Integrand size = 10, antiderivative size = 276 \[ \int x^5 \text {arcsinh}(a x)^4 \, dx=\frac {245 x^2}{1152 a^4}-\frac {65 x^4}{3456 a^2}+\frac {x^6}{324}-\frac {245 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{576 a^5}+\frac {65 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{864 a^3}-\frac {x^5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{54 a}+\frac {245 \text {arcsinh}(a x)^2}{1152 a^6}+\frac {5 x^2 \text {arcsinh}(a x)^2}{16 a^4}-\frac {5 x^4 \text {arcsinh}(a x)^2}{48 a^2}+\frac {1}{18} x^6 \text {arcsinh}(a x)^2-\frac {5 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{24 a^5}+\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{36 a^3}-\frac {x^5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{9 a}+\frac {5 \text {arcsinh}(a x)^4}{96 a^6}+\frac {1}{6} x^6 \text {arcsinh}(a x)^4 \]
245/1152*x^2/a^4-65/3456*x^4/a^2+1/324*x^6+245/1152*arcsinh(a*x)^2/a^6+5/1 6*x^2*arcsinh(a*x)^2/a^4-5/48*x^4*arcsinh(a*x)^2/a^2+1/18*x^6*arcsinh(a*x) ^2+5/96*arcsinh(a*x)^4/a^6+1/6*x^6*arcsinh(a*x)^4-245/576*x*arcsinh(a*x)*( a^2*x^2+1)^(1/2)/a^5+65/864*x^3*arcsinh(a*x)*(a^2*x^2+1)^(1/2)/a^3-1/54*x^ 5*arcsinh(a*x)*(a^2*x^2+1)^(1/2)/a-5/24*x*arcsinh(a*x)^3*(a^2*x^2+1)^(1/2) /a^5+5/36*x^3*arcsinh(a*x)^3*(a^2*x^2+1)^(1/2)/a^3-1/9*x^5*arcsinh(a*x)^3* (a^2*x^2+1)^(1/2)/a
Time = 0.08 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.60 \[ \int x^5 \text {arcsinh}(a x)^4 \, dx=\frac {a^2 x^2 \left (2205-195 a^2 x^2+32 a^4 x^4\right )-6 a x \sqrt {1+a^2 x^2} \left (735-130 a^2 x^2+32 a^4 x^4\right ) \text {arcsinh}(a x)+9 \left (245+360 a^2 x^2-120 a^4 x^4+64 a^6 x^6\right ) \text {arcsinh}(a x)^2-144 a x \sqrt {1+a^2 x^2} \left (15-10 a^2 x^2+8 a^4 x^4\right ) \text {arcsinh}(a x)^3+108 \left (5+16 a^6 x^6\right ) \text {arcsinh}(a x)^4}{10368 a^6} \]
(a^2*x^2*(2205 - 195*a^2*x^2 + 32*a^4*x^4) - 6*a*x*Sqrt[1 + a^2*x^2]*(735 - 130*a^2*x^2 + 32*a^4*x^4)*ArcSinh[a*x] + 9*(245 + 360*a^2*x^2 - 120*a^4* x^4 + 64*a^6*x^6)*ArcSinh[a*x]^2 - 144*a*x*Sqrt[1 + a^2*x^2]*(15 - 10*a^2* x^2 + 8*a^4*x^4)*ArcSinh[a*x]^3 + 108*(5 + 16*a^6*x^6)*ArcSinh[a*x]^4)/(10 368*a^6)
Time = 2.75 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.78, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.300, Rules used = {6191, 6227, 6191, 6227, 15, 6191, 6227, 15, 6191, 6198, 6227, 15, 6198}
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.
| \(\displaystyle \int x^5 \text {arcsinh}(a x)^4 \, dx\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {1}{6} x^6 \text {arcsinh}(a x)^4-\frac {2}{3} a \int \frac {x^6 \text {arcsinh}(a x)^3}{\sqrt {a^2 x^2+1}}dx\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {1}{6} x^6 \text {arcsinh}(a x)^4-\frac {2}{3} a \left (-\frac {5 \int \frac {x^4 \text {arcsinh}(a x)^3}{\sqrt {a^2 x^2+1}}dx}{6 a^2}-\frac {\int x^5 \text {arcsinh}(a x)^2dx}{2 a}+\frac {x^5 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{6 a^2}\right )\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {1}{6} x^6 \text {arcsinh}(a x)^4-\frac {2}{3} a \left (-\frac {\frac {1}{6} x^6 \text {arcsinh}(a x)^2-\frac {1}{3} a \int \frac {x^6 \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{2 a}-\frac {5 \int \frac {x^4 \text {arcsinh}(a x)^3}{\sqrt {a^2 x^2+1}}dx}{6 a^2}+\frac {x^5 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{6 a^2}\right )\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {1}{6} x^6 \text {arcsinh}(a x)^4-\frac {2}{3} a \left (-\frac {5 \left (-\frac {3 \int \frac {x^2 \text {arcsinh}(a x)^3}{\sqrt {a^2 x^2+1}}dx}{4 a^2}-\frac {3 \int x^3 \text {arcsinh}(a x)^2dx}{4 a}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{4 a^2}\right )}{6 a^2}-\frac {\frac {1}{6} x^6 \text {arcsinh}(a x)^2-\frac {1}{3} a \left (-\frac {5 \int \frac {x^4 \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{6 a^2}-\frac {\int x^5dx}{6 a}+\frac {x^5 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{6 a^2}\right )}{2 a}+\frac {x^5 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{6 a^2}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {1}{6} x^6 \text {arcsinh}(a x)^4-\frac {2}{3} a \left (-\frac {5 \left (-\frac {3 \int \frac {x^2 \text {arcsinh}(a x)^3}{\sqrt {a^2 x^2+1}}dx}{4 a^2}-\frac {3 \int x^3 \text {arcsinh}(a x)^2dx}{4 a}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{4 a^2}\right )}{6 a^2}-\frac {\frac {1}{6} x^6 \text {arcsinh}(a x)^2-\frac {1}{3} a \left (-\frac {5 \int \frac {x^4 \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{6 a^2}+\frac {x^5 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{6 a^2}-\frac {x^6}{36 a}\right )}{2 a}+\frac {x^5 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{6 a^2}\right )\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {1}{6} x^6 \text {arcsinh}(a x)^4-\frac {2}{3} a \left (-\frac {5 \left (-\frac {3 \int \frac {x^2 \text {arcsinh}(a x)^3}{\sqrt {a^2 x^2+1}}dx}{4 a^2}-\frac {3 \left (\frac {1}{4} x^4 \text {arcsinh}(a x)^2-\frac {1}{2} a \int \frac {x^4 \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx\right )}{4 a}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{4 a^2}\right )}{6 a^2}-\frac {\frac {1}{6} x^6 \text {arcsinh}(a x)^2-\frac {1}{3} a \left (-\frac {5 \int \frac {x^4 \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{6 a^2}+\frac {x^5 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{6 a^2}-\frac {x^6}{36 a}\right )}{2 a}+\frac {x^5 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{6 a^2}\right )\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {1}{6} x^6 \text {arcsinh}(a x)^4-\frac {2}{3} a \left (-\frac {5 \left (-\frac {3 \left (-\frac {\int \frac {\text {arcsinh}(a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 a^2}-\frac {3 \int x \text {arcsinh}(a x)^2dx}{2 a}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 a^2}\right )}{4 a^2}-\frac {3 \left (\frac {1}{4} x^4 \text {arcsinh}(a x)^2-\frac {1}{2} a \left (-\frac {3 \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{4 a^2}-\frac {\int x^3dx}{4 a}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{4 a^2}\right )\right )}{4 a}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{4 a^2}\right )}{6 a^2}-\frac {\frac {1}{6} x^6 \text {arcsinh}(a x)^2-\frac {1}{3} a \left (-\frac {5 \left (-\frac {3 \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{4 a^2}-\frac {\int x^3dx}{4 a}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{4 a^2}\right )}{6 a^2}+\frac {x^5 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{6 a^2}-\frac {x^6}{36 a}\right )}{2 a}+\frac {x^5 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{6 a^2}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {1}{6} x^6 \text {arcsinh}(a x)^4-\frac {2}{3} a \left (-\frac {5 \left (-\frac {3 \left (-\frac {\int \frac {\text {arcsinh}(a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 a^2}-\frac {3 \int x \text {arcsinh}(a x)^2dx}{2 a}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 a^2}\right )}{4 a^2}-\frac {3 \left (\frac {1}{4} x^4 \text {arcsinh}(a x)^2-\frac {1}{2} a \left (-\frac {3 \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{4 a^2}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{4 a^2}-\frac {x^4}{16 a}\right )\right )}{4 a}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{4 a^2}\right )}{6 a^2}-\frac {\frac {1}{6} x^6 \text {arcsinh}(a x)^2-\frac {1}{3} a \left (-\frac {5 \left (-\frac {3 \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{4 a^2}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{4 a^2}-\frac {x^4}{16 a}\right )}{6 a^2}+\frac {x^5 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{6 a^2}-\frac {x^6}{36 a}\right )}{2 a}+\frac {x^5 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{6 a^2}\right )\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {1}{6} x^6 \text {arcsinh}(a x)^4-\frac {2}{3} a \left (-\frac {5 \left (-\frac {3 \left (-\frac {3 \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^2-a \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx\right )}{2 a}-\frac {\int \frac {\text {arcsinh}(a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 a^2}\right )}{4 a^2}-\frac {3 \left (\frac {1}{4} x^4 \text {arcsinh}(a x)^2-\frac {1}{2} a \left (-\frac {3 \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{4 a^2}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{4 a^2}-\frac {x^4}{16 a}\right )\right )}{4 a}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{4 a^2}\right )}{6 a^2}-\frac {\frac {1}{6} x^6 \text {arcsinh}(a x)^2-\frac {1}{3} a \left (-\frac {5 \left (-\frac {3 \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{4 a^2}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{4 a^2}-\frac {x^4}{16 a}\right )}{6 a^2}+\frac {x^5 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{6 a^2}-\frac {x^6}{36 a}\right )}{2 a}+\frac {x^5 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{6 a^2}\right )\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {1}{6} x^6 \text {arcsinh}(a x)^4-\frac {2}{3} a \left (-\frac {\frac {1}{6} x^6 \text {arcsinh}(a x)^2-\frac {1}{3} a \left (-\frac {5 \left (-\frac {3 \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{4 a^2}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{4 a^2}-\frac {x^4}{16 a}\right )}{6 a^2}+\frac {x^5 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{6 a^2}-\frac {x^6}{36 a}\right )}{2 a}-\frac {5 \left (-\frac {3 \left (\frac {1}{4} x^4 \text {arcsinh}(a x)^2-\frac {1}{2} a \left (-\frac {3 \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{4 a^2}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{4 a^2}-\frac {x^4}{16 a}\right )\right )}{4 a}-\frac {3 \left (-\frac {3 \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^2-a \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx\right )}{2 a}-\frac {\text {arcsinh}(a x)^4}{8 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 a^2}\right )}{4 a^2}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{4 a^2}\right )}{6 a^2}+\frac {x^5 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{6 a^2}\right )\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {1}{6} x^6 \text {arcsinh}(a x)^4-\frac {2}{3} a \left (-\frac {\frac {1}{6} x^6 \text {arcsinh}(a x)^2-\frac {1}{3} a \left (-\frac {5 \left (-\frac {3 \left (-\frac {\int \frac {\text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2}-\frac {\int xdx}{2 a}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 a^2}\right )}{4 a^2}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{4 a^2}-\frac {x^4}{16 a}\right )}{6 a^2}+\frac {x^5 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{6 a^2}-\frac {x^6}{36 a}\right )}{2 a}-\frac {5 \left (-\frac {3 \left (\frac {1}{4} x^4 \text {arcsinh}(a x)^2-\frac {1}{2} a \left (-\frac {3 \left (-\frac {\int \frac {\text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2}-\frac {\int xdx}{2 a}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 a^2}\right )}{4 a^2}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{4 a^2}-\frac {x^4}{16 a}\right )\right )}{4 a}-\frac {3 \left (-\frac {3 \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^2-a \left (-\frac {\int \frac {\text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2}-\frac {\int xdx}{2 a}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 a^2}\right )\right )}{2 a}-\frac {\text {arcsinh}(a x)^4}{8 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 a^2}\right )}{4 a^2}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{4 a^2}\right )}{6 a^2}+\frac {x^5 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{6 a^2}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {1}{6} x^6 \text {arcsinh}(a x)^4-\frac {2}{3} a \left (-\frac {\frac {1}{6} x^6 \text {arcsinh}(a x)^2-\frac {1}{3} a \left (-\frac {5 \left (-\frac {3 \left (-\frac {\int \frac {\text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )}{4 a^2}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{4 a^2}-\frac {x^4}{16 a}\right )}{6 a^2}+\frac {x^5 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{6 a^2}-\frac {x^6}{36 a}\right )}{2 a}-\frac {5 \left (-\frac {3 \left (\frac {1}{4} x^4 \text {arcsinh}(a x)^2-\frac {1}{2} a \left (-\frac {3 \left (-\frac {\int \frac {\text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )}{4 a^2}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{4 a^2}-\frac {x^4}{16 a}\right )\right )}{4 a}-\frac {3 \left (-\frac {3 \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^2-a \left (-\frac {\int \frac {\text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )\right )}{2 a}-\frac {\text {arcsinh}(a x)^4}{8 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 a^2}\right )}{4 a^2}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{4 a^2}\right )}{6 a^2}+\frac {x^5 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{6 a^2}\right )\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {1}{6} x^6 \text {arcsinh}(a x)^4-\frac {2}{3} a \left (\frac {x^5 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{6 a^2}-\frac {5 \left (\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{4 a^2}-\frac {3 \left (-\frac {\text {arcsinh}(a x)^4}{8 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 a^2}-\frac {3 \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^2-a \left (-\frac {\text {arcsinh}(a x)^2}{4 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )\right )}{2 a}\right )}{4 a^2}-\frac {3 \left (\frac {1}{4} x^4 \text {arcsinh}(a x)^2-\frac {1}{2} a \left (\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{4 a^2}-\frac {3 \left (-\frac {\text {arcsinh}(a x)^2}{4 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )}{4 a^2}-\frac {x^4}{16 a}\right )\right )}{4 a}\right )}{6 a^2}-\frac {\frac {1}{6} x^6 \text {arcsinh}(a x)^2-\frac {1}{3} a \left (\frac {x^5 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{6 a^2}-\frac {5 \left (\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{4 a^2}-\frac {3 \left (-\frac {\text {arcsinh}(a x)^2}{4 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )}{4 a^2}-\frac {x^4}{16 a}\right )}{6 a^2}-\frac {x^6}{36 a}\right )}{2 a}\right )\) |
(x^6*ArcSinh[a*x]^4)/6 - (2*a*((x^5*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(6*a ^2) - ((x^6*ArcSinh[a*x]^2)/6 - (a*(-1/36*x^6/a + (x^5*Sqrt[1 + a^2*x^2]*A rcSinh[a*x])/(6*a^2) - (5*(-1/16*x^4/a + (x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a* x])/(4*a^2) - (3*(-1/4*x^2/a + (x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(2*a^2) - ArcSinh[a*x]^2/(4*a^3)))/(4*a^2)))/(6*a^2)))/3)/(2*a) - (5*((x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(4*a^2) - (3*((x^4*ArcSinh[a*x]^2)/4 - (a*(-1/1 6*x^4/a + (x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(4*a^2) - (3*(-1/4*x^2/a + (x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(2*a^2) - ArcSinh[a*x]^2/(4*a^3)))/(4*a ^2)))/2))/(4*a) - (3*((x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(2*a^2) - ArcSi nh[a*x]^4/(8*a^3) - (3*((x^2*ArcSinh[a*x]^2)/2 - a*(-1/4*x^2/a + (x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(2*a^2) - ArcSinh[a*x]^2/(4*a^3))))/(2*a)))/(4*a ^2)))/(6*a^2)))/3
3.1.32.3.1 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_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c ^2*d] && NeQ[n, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int [(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] ) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ m, 1] && NeQ[m + 2*p + 1, 0]
Time = 0.22 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.88
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{6} x^{6} \operatorname {arcsinh}\left (a x \right )^{4}}{6}-\frac {a^{5} x^{5} \operatorname {arcsinh}\left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{9}+\frac {5 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{36}-\frac {5 \operatorname {arcsinh}\left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}\, a x}{24}+\frac {5 \operatorname {arcsinh}\left (a x \right )^{4}}{96}+\frac {\operatorname {arcsinh}\left (a x \right )^{2} a^{6} x^{6}}{18}-\frac {\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{5} x^{5}}{54}+\frac {65 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}}{864}-\frac {245 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x}{576}-\frac {115 \operatorname {arcsinh}\left (a x \right )^{2}}{1152}+\frac {a^{6} x^{6}}{324}-\frac {65 a^{4} x^{4}}{3456}+\frac {245 a^{2} x^{2}}{1152}+\frac {245}{1152}-\frac {5 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )^{2}}{48}+\frac {5 \operatorname {arcsinh}\left (a x \right )^{2} \left (a^{2} x^{2}+1\right )}{16}}{a^{6}}\) | \(242\) |
| default | \(\frac {\frac {a^{6} x^{6} \operatorname {arcsinh}\left (a x \right )^{4}}{6}-\frac {a^{5} x^{5} \operatorname {arcsinh}\left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{9}+\frac {5 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{36}-\frac {5 \operatorname {arcsinh}\left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}\, a x}{24}+\frac {5 \operatorname {arcsinh}\left (a x \right )^{4}}{96}+\frac {\operatorname {arcsinh}\left (a x \right )^{2} a^{6} x^{6}}{18}-\frac {\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{5} x^{5}}{54}+\frac {65 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}}{864}-\frac {245 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x}{576}-\frac {115 \operatorname {arcsinh}\left (a x \right )^{2}}{1152}+\frac {a^{6} x^{6}}{324}-\frac {65 a^{4} x^{4}}{3456}+\frac {245 a^{2} x^{2}}{1152}+\frac {245}{1152}-\frac {5 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )^{2}}{48}+\frac {5 \operatorname {arcsinh}\left (a x \right )^{2} \left (a^{2} x^{2}+1\right )}{16}}{a^{6}}\) | \(242\) |
1/a^6*(1/6*a^6*x^6*arcsinh(a*x)^4-1/9*a^5*x^5*arcsinh(a*x)^3*(a^2*x^2+1)^( 1/2)+5/36*a^3*x^3*arcsinh(a*x)^3*(a^2*x^2+1)^(1/2)-5/24*arcsinh(a*x)^3*(a^ 2*x^2+1)^(1/2)*a*x+5/96*arcsinh(a*x)^4+1/18*arcsinh(a*x)^2*a^6*x^6-1/54*ar csinh(a*x)*(a^2*x^2+1)^(1/2)*a^5*x^5+65/864*a^3*x^3*arcsinh(a*x)*(a^2*x^2+ 1)^(1/2)-245/576*arcsinh(a*x)*(a^2*x^2+1)^(1/2)*a*x-115/1152*arcsinh(a*x)^ 2+1/324*a^6*x^6-65/3456*a^4*x^4+245/1152*a^2*x^2+245/1152-5/48*a^4*x^4*arc sinh(a*x)^2+5/16*arcsinh(a*x)^2*(a^2*x^2+1))
Time = 0.27 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.75 \[ \int x^5 \text {arcsinh}(a x)^4 \, dx=\frac {32 \, a^{6} x^{6} - 195 \, a^{4} x^{4} + 108 \, {\left (16 \, a^{6} x^{6} + 5\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{4} - 144 \, {\left (8 \, a^{5} x^{5} - 10 \, a^{3} x^{3} + 15 \, a x\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} + 2205 \, a^{2} x^{2} + 9 \, {\left (64 \, a^{6} x^{6} - 120 \, a^{4} x^{4} + 360 \, a^{2} x^{2} + 245\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - 6 \, {\left (32 \, a^{5} x^{5} - 130 \, a^{3} x^{3} + 735 \, a x\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{10368 \, a^{6}} \]
1/10368*(32*a^6*x^6 - 195*a^4*x^4 + 108*(16*a^6*x^6 + 5)*log(a*x + sqrt(a^ 2*x^2 + 1))^4 - 144*(8*a^5*x^5 - 10*a^3*x^3 + 15*a*x)*sqrt(a^2*x^2 + 1)*lo g(a*x + sqrt(a^2*x^2 + 1))^3 + 2205*a^2*x^2 + 9*(64*a^6*x^6 - 120*a^4*x^4 + 360*a^2*x^2 + 245)*log(a*x + sqrt(a^2*x^2 + 1))^2 - 6*(32*a^5*x^5 - 130* a^3*x^3 + 735*a*x)*sqrt(a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 + 1)))/a^6
Time = 1.32 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.97 \[ \int x^5 \text {arcsinh}(a x)^4 \, dx=\begin {cases} \frac {x^{6} \operatorname {asinh}^{4}{\left (a x \right )}}{6} + \frac {x^{6} \operatorname {asinh}^{2}{\left (a x \right )}}{18} + \frac {x^{6}}{324} - \frac {x^{5} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{9 a} - \frac {x^{5} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{54 a} - \frac {5 x^{4} \operatorname {asinh}^{2}{\left (a x \right )}}{48 a^{2}} - \frac {65 x^{4}}{3456 a^{2}} + \frac {5 x^{3} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{36 a^{3}} + \frac {65 x^{3} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{864 a^{3}} + \frac {5 x^{2} \operatorname {asinh}^{2}{\left (a x \right )}}{16 a^{4}} + \frac {245 x^{2}}{1152 a^{4}} - \frac {5 x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{24 a^{5}} - \frac {245 x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{576 a^{5}} + \frac {5 \operatorname {asinh}^{4}{\left (a x \right )}}{96 a^{6}} + \frac {245 \operatorname {asinh}^{2}{\left (a x \right )}}{1152 a^{6}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((x**6*asinh(a*x)**4/6 + x**6*asinh(a*x)**2/18 + x**6/324 - x**5* sqrt(a**2*x**2 + 1)*asinh(a*x)**3/(9*a) - x**5*sqrt(a**2*x**2 + 1)*asinh(a *x)/(54*a) - 5*x**4*asinh(a*x)**2/(48*a**2) - 65*x**4/(3456*a**2) + 5*x**3 *sqrt(a**2*x**2 + 1)*asinh(a*x)**3/(36*a**3) + 65*x**3*sqrt(a**2*x**2 + 1) *asinh(a*x)/(864*a**3) + 5*x**2*asinh(a*x)**2/(16*a**4) + 245*x**2/(1152*a **4) - 5*x*sqrt(a**2*x**2 + 1)*asinh(a*x)**3/(24*a**5) - 245*x*sqrt(a**2*x **2 + 1)*asinh(a*x)/(576*a**5) + 5*asinh(a*x)**4/(96*a**6) + 245*asinh(a*x )**2/(1152*a**6), Ne(a, 0)), (0, True))
\[ \int x^5 \text {arcsinh}(a x)^4 \, dx=\int { x^{5} \operatorname {arsinh}\left (a x\right )^{4} \,d x } \]
1/6*x^6*log(a*x + sqrt(a^2*x^2 + 1))^4 - integrate(2/3*(a^3*x^8 + sqrt(a^2 *x^2 + 1)*a^2*x^7 + a*x^6)*log(a*x + sqrt(a^2*x^2 + 1))^3/(a^3*x^3 + a*x + (a^2*x^2 + 1)^(3/2)), x)
Exception generated. \[ \int x^5 \text {arcsinh}(a x)^4 \, dx=\text {Exception raised: TypeError} \]
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. \[ \int x^5 \text {arcsinh}(a x)^4 \, dx=\int x^5\,{\mathrm {asinh}\left (a\,x\right )}^4 \,d x \]