Integrand size = 26, antiderivative size = 486 \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {245 b^2 d^2 x \sqrt {d-c^2 d x^2}}{1152}+\frac {65 b^2 d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 d^2 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2}+\frac {115 b^2 d^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{1152 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 b c d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{48 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{18 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{48 b c \sqrt {-1+c x} \sqrt {1+c x}} \]
5/24*d*x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2+1/6*x*(-c^2*d*x^2+d)^(5 /2)*(a+b*arccosh(c*x))^2+245/1152*b^2*d^2*x*(-c^2*d*x^2+d)^(1/2)+65/1728*b ^2*d^2*x*(-c*x+1)*(c*x+1)*(-c^2*d*x^2+d)^(1/2)+1/108*b^2*d^2*x*(-c*x+1)^2* (c*x+1)^2*(-c^2*d*x^2+d)^(1/2)+5/16*d^2*x*(a+b*arccosh(c*x))^2*(-c^2*d*x^2 +d)^(1/2)+115/1152*b^2*d^2*arccosh(c*x)*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/ 2)/(c*x+1)^(1/2)-5/16*b*c*d^2*x^2*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/ (c*x-1)^(1/2)/(c*x+1)^(1/2)+5/48*b*d^2*(-c^2*x^2+1)^2*(a+b*arccosh(c*x))*( -c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/18*b*d^2*(-c^2*x^2+1)^ 3*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-5/ 48*d^2*(a+b*arccosh(c*x))^3*(-c^2*d*x^2+d)^(1/2)/b/c/(c*x-1)^(1/2)/(c*x+1) ^(1/2)
Time = 3.65 (sec) , antiderivative size = 740, normalized size of antiderivative = 1.52 \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {d^2 \left (9504 a^2 c x \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}+9504 a^2 c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}-7488 a^2 c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}-7488 a^2 c^4 x^4 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}+2304 a^2 c^5 x^5 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}+2304 a^2 c^6 x^6 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}-1440 b^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)^3-4320 a^2 \sqrt {d} \sqrt {\frac {-1+c x}{1+c x}} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-4320 a^2 c \sqrt {d} x \sqrt {\frac {-1+c x}{1+c x}} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-3240 a b \sqrt {d-c^2 d x^2} \cosh (2 \text {arccosh}(c x))+324 a b \sqrt {d-c^2 d x^2} \cosh (4 \text {arccosh}(c x))-24 a b \sqrt {d-c^2 d x^2} \cosh (6 \text {arccosh}(c x))+1620 b^2 \sqrt {d-c^2 d x^2} \sinh (2 \text {arccosh}(c x))-81 b^2 \sqrt {d-c^2 d x^2} \sinh (4 \text {arccosh}(c x))+4 b^2 \sqrt {d-c^2 d x^2} \sinh (6 \text {arccosh}(c x))-12 b \sqrt {d-c^2 d x^2} \text {arccosh}(c x) (270 b \cosh (2 \text {arccosh}(c x))-27 b \cosh (4 \text {arccosh}(c x))+2 b \cosh (6 \text {arccosh}(c x))-540 a \sinh (2 \text {arccosh}(c x))+108 a \sinh (4 \text {arccosh}(c x))-12 a \sinh (6 \text {arccosh}(c x)))+72 b \sqrt {d-c^2 d x^2} \text {arccosh}(c x)^2 (-60 a+45 b \sinh (2 \text {arccosh}(c x))-9 b \sinh (4 \text {arccosh}(c x))+b \sinh (6 \text {arccosh}(c x)))\right )}{13824 c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \]
(d^2*(9504*a^2*c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] + 9504*a ^2*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] - 7488*a^2*c^3*x ^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] - 7488*a^2*c^4*x^4*Sqrt[ (-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] + 2304*a^2*c^5*x^5*Sqrt[(-1 + c* x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] + 2304*a^2*c^6*x^6*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] - 1440*b^2*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]^3 - 4320*a^2*Sqrt[d]*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2 ])/(Sqrt[d]*(-1 + c^2*x^2))] - 4320*a^2*c*Sqrt[d]*x*Sqrt[(-1 + c*x)/(1 + c *x)]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 3240*a*b *Sqrt[d - c^2*d*x^2]*Cosh[2*ArcCosh[c*x]] + 324*a*b*Sqrt[d - c^2*d*x^2]*Co sh[4*ArcCosh[c*x]] - 24*a*b*Sqrt[d - c^2*d*x^2]*Cosh[6*ArcCosh[c*x]] + 162 0*b^2*Sqrt[d - c^2*d*x^2]*Sinh[2*ArcCosh[c*x]] - 81*b^2*Sqrt[d - c^2*d*x^2 ]*Sinh[4*ArcCosh[c*x]] + 4*b^2*Sqrt[d - c^2*d*x^2]*Sinh[6*ArcCosh[c*x]] - 12*b*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]*(270*b*Cosh[2*ArcCosh[c*x]] - 27*b*C osh[4*ArcCosh[c*x]] + 2*b*Cosh[6*ArcCosh[c*x]] - 540*a*Sinh[2*ArcCosh[c*x] ] + 108*a*Sinh[4*ArcCosh[c*x]] - 12*a*Sinh[6*ArcCosh[c*x]]) + 72*b*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]^2*(-60*a + 45*b*Sinh[2*ArcCosh[c*x]] - 9*b*Sinh[ 4*ArcCosh[c*x]] + b*Sinh[6*ArcCosh[c*x]])))/(13824*c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))
Time = 2.37 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.15, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {6312, 6312, 25, 6310, 6298, 101, 43, 6308, 6327, 6329, 40, 40, 40, 43}
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.
| \(\displaystyle \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6312 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x (1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{6} d \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6312 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x (1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{6} d \left (\frac {b c d \sqrt {d-c^2 d x^2} \int -x (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x (1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \int x (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6310 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x (1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \int x (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \left (-\frac {b c \sqrt {d-c^2 d x^2} \int x (a+b \text {arccosh}(c x))dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x (1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \int x (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \left (-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x (1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \int x (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \left (-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x (1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \int x (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6308 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x (1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \int x (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6329 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b \int (c x-1)^{5/2} (c x+1)^{5/2}dx}{6 c}-\frac {\left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))}{6 c^2}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \int (c x-1)^{3/2} (c x+1)^{3/2}dx}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 40 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b \left (\frac {1}{6} x (c x-1)^{5/2} (c x+1)^{5/2}-\frac {5}{6} \int (c x-1)^{3/2} (c x+1)^{3/2}dx\right )}{6 c}-\frac {\left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))}{6 c^2}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \int \sqrt {c x-1} \sqrt {c x+1}dx\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 40 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b \left (\frac {1}{6} x (c x-1)^{5/2} (c x+1)^{5/2}-\frac {5}{6} \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \int \sqrt {c x-1} \sqrt {c x+1}dx\right )\right )}{6 c}-\frac {\left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))}{6 c^2}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {1}{2} \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 40 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b \left (\frac {1}{6} x (c x-1)^{5/2} (c x+1)^{5/2}-\frac {5}{6} \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {1}{2} \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )\right )\right )}{6 c}-\frac {\left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))}{6 c^2}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {1}{2} \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {\left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))}{6 c^2}-\frac {b \left (\frac {1}{6} x (c x-1)^{5/2} (c x+1)^{5/2}-\frac {5}{6} \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )\right )}{6 c}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2+\frac {5}{6} d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
(x*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/6 - (b*c*d^2*Sqrt[d - c^2 *d*x^2]*(-1/6*((1 - c^2*x^2)^3*(a + b*ArcCosh[c*x]))/c^2 - (b*((x*(-1 + c* x)^(5/2)*(1 + c*x)^(5/2))/6 - (5*((x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/4 - (3*((x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/2 - ArcCosh[c*x]/(2*c)))/4))/6))/(6* c)))/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*d*((x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/4 + (3*d*((x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) ^2)/2 - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^3)/(6*b*c*Sqrt[-1 + c*x] *Sqrt[1 + c*x]) - (b*c*Sqrt[d - c^2*d*x^2]*((x^2*(a + b*ArcCosh[c*x]))/2 - (b*c*((x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*c^2) + ArcCosh[c*x]/(2*c^3)))/2 ))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/4 - (b*c*d*Sqrt[d - c^2*d*x^2]*(-1/4*( (1 - c^2*x^2)^2*(a + b*ArcCosh[c*x]))/c^2 + (b*((x*(-1 + c*x)^(3/2)*(1 + c *x)^(3/2))/4 - (3*((x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/2 - ArcCosh[c*x]/(2*c) ))/4))/(4*c)))/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/6
3.2.89.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[x* (a + b*x)^m*((c + d*x)^m/(2*m + 1)), x] + Simp[2*a*c*(m/(2*m + 1)) Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[ b*c + a*d, 0] && IGtQ[m + 1/2, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a *d, 0] && GtQ[a, 0] && GtQ[d/b, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 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_.)*((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_.)*((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]
Leaf count of result is larger than twice the leaf count of optimal. \(1736\) vs. \(2(422)=844\).
Time = 1.05 (sec) , antiderivative size = 1737, normalized size of antiderivative = 3.57
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1737\) |
| parts | \(\text {Expression too large to display}\) | \(1737\) |
1/6*x*(-c^2*d*x^2+d)^(5/2)*a^2+5/24*a^2*d*x*(-c^2*d*x^2+d)^(3/2)+5/16*a^2* d^2*x*(-c^2*d*x^2+d)^(1/2)+5/16*a^2*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2) *x/(-c^2*d*x^2+d)^(1/2))+b^2*(-5/48*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/( c*x+1)^(1/2)/c*arccosh(c*x)^3*d^2+1/6912*(-d*(c^2*x^2-1))^(1/2)*(32*c^7*x^ 7-64*c^5*x^5+32*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^6*x^6+38*c^3*x^3-48*(c*x+1)^ (1/2)*(c*x-1)^(1/2)*c^4*x^4-6*c*x+18*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-( c*x-1)^(1/2)*(c*x+1)^(1/2))*(18*arccosh(c*x)^2-6*arccosh(c*x)+1)*d^2/(c*x- 1)/(c*x+1)/c-3/1024*(-d*(c^2*x^2-1))^(1/2)*(8*c^5*x^5-12*c^3*x^3+8*(c*x+1) ^(1/2)*(c*x-1)^(1/2)*c^4*x^4+4*c*x-8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+( c*x-1)^(1/2)*(c*x+1)^(1/2))*(8*arccosh(c*x)^2-4*arccosh(c*x)+1)*d^2/(c*x-1 )/(c*x+1)/c+15/256*(-d*(c^2*x^2-1))^(1/2)*(2*c^3*x^3-2*c*x+2*(c*x-1)^(1/2) *(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1)^(1/2))*(2*arccosh(c*x)^2-2*ar ccosh(c*x)+1)*d^2/(c*x-1)/(c*x+1)/c+15/256*(-d*(c^2*x^2-1))^(1/2)*(-2*(c*x -1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+2*c^3*x^3+(c*x-1)^(1/2)*(c*x+1)^(1/2)-2*c* x)*(2*arccosh(c*x)^2+2*arccosh(c*x)+1)*d^2/(c*x-1)/(c*x+1)/c-3/1024*(-d*(c ^2*x^2-1))^(1/2)*(-8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+8*c^5*x^5+8*(c*x- 1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-12*c^3*x^3-(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c* x)*(8*arccosh(c*x)^2+4*arccosh(c*x)+1)*d^2/(c*x-1)/(c*x+1)/c+1/6912*(-d*(c ^2*x^2-1))^(1/2)*(-32*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^6*x^6+32*c^7*x^7+48*(c *x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4-64*c^5*x^5-18*(c*x-1)^(1/2)*(c*x+1)^(...
\[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \,d x } \]
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)
Timed out. \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\text {Timed out} \]
\[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \,d x } \]
1/48*(8*(-c^2*d*x^2 + d)^(5/2)*x + 10*(-c^2*d*x^2 + d)^(3/2)*d*x + 15*sqrt (-c^2*d*x^2 + d)*d^2*x + 15*d^(5/2)*arcsin(c*x)/c)*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 + 2*(-c^2*d*x ^2 + d)^(5/2)*a*b*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)), x)
Exception generated. \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]