Integrand size = 27, antiderivative size = 454 \[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {3 b d^2 x^2 \sqrt {d-c^2 d x^2}}{512 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 x^4 \sqrt {d-c^2 d x^2}}{512 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {31 b c d^2 x^6 \sqrt {d-c^2 d x^2}}{960 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {21 b c^3 d^2 x^8 \sqrt {d-c^2 d x^2}}{640 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^{10} \sqrt {d-c^2 d x^2}}{100 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{256 c^4}-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{128 c^2}+\frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {3 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{512 b c^5 \sqrt {-1+c x} \sqrt {1+c x}} \]
1/16*d*x^5*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))+1/10*x^5*(-c^2*d*x^2+d) ^(5/2)*(a+b*arccosh(c*x))-3/256*d^2*x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1 /2)/c^4-1/128*d^2*x^3*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+1/32*d^2 *x^5*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)+3/512*b*d^2*x^2*(-c^2*d*x^2+d )^(1/2)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/512*b*d^2*x^4*(-c^2*d*x^2+d)^(1/ 2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-31/960*b*c*d^2*x^6*(-c^2*d*x^2+d)^(1/2)/( c*x-1)^(1/2)/(c*x+1)^(1/2)+21/640*b*c^3*d^2*x^8*(-c^2*d*x^2+d)^(1/2)/(c*x- 1)^(1/2)/(c*x+1)^(1/2)-1/100*b*c^5*d^2*x^10*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^( 1/2)/(c*x+1)^(1/2)-3/512*d^2*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c ^5/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 6.04 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.10 \[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {2880 a c d^2 x \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2} \left (-15-10 c^2 x^2+248 c^4 x^4-336 c^6 x^6+128 c^8 x^8\right )-43200 a d^{5/2} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+1600 b d^2 \sqrt {d-c^2 d x^2} \left (-72 \text {arccosh}(c x)^2+18 \cosh (2 \text {arccosh}(c x))-9 \cosh (4 \text {arccosh}(c x))-2 \cosh (6 \text {arccosh}(c x))+12 \text {arccosh}(c x) (-3 \sinh (2 \text {arccosh}(c x))+3 \sinh (4 \text {arccosh}(c x))+\sinh (6 \text {arccosh}(c x)))\right )+100 b d^2 \sqrt {d-c^2 d x^2} \left (1440 \text {arccosh}(c x)^2-576 \cosh (2 \text {arccosh}(c x))+144 \cosh (4 \text {arccosh}(c x))+64 \cosh (6 \text {arccosh}(c x))+9 \cosh (8 \text {arccosh}(c x))-24 \text {arccosh}(c x) (-48 \sinh (2 \text {arccosh}(c x))+24 \sinh (4 \text {arccosh}(c x))+16 \sinh (6 \text {arccosh}(c x))+3 \sinh (8 \text {arccosh}(c x)))\right )+b d^2 \sqrt {d-c^2 d x^2} \left (-50400 \text {arccosh}(c x)^2+25200 \cosh (2 \text {arccosh}(c x))-3600 \cosh (4 \text {arccosh}(c x))-2600 \cosh (6 \text {arccosh}(c x))-675 \cosh (8 \text {arccosh}(c x))-72 \cosh (10 \text {arccosh}(c x))+120 \text {arccosh}(c x) (-420 \sinh (2 \text {arccosh}(c x))+120 \sinh (4 \text {arccosh}(c x))+130 \sinh (6 \text {arccosh}(c x))+45 \sinh (8 \text {arccosh}(c x))+6 \sinh (10 \text {arccosh}(c x)))\right )}{3686400 c^5 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \]
(2880*a*c*d^2*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*( -15 - 10*c^2*x^2 + 248*c^4*x^4 - 336*c^6*x^6 + 128*c^8*x^8) - 43200*a*d^(5 /2)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/ (Sqrt[d]*(-1 + c^2*x^2))] + 1600*b*d^2*Sqrt[d - c^2*d*x^2]*(-72*ArcCosh[c* x]^2 + 18*Cosh[2*ArcCosh[c*x]] - 9*Cosh[4*ArcCosh[c*x]] - 2*Cosh[6*ArcCosh [c*x]] + 12*ArcCosh[c*x]*(-3*Sinh[2*ArcCosh[c*x]] + 3*Sinh[4*ArcCosh[c*x]] + Sinh[6*ArcCosh[c*x]])) + 100*b*d^2*Sqrt[d - c^2*d*x^2]*(1440*ArcCosh[c* x]^2 - 576*Cosh[2*ArcCosh[c*x]] + 144*Cosh[4*ArcCosh[c*x]] + 64*Cosh[6*Arc Cosh[c*x]] + 9*Cosh[8*ArcCosh[c*x]] - 24*ArcCosh[c*x]*(-48*Sinh[2*ArcCosh[ c*x]] + 24*Sinh[4*ArcCosh[c*x]] + 16*Sinh[6*ArcCosh[c*x]] + 3*Sinh[8*ArcCo sh[c*x]])) + b*d^2*Sqrt[d - c^2*d*x^2]*(-50400*ArcCosh[c*x]^2 + 25200*Cosh [2*ArcCosh[c*x]] - 3600*Cosh[4*ArcCosh[c*x]] - 2600*Cosh[6*ArcCosh[c*x]] - 675*Cosh[8*ArcCosh[c*x]] - 72*Cosh[10*ArcCosh[c*x]] + 120*ArcCosh[c*x]*(- 420*Sinh[2*ArcCosh[c*x]] + 120*Sinh[4*ArcCosh[c*x]] + 130*Sinh[6*ArcCosh[c *x]] + 45*Sinh[8*ArcCosh[c*x]] + 6*Sinh[10*ArcCosh[c*x]])))/(3686400*c^5*S qrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))
Time = 2.02 (sec) , antiderivative size = 431, normalized size of antiderivative = 0.95, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {6345, 82, 243, 49, 2009, 6345, 25, 82, 244, 2009, 6341, 15, 6354, 15, 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.
\(\displaystyle \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx\) |
\(\Big \downarrow \) 6345 |
\(\displaystyle \frac {1}{2} d \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x^5 (1-c x)^2 (c x+1)^2dx}{10 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 82 |
\(\displaystyle \frac {1}{2} d \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x^5 \left (1-c^2 x^2\right )^2dx}{10 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} d \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x^4 \left (1-c^2 x^2\right )^2dx^2}{20 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{2} d \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \left (c^4 x^8-2 c^2 x^6+x^4\right )dx^2}{20 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} d \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))dx+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right ) \sqrt {d-c^2 d x^2}}{20 \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 6345 |
\(\displaystyle \frac {1}{2} d \left (\frac {3}{8} d \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx+\frac {b c d \sqrt {d-c^2 d x^2} \int -x^5 (1-c x) (c x+1)dx}{8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))\right )+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right ) \sqrt {d-c^2 d x^2}}{20 \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} d \left (\frac {3}{8} d \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx-\frac {b c d \sqrt {d-c^2 d x^2} \int x^5 (1-c x) (c x+1)dx}{8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))\right )+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right ) \sqrt {d-c^2 d x^2}}{20 \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 82 |
\(\displaystyle \frac {1}{2} d \left (\frac {3}{8} d \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx-\frac {b c d \sqrt {d-c^2 d x^2} \int x^5 \left (1-c^2 x^2\right )dx}{8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))\right )+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right ) \sqrt {d-c^2 d x^2}}{20 \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {1}{2} d \left (\frac {3}{8} d \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx-\frac {b c d \sqrt {d-c^2 d x^2} \int \left (x^5-c^2 x^7\right )dx}{8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))\right )+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right ) \sqrt {d-c^2 d x^2}}{20 \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} d \left (\frac {3}{8} d \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right ) \sqrt {d-c^2 d x^2}}{8 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right ) \sqrt {d-c^2 d x^2}}{20 \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 6341 |
\(\displaystyle \frac {1}{2} d \left (\frac {3}{8} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^4 (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \int x^5dx}{6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))\right )+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right ) \sqrt {d-c^2 d x^2}}{8 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right ) \sqrt {d-c^2 d x^2}}{20 \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {1}{2} d \left (\frac {3}{8} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^4 (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right ) \sqrt {d-c^2 d x^2}}{8 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right ) \sqrt {d-c^2 d x^2}}{20 \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 6354 |
\(\displaystyle \frac {1}{2} d \left (\frac {3}{8} d \left (-\frac {\sqrt {d-c^2 d x^2} \left (\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 c^2}-\frac {b \int x^3dx}{4 c}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{4 c^2}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right ) \sqrt {d-c^2 d x^2}}{8 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right ) \sqrt {d-c^2 d x^2}}{20 \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {1}{2} d \left (\frac {3}{8} d \left (-\frac {\sqrt {d-c^2 d x^2} \left (\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right ) \sqrt {d-c^2 d x^2}}{8 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right ) \sqrt {d-c^2 d x^2}}{20 \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 6354 |
\(\displaystyle \frac {1}{2} d \left (\frac {3}{8} d \left (-\frac {\sqrt {d-c^2 d x^2} \left (\frac {3 \left (\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}-\frac {b \int xdx}{2 c}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right ) \sqrt {d-c^2 d x^2}}{8 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right ) \sqrt {d-c^2 d x^2}}{20 \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {1}{2} d \left (\frac {3}{8} d \left (-\frac {\sqrt {d-c^2 d x^2} \left (\frac {3 \left (\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right ) \sqrt {d-c^2 d x^2}}{8 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {b c d^2 \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right ) \sqrt {d-c^2 d x^2}}{20 \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 6308 |
\(\displaystyle \frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))+\frac {1}{2} d \left (\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))+\frac {3}{8} d \left (\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\sqrt {d-c^2 d x^2} \left (\frac {x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{4 c^2}+\frac {3 \left (\frac {(a+b \text {arccosh}(c x))^2}{4 b c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 c^2}-\frac {b x^4}{16 c}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {b c d \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right ) \sqrt {d-c^2 d x^2}}{8 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {b c d^2 \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right ) \sqrt {d-c^2 d x^2}}{20 \sqrt {c x-1} \sqrt {c x+1}}\) |
-1/20*(b*c*d^2*Sqrt[d - c^2*d*x^2]*(x^6/3 - (c^2*x^8)/2 + (c^4*x^10)/5))/( Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x^5*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[ c*x]))/10 + (d*(-1/8*(b*c*d*Sqrt[d - c^2*d*x^2]*(x^6/6 - (c^2*x^8)/8))/(Sq rt[-1 + c*x]*Sqrt[1 + c*x]) + (x^5*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c* x]))/8 + (3*d*(-1/36*(b*c*x^6*Sqrt[d - c^2*d*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/6 - (Sqrt[d - c^2 *d*x^2]*(-1/16*(b*x^4)/c + (x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCos h[c*x]))/(4*c^2) + (3*(-1/4*(b*x^2)/c + (x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(2*c^2) + (a + b*ArcCosh[c*x])^2/(4*b*c^3)))/(4*c^2))) /(6*Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/8))/2
3.1.87.3.1 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_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
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_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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_.)/(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_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc Cosh[c*x])^n/(f*(m + 2))), x] + (-Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/(Sq rt[1 + c*x]*Sqrt[-1 + c*x])] Int[(f*x)^m*((a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e* x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])] Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x ])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*Arc Cosh[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1)) Int[(f* x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*c*(n/(f*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[(f*x)^(m + 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1) , x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((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]
Leaf count of result is larger than twice the leaf count of optimal. \(1742\) vs. \(2(386)=772\).
Time = 0.90 (sec) , antiderivative size = 1743, normalized size of antiderivative = 3.84
method | result | size |
default | \(\text {Expression too large to display}\) | \(1743\) |
parts | \(\text {Expression too large to display}\) | \(1743\) |
-1/10*a*x^3*(-c^2*d*x^2+d)^(7/2)/c^2/d-3/80*a/c^4*x*(-c^2*d*x^2+d)^(7/2)/d +1/160*a/c^4*x*(-c^2*d*x^2+d)^(5/2)+1/128*a/c^4*d*x*(-c^2*d*x^2+d)^(3/2)+3 /256*a/c^4*d^2*x*(-c^2*d*x^2+d)^(1/2)+3/256*a/c^4*d^3/(c^2*d)^(1/2)*arctan ((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b*(-3/512*(-d*(c^2*x^2-1))^(1/2)/(c *x-1)^(1/2)/(c*x+1)^(1/2)/c^5*arccosh(c*x)^2*d^2+1/102400*(-d*(c^2*x^2-1)) ^(1/2)*(-1280*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^8*x^8+512*(c*x+1)^(1/2)*(c*x-1 )^(1/2)*c^10*x^10+50*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c* x+1)^(1/2)-832*c^5*x^5+170*c^3*x^3+1696*c^7*x^7+1120*(c*x+1)^(1/2)*(c*x-1) ^(1/2)*c^6*x^6-400*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4-1536*c^9*x^9-10*c*x +512*c^11*x^11)*(-1+10*arccosh(c*x))*d^2/(c*x+1)/c^5/(c*x-1)-1/32768*(-d*( c^2*x^2-1))^(1/2)*(128*c^9*x^9-320*c^7*x^7+128*(c*x-1)^(1/2)*(c*x+1)^(1/2) *c^8*x^8+272*c^5*x^5-256*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^6*x^6-88*c^3*x^3+16 0*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+8*c*x-32*(c*x-1)^(1/2)*(c*x+1)^(1/2) *c^2*x^2+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(-1+8*arccosh(c*x))*d^2/(c*x+1)/c^5/ (c*x-1)-1/12288*(-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 ))*(-1+6*arccosh(c*x))*d^2/(c*x+1)/c^5/(c*x-1)+1/2048*(-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))*(-1+4*arc...
\[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4} \,d x } \]
integral((a*c^4*d^2*x^8 - 2*a*c^2*d^2*x^6 + a*d^2*x^4 + (b*c^4*d^2*x^8 - 2 *b*c^2*d^2*x^6 + b*d^2*x^4)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d), x)
Timed out. \[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\text {Timed out} \]
\[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4} \,d x } \]
-1/1280*(128*(-c^2*d*x^2 + d)^(7/2)*x^3/(c^2*d) - 8*(-c^2*d*x^2 + d)^(5/2) *x/c^4 + 48*(-c^2*d*x^2 + d)^(7/2)*x/(c^4*d) - 10*(-c^2*d*x^2 + d)^(3/2)*d *x/c^4 - 15*sqrt(-c^2*d*x^2 + d)*d^2*x/c^4 - 15*d^(5/2)*arcsin(c*x)/c^5)*a + b*integrate((-c^2*d*x^2 + d)^(5/2)*x^4*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)), x)
\[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4} \,d x } \]
Timed out. \[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]