frac {1}{6} x (a x-1)^{5/2} (a x+1)^{5/2} \text {arccosh}(a x)}{3 a}\right ) \sqrt {c-a^2 c x^2}}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 6354 |
| \(\displaystyle \frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \left (\frac {\int \frac {\text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}-\frac {\int xdx}{2 a}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{2 a^2}\right )\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {3 a c \left (\frac {\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )-\frac {3}{4} \left (\frac {1}{2} x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)-\frac {\text {arccosh}(a x)^2}{4 a}-\frac {a x^2}{4}\right )+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2}{4 a^2}\right ) \sqrt {c-a^2 c x^2}}{4 \sqrt {a x-1} \sqrt {a x+1}}\right )-\frac {a c^2 \left (-\frac {\left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2}{6 a^2}-\frac {-\frac {5}{6} \left (\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )-\frac {3}{4} \left (\frac {1}{2} x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)-\frac {\text {arccosh}(a x)^2}{4 a}-\frac {a x^2}{4}\right )+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)\right )+\frac {\left (1-a^2 x^2\right )^3}{36 a}+\frac {1}{6} x (a x-1)^{5/2} (a x+1)^{5/2} \text {arccosh}(a x)}{3 a}\right ) \sqrt {c-a^2 c x^2}}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \left (\frac {\int \frac {\text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {3 a c \left (\frac {\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )-\frac {3}{4} \left (\frac {1}{2} x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)-\frac {\text {arccosh}(a x)^2}{4 a}-\frac {a x^2}{4}\right )+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2}{4 a^2}\right ) \sqrt {c-a^2 c x^2}}{4 \sqrt {a x-1} \sqrt {a x+1}}\right )-\frac {a c^2 \left (-\frac {\left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2}{6 a^2}-\frac {-\frac {5}{6} \left (\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )-\frac {3}{4} \left (\frac {1}{2} x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)-\frac {\text {arccosh}(a x)^2}{4 a}-\frac {a x^2}{4}\right )+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)\right )+\frac {\left (1-a^2 x^2\right )^3}{36 a}+\frac {1}{6} x (a x-1)^{5/2} (a x+1)^{5/2} \text {arccosh}(a x)}{3 a}\right ) \sqrt {c-a^2 c x^2}}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}\) |
| \(\Big \downarrow \) 6308 |
| \(\displaystyle -\frac {a c^2 \left (-\frac {\left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2}{6 a^2}-\frac {-\frac {5}{6} \left (\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )-\frac {3}{4} \left (\frac {1}{2} x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)-\frac {\text {arccosh}(a x)^2}{4 a}-\frac {a x^2}{4}\right )+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)\right )+\frac {\left (1-a^2 x^2\right )^3}{36 a}+\frac {1}{6} x (a x-1)^{5/2} (a x+1)^{5/2} \text {arccosh}(a x)}{3 a}\right ) \sqrt {c-a^2 c x^2}}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}+\frac {5}{6} c \left (\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {3 a c \left (\frac {\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )-\frac {3}{4} \left (\frac {1}{2} x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)-\frac {\text {arccosh}(a x)^2}{4 a}-\frac {a x^2}{4}\right )+\frac {1}{4} x (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2}{4 a^2}\right ) \sqrt {c-a^2 c x^2}}{4 \sqrt {a x-1} \sqrt {a x+1}}+\frac {3}{4} c \left (-\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}-\frac {3 a \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \left (\frac {\text {arccosh}(a x)^2}{4 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )\right ) \sqrt {c-a^2 c x^2}}{2 \sqrt {a x-1} \sqrt {a x+1}}\right )\right )\) |
(x*(c - a^2*c*x^2)^(5/2)*ArcCosh[a*x]^3)/6 - (a*c^2*Sqrt[c - a^2*c*x^2]*(- 1/6*((1 - a^2*x^2)^3*ArcCosh[a*x]^2)/a^2 - ((1 - a^2*x^2)^3/(36*a) + (x*(- 1 + a*x)^(5/2)*(1 + a*x)^(5/2)*ArcCosh[a*x])/6 - (5*((a*(x^2/2 - (a^2*x^4) /4))/4 + (x*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x])/4 - (3*(-1/4*(a *x^2) + (x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/2 - ArcCosh[a*x]^2/( 4*a)))/4))/6)/(3*a)))/(2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (5*c*((x*(c - a^2 *c*x^2)^(3/2)*ArcCosh[a*x]^3)/4 + (3*c*((x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x ]^3)/2 - (Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^4)/(8*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (3*a*Sqrt[c - a^2*c*x^2]*((x^2*ArcCosh[a*x]^2)/2 - a*(-1/4*x^2/a + (x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(2*a^2) + ArcCosh[a*x]^2/( 4*a^3))))/(2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])))/4 - (3*a*c*Sqrt[c - a^2*c*x^2 ]*(-1/4*((1 - a^2*x^2)^2*ArcCosh[a*x]^2)/a^2 + ((a*(x^2/2 - (a^2*x^4)/4))/ 4 + (x*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x])/4 - (3*(-1/4*(a*x^2) + (x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/2 - ArcCosh[a*x]^2/(4*a)) )/4)/(2*a)))/(4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])))/6
3.3.46.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_) )^(p_.), x_] :> Int[(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && IntegerQ[m]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 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_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 ] && EqQ[e2, (-c)*d2] && NeQ[n, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcCosh[c*x])^n/2), x] + (-Simp[( 1/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])] Int[(a + b*ArcC osh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Simp[b*c*(n/2)*Simp[Sq rt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])] Int[x*(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]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d1_) + (e1_.)*(x_)]*Sqr t[(d2_) + (e2_.)*(x_)], x_Symbol] :> Simp[x*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x] *((a + b*ArcCosh[c*x])^n/2), x] + (-Simp[(1/2)*Simp[Sqrt[d1 + e1*x]/Sqrt[1 + c*x]]*Simp[Sqrt[d2 + e2*x]/Sqrt[-1 + c*x]] Int[(a + b*ArcCosh[c*x])^n/( Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d1 + e1*x] /Sqrt[1 + c*x]]*Simp[Sqrt[d2 + e2*x]/Sqrt[-1 + c*x]] Int[x*(a + b*ArcCosh [c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c *d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcCosh[c*x])^n/(2*p + 1)), x] + (Simp[2*d*(p/(2*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] && GtQ[p, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*( (d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Simp[x*(d1 + e1*x)^p*(d2 + e2*x)^p *((a + b*ArcCosh[c*x])^n/(2*p + 1)), x] + (Simp[2*d1*d2*(p/(2*p + 1)) Int [(d1 + e1*x)^(p - 1)*(d2 + e2*x)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - S imp[b*c*(n/(2*p + 1))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*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, d1, e1, d2, e2}, x] && EqQ[e1, c *d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && GtQ[p, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + ( e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1 *d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2 , e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(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_)*((d1_) + (e 1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1*e2*( m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*f *(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*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, d1, e1, d2, e2, f, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IGtQ[m, 1] && N eQ[m + 2*p + 1, 0]
Time = 1.31 (sec) , antiderivative size = 887, normalized size of antiderivative = 1.47
| method | result | size |
| default | \(-\frac {5 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (a x \right )^{4} c^{2}}{64 \sqrt {a x -1}\, \sqrt {a x +1}\, a}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (32 a^{7} x^{7}-64 a^{5} x^{5}+32 x^{6} a^{6} \sqrt {a x -1}\, \sqrt {a x +1}+38 a^{3} x^{3}-48 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}-6 a x +18 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (36 \operatorname {arccosh}\left (a x \right )^{3}-18 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )-1\right ) c^{2}}{13824 \left (a x -1\right ) \left (a x +1\right ) a}-\frac {3 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (8 a^{5} x^{5}-12 a^{3} x^{3}+8 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}+4 a x -8 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (32 \operatorname {arccosh}\left (a x \right )^{3}-24 \operatorname {arccosh}\left (a x \right )^{2}+12 \,\operatorname {arccosh}\left (a x \right )-3\right ) c^{2}}{4096 \left (a x -1\right ) \left (a x +1\right ) a}+\frac {15 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 a^{3} x^{3}-2 a x +2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (4 \operatorname {arccosh}\left (a x \right )^{3}-6 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )-3\right ) c^{2}}{512 \left (a x -1\right ) \left (a x +1\right ) a}+\frac {15 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 a^{3} x^{3}-2 a x -2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (4 \operatorname {arccosh}\left (a x \right )^{3}+6 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )+3\right ) c^{2}}{512 \left (a x -1\right ) \left (a x +1\right ) a}-\frac {3 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (8 a^{5} x^{5}-12 a^{3} x^{3}-8 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}+4 a x +8 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (32 \operatorname {arccosh}\left (a x \right )^{3}+24 \operatorname {arccosh}\left (a x \right )^{2}+12 \,\operatorname {arccosh}\left (a x \right )+3\right ) c^{2}}{4096 \left (a x -1\right ) \left (a x +1\right ) a}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-32 x^{6} a^{6} \sqrt {a x -1}\, \sqrt {a x +1}+32 a^{7} x^{7}+48 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}-64 a^{5} x^{5}-18 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+38 a^{3} x^{3}+\sqrt {a x -1}\, \sqrt {a x +1}-6 a x \right ) \left (36 \operatorname {arccosh}\left (a x \right )^{3}+18 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )+1\right ) c^{2}}{13824 \left (a x -1\right ) \left (a x +1\right ) a}\) | \(887\) |
-5/64*(-c*(a^2*x^2-1))^(1/2)/(a*x-1)^(1/2)/(a*x+1)^(1/2)/a*arccosh(a*x)^4* c^2+1/13824*(-c*(a^2*x^2-1))^(1/2)*(32*a^7*x^7-64*a^5*x^5+32*x^6*a^6*(a*x- 1)^(1/2)*(a*x+1)^(1/2)+38*a^3*x^3-48*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^4*x^4-6 *a*x+18*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-(a*x-1)^(1/2)*(a*x+1)^(1/2))*( 36*arccosh(a*x)^3-18*arccosh(a*x)^2+6*arccosh(a*x)-1)*c^2/(a*x-1)/(a*x+1)/ a-3/4096*(-c*(a^2*x^2-1))^(1/2)*(8*a^5*x^5-12*a^3*x^3+8*(a*x-1)^(1/2)*(a*x +1)^(1/2)*a^4*x^4+4*a*x-8*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+(a*x-1)^(1/2 )*(a*x+1)^(1/2))*(32*arccosh(a*x)^3-24*arccosh(a*x)^2+12*arccosh(a*x)-3)*c ^2/(a*x-1)/(a*x+1)/a+15/512*(-c*(a^2*x^2-1))^(1/2)*(2*a^3*x^3-2*a*x+2*a^2* x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-(a*x-1)^(1/2)*(a*x+1)^(1/2))*(4*arccosh(a* x)^3-6*arccosh(a*x)^2+6*arccosh(a*x)-3)*c^2/(a*x-1)/(a*x+1)/a+15/512*(-c*( a^2*x^2-1))^(1/2)*(2*a^3*x^3-2*a*x-2*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+( a*x-1)^(1/2)*(a*x+1)^(1/2))*(4*arccosh(a*x)^3+6*arccosh(a*x)^2+6*arccosh(a *x)+3)*c^2/(a*x-1)/(a*x+1)/a-3/4096*(-c*(a^2*x^2-1))^(1/2)*(8*a^5*x^5-12*a ^3*x^3-8*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^4*x^4+4*a*x+8*a^2*x^2*(a*x-1)^(1/2) *(a*x+1)^(1/2)-(a*x-1)^(1/2)*(a*x+1)^(1/2))*(32*arccosh(a*x)^3+24*arccosh( a*x)^2+12*arccosh(a*x)+3)*c^2/(a*x-1)/(a*x+1)/a+1/13824*(-c*(a^2*x^2-1))^( 1/2)*(-32*x^6*a^6*(a*x-1)^(1/2)*(a*x+1)^(1/2)+32*a^7*x^7+48*(a*x-1)^(1/2)* (a*x+1)^(1/2)*a^4*x^4-64*a^5*x^5-18*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+38 *a^3*x^3+(a*x-1)^(1/2)*(a*x+1)^(1/2)-6*a*x)*(36*arccosh(a*x)^3+18*arcco...
\[ \int \left (c-a^2 c x^2\right )^{5/2} \text {arccosh}(a x)^3 \, dx=\int { {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \operatorname {arcosh}\left (a x\right )^{3} \,d x } \]
Timed out. \[ \int \left (c-a^2 c x^2\right )^{5/2} \text {arccosh}(a x)^3 \, dx=\text {Timed out} \]
Exception generated. \[ \int \left (c-a^2 c x^2\right )^{5/2} \text {arccosh}(a x)^3 \, dx=\text {Exception raised: RuntimeError} \]
Exception generated. \[ \int \left (c-a^2 c x^2\right )^{5/2} \text {arccosh}(a x)^3 \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \left (c-a^2 c x^2\right )^{5/2} \text {arccosh}(a x)^3 \, dx=\int {\mathrm {acosh}\left (a\,x\right )}^3\,{\left (c-a^2\,c\,x^2\right )}^{5/2} \,d x \]