Integrand size = 26, antiderivative size = 296 \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {73 b^2 d^2 x^2}{3072 c^2}+\frac {73 b^2 d^2 x^4}{9216}+\frac {43 b^2 c^2 d^2 x^6}{3456}+\frac {1}{256} b^2 c^4 d^2 x^8+\frac {73 b d^2 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{1536 c^3}-\frac {73 b d^2 x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2304 c}-\frac {25}{576} b c d^2 x^5 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {1}{32} b c d^2 x^5 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {73 d^2 (a+b \text {arcsinh}(c x))^2}{3072 c^4}+\frac {1}{24} d^2 x^4 (a+b \text {arcsinh}(c x))^2+\frac {1}{12} d^2 x^4 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{8} d^2 x^4 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \] Output:
-73/3072*b^2*d^2*x^2/c^2+73/9216*b^2*d^2*x^4+43/3456*b^2*c^2*d^2*x^6+1/256 *b^2*c^4*d^2*x^8+73/1536*b*d^2*x*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/c^3- 73/2304*b*d^2*x^3*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/c-25/576*b*c*d^2*x^ 5*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))-1/32*b*c*d^2*x^5*(c^2*x^2+1)^(3/2)* (a+b*arcsinh(c*x))-73/3072*d^2*(a+b*arcsinh(c*x))^2/c^4+1/24*d^2*x^4*(a+b* arcsinh(c*x))^2+1/12*d^2*x^4*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2+1/8*d^2*x^4* (c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2
Time = 0.23 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.80 \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^2 \left (c x \left (1152 a^2 c^3 x^3 \left (6+8 c^2 x^2+3 c^4 x^4\right )+b^2 c x \left (-657+219 c^2 x^2+344 c^4 x^4+108 c^6 x^6\right )-6 a b \sqrt {1+c^2 x^2} \left (-219+146 c^2 x^2+344 c^4 x^4+144 c^6 x^6\right )\right )+6 b \left (-b c x \sqrt {1+c^2 x^2} \left (-219+146 c^2 x^2+344 c^4 x^4+144 c^6 x^6\right )+3 a \left (-73+768 c^4 x^4+1024 c^6 x^6+384 c^8 x^8\right )\right ) \text {arcsinh}(c x)+9 b^2 \left (-73+768 c^4 x^4+1024 c^6 x^6+384 c^8 x^8\right ) \text {arcsinh}(c x)^2\right )}{27648 c^4} \] Input:
Integrate[x^3*(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2,x]
Output:
(d^2*(c*x*(1152*a^2*c^3*x^3*(6 + 8*c^2*x^2 + 3*c^4*x^4) + b^2*c*x*(-657 + 219*c^2*x^2 + 344*c^4*x^4 + 108*c^6*x^6) - 6*a*b*Sqrt[1 + c^2*x^2]*(-219 + 146*c^2*x^2 + 344*c^4*x^4 + 144*c^6*x^6)) + 6*b*(-(b*c*x*Sqrt[1 + c^2*x^2 ]*(-219 + 146*c^2*x^2 + 344*c^4*x^4 + 144*c^6*x^6)) + 3*a*(-73 + 768*c^4*x ^4 + 1024*c^6*x^6 + 384*c^8*x^8))*ArcSinh[c*x] + 9*b^2*(-73 + 768*c^4*x^4 + 1024*c^6*x^6 + 384*c^8*x^8)*ArcSinh[c*x]^2))/(27648*c^4)
Time = 2.54 (sec) , antiderivative size = 583, normalized size of antiderivative = 1.97, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6223, 27, 6223, 244, 2009, 6191, 6221, 15, 6227, 15, 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.
| \(\displaystyle \int x^3 \left (c^2 d x^2+d\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle -\frac {1}{4} b c d^2 \int x^4 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {1}{2} d \int d x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx+\frac {1}{8} d^2 x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{4} b c d^2 \int x^4 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {1}{2} d^2 \int x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx+\frac {1}{8} d^2 x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle -\frac {1}{4} b c d^2 \left (\frac {3}{8} \int x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {1}{8} b c \int x^5 \left (c^2 x^2+1\right )dx+\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))\right )+\frac {1}{2} d^2 \left (-\frac {1}{3} b c \int x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{3} \int x^3 (a+b \text {arcsinh}(c x))^2dx+\frac {1}{6} x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{8} d^2 x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {1}{2} d^2 \left (-\frac {1}{3} b c \int x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{3} \int x^3 (a+b \text {arcsinh}(c x))^2dx+\frac {1}{6} x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {1}{4} b c d^2 \left (\frac {3}{8} \int x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {1}{8} b c \int \left (c^2 x^7+x^5\right )dx+\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))\right )+\frac {1}{8} d^2 x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} d^2 \left (-\frac {1}{3} b c \int x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{3} \int x^3 (a+b \text {arcsinh}(c x))^2dx+\frac {1}{6} x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {1}{4} b c d^2 \left (\frac {3}{8} \int x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )\right )+\frac {1}{8} d^2 x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {1}{2} d^2 \left (\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {1}{3} b c \int x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{6} x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {1}{4} b c d^2 \left (\frac {3}{8} \int x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )\right )+\frac {1}{8} d^2 x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6221 |
| \(\displaystyle \frac {1}{2} d^2 \left (\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {1}{3} b c \left (\frac {1}{6} \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx-\frac {1}{6} b c \int x^5dx+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))\right )+\frac {1}{6} x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {1}{4} b c d^2 \left (\frac {3}{8} \left (\frac {1}{6} \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx-\frac {1}{6} b c \int x^5dx+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))\right )+\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )\right )+\frac {1}{8} d^2 x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {1}{2} d^2 \left (\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {1}{3} b c \left (\frac {1}{6} \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{36} b c x^6\right )+\frac {1}{6} x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {1}{4} b c d^2 \left (\frac {3}{8} \left (\frac {1}{6} \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{36} b c x^6\right )+\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )\right )+\frac {1}{8} d^2 x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle -\frac {1}{4} b c d^2 \left (\frac {3}{8} \left (\frac {1}{6} \left (-\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{4 c^2}-\frac {b \int x^3dx}{4 c}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}\right )+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{36} b c x^6\right )+\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )\right )+\frac {1}{2} d^2 \left (\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (-\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{4 c^2}-\frac {b \int x^3dx}{4 c}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}\right )\right )-\frac {1}{3} b c \left (\frac {1}{6} \left (-\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{4 c^2}-\frac {b \int x^3dx}{4 c}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}\right )+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{36} b c x^6\right )+\frac {1}{6} x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{8} d^2 x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {1}{2} d^2 \left (\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (-\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )\right )-\frac {1}{3} b c \left (\frac {1}{6} \left (-\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{36} b c x^6\right )+\frac {1}{6} x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {1}{4} b c d^2 \left (\frac {3}{8} \left (\frac {1}{6} \left (-\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{36} b c x^6\right )+\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )\right )+\frac {1}{8} d^2 x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {1}{2} d^2 \left (\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (-\frac {3 \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}-\frac {b \int xdx}{2 c}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )\right )-\frac {1}{3} b c \left (\frac {1}{6} \left (-\frac {3 \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}-\frac {b \int xdx}{2 c}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{36} b c x^6\right )+\frac {1}{6} x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {1}{4} b c d^2 \left (\frac {3}{8} \left (\frac {1}{6} \left (-\frac {3 \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}-\frac {b \int xdx}{2 c}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{36} b c x^6\right )+\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )\right )+\frac {1}{8} d^2 x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {1}{2} d^2 \left (\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (-\frac {3 \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )\right )-\frac {1}{3} b c \left (\frac {1}{6} \left (-\frac {3 \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{36} b c x^6\right )+\frac {1}{6} x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {1}{4} b c d^2 \left (\frac {3}{8} \left (\frac {1}{6} \left (-\frac {3 \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{36} b c x^6\right )+\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )\right )+\frac {1}{8} d^2 x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {1}{8} d^2 x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d^2 \left (\frac {1}{6} x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {3 \left (-\frac {(a+b \text {arcsinh}(c x))^2}{4 b c^3}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 c^2}-\frac {b x^4}{16 c}\right )\right )-\frac {1}{3} b c \left (\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {1}{6} \left (\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {3 \left (-\frac {(a+b \text {arcsinh}(c x))^2}{4 b c^3}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 c^2}-\frac {b x^4}{16 c}\right )-\frac {1}{36} b c x^6\right )\right )-\frac {1}{4} b c d^2 \left (\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{8} \left (\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {1}{6} \left (\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {3 \left (-\frac {(a+b \text {arcsinh}(c x))^2}{4 b c^3}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 c^2}-\frac {b x^4}{16 c}\right )-\frac {1}{36} b c x^6\right )-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )\right )\) |
Input:
Int[x^3*(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2,x]
Output:
(d^2*x^4*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/8 - (b*c*d^2*(-1/8*(b*c*( x^6/6 + (c^2*x^8)/8)) + (x^5*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/8 + (3*(-1/36*(b*c*x^6) + (x^5*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/6 + (- 1/16*(b*x^4)/c + (x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(4*c^2) - (3 *(-1/4*(b*x^2)/c + (x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2*c^2) - (a + b*ArcSinh[c*x])^2/(4*b*c^3)))/(4*c^2))/6))/8))/4 + (d^2*((x^4*(1 + c^2* x^2)*(a + b*ArcSinh[c*x])^2)/6 - (b*c*(-1/36*(b*c*x^6) + (x^5*Sqrt[1 + c^2 *x^2]*(a + b*ArcSinh[c*x]))/6 + (-1/16*(b*x^4)/c + (x^3*Sqrt[1 + c^2*x^2]* (a + b*ArcSinh[c*x]))/(4*c^2) - (3*(-1/4*(b*x^2)/c + (x*Sqrt[1 + c^2*x^2]* (a + b*ArcSinh[c*x]))/(2*c^2) - (a + b*ArcSinh[c*x])^2/(4*b*c^3)))/(4*c^2) )/6))/3 + ((x^4*(a + b*ArcSinh[c*x])^2)/4 - (b*c*(-1/16*(b*x^4)/c + (x^3*S qrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(4*c^2) - (3*(-1/4*(b*x^2)/c + (x*S qrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2*c^2) - (a + b*ArcSinh[c*x])^2/(4 *b*c^3)))/(4*c^2)))/2)/3))/2
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_.) + 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_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc Sinh[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt [1 + c^2*x^2]] Int[(f*x)^m*((a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2]), x] , x] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] I nt[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d , e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
Int[((a_.) + ArcSinh[(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 Sinh[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*ArcSinh[c*x])^n, x], x] - Simp[b*c*(n/(f*(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, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -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 = 1.67 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.13
| method | result | size |
| derivativedivides | \(\frac {d^{2} a^{2} \left (\frac {\left (c^{2} x^{2}+1\right )^{4}}{8}-\frac {\left (c^{2} x^{2}+1\right )^{3}}{6}\right )+b^{2} d^{2} \left (\frac {x^{2} c^{2} \left (c^{2} x^{2}+1\right )^{3} \operatorname {arcsinh}\left (x c \right )^{2}}{8}-\frac {\operatorname {arcsinh}\left (x c \right )^{2} \left (c^{2} x^{2}+1\right )^{3}}{24}-\frac {\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{32}+\frac {11 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{576}+\frac {55 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{2304}+\frac {55 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c}{1536}+\frac {55 \operatorname {arcsinh}\left (x c \right )^{2}}{3072}+\frac {\left (c^{2} x^{2}+1\right )^{4}}{256}-\frac {11 \left (c^{2} x^{2}+1\right )^{3}}{3456}-\frac {55 \left (c^{2} x^{2}+1\right )^{2}}{9216}-\frac {55 c^{2} x^{2}}{3072}-\frac {55}{3072}\right )+2 d^{2} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}}{8}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{3}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}-\frac {73 \,\operatorname {arcsinh}\left (x c \right )}{3072}+\frac {11 x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{1152}+\frac {55 x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{4608}+\frac {55 \sqrt {c^{2} x^{2}+1}\, x c}{3072}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{64}\right )}{c^{4}}\) | \(334\) |
| default | \(\frac {d^{2} a^{2} \left (\frac {\left (c^{2} x^{2}+1\right )^{4}}{8}-\frac {\left (c^{2} x^{2}+1\right )^{3}}{6}\right )+b^{2} d^{2} \left (\frac {x^{2} c^{2} \left (c^{2} x^{2}+1\right )^{3} \operatorname {arcsinh}\left (x c \right )^{2}}{8}-\frac {\operatorname {arcsinh}\left (x c \right )^{2} \left (c^{2} x^{2}+1\right )^{3}}{24}-\frac {\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{32}+\frac {11 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{576}+\frac {55 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{2304}+\frac {55 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c}{1536}+\frac {55 \operatorname {arcsinh}\left (x c \right )^{2}}{3072}+\frac {\left (c^{2} x^{2}+1\right )^{4}}{256}-\frac {11 \left (c^{2} x^{2}+1\right )^{3}}{3456}-\frac {55 \left (c^{2} x^{2}+1\right )^{2}}{9216}-\frac {55 c^{2} x^{2}}{3072}-\frac {55}{3072}\right )+2 d^{2} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}}{8}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{3}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}-\frac {73 \,\operatorname {arcsinh}\left (x c \right )}{3072}+\frac {11 x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{1152}+\frac {55 x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{4608}+\frac {55 \sqrt {c^{2} x^{2}+1}\, x c}{3072}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{64}\right )}{c^{4}}\) | \(334\) |
| parts | \(d^{2} a^{2} \left (\frac {1}{8} c^{4} x^{8}+\frac {1}{3} x^{6} c^{2}+\frac {1}{4} x^{4}\right )+\frac {b^{2} d^{2} \left (\frac {x^{2} c^{2} \left (c^{2} x^{2}+1\right )^{3} \operatorname {arcsinh}\left (x c \right )^{2}}{8}-\frac {\operatorname {arcsinh}\left (x c \right )^{2} \left (c^{2} x^{2}+1\right )^{3}}{24}-\frac {\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{32}+\frac {11 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{576}+\frac {55 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{2304}+\frac {55 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c}{1536}+\frac {55 \operatorname {arcsinh}\left (x c \right )^{2}}{3072}+\frac {\left (c^{2} x^{2}+1\right )^{4}}{256}-\frac {11 \left (c^{2} x^{2}+1\right )^{3}}{3456}-\frac {55 \left (c^{2} x^{2}+1\right )^{2}}{9216}-\frac {55 c^{2} x^{2}}{3072}-\frac {55}{3072}\right )}{c^{4}}+\frac {2 d^{2} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}}{8}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{3}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}-\frac {73 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{4608}+\frac {73 \sqrt {c^{2} x^{2}+1}\, x c}{3072}-\frac {73 \,\operatorname {arcsinh}\left (x c \right )}{3072}-\frac {43 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}}{1152}-\frac {x^{7} c^{7} \sqrt {c^{2} x^{2}+1}}{64}\right )}{c^{4}}\) | \(343\) |
| orering | \(\frac {\left (18252 c^{10} x^{10}+69716 c^{8} x^{8}+87751 c^{6} x^{6}+492 c^{4} x^{4}-36135 c^{2} x^{2}-13140\right ) \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{55296 c^{4} \left (c^{2} x^{2}+1\right )^{3}}-\frac {\left (2268 c^{8} x^{8}+8048 c^{6} x^{6}+7851 c^{4} x^{4}-7665 c^{2} x^{2}-5256\right ) \left (3 x^{2} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+4 x^{4} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d +\frac {2 x^{3} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}\right )}{55296 c^{4} x^{2} \left (c^{2} x^{2}+1\right )^{2}}+\frac {\left (108 c^{6} x^{6}+344 c^{4} x^{4}+219 c^{2} x^{2}-657\right ) \left (6 x \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+28 x^{3} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d +\frac {12 x^{2} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}+8 x^{5} c^{4} d^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+\frac {16 x^{4} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{3} d b}{\sqrt {c^{2} x^{2}+1}}+\frac {2 x^{3} \left (c^{2} d \,x^{2}+d \right )^{2} b^{2} c^{2}}{c^{2} x^{2}+1}-\frac {2 x^{4} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b \,c^{3}}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{55296 x \,c^{4} \left (c^{2} x^{2}+1\right )}\) | \(502\) |
Input:
int(x^3*(c^2*d*x^2+d)^2*(a+b*arcsinh(x*c))^2,x,method=_RETURNVERBOSE)
Output:
1/c^4*(d^2*a^2*(1/8*(c^2*x^2+1)^4-1/6*(c^2*x^2+1)^3)+b^2*d^2*(1/8*x^2*c^2* (c^2*x^2+1)^3*arcsinh(x*c)^2-1/24*arcsinh(x*c)^2*(c^2*x^2+1)^3-1/32*arcsin h(x*c)*x*c*(c^2*x^2+1)^(7/2)+11/576*arcsinh(x*c)*x*c*(c^2*x^2+1)^(5/2)+55/ 2304*arcsinh(x*c)*x*c*(c^2*x^2+1)^(3/2)+55/1536*arcsinh(x*c)*(c^2*x^2+1)^( 1/2)*x*c+55/3072*arcsinh(x*c)^2+1/256*(c^2*x^2+1)^4-11/3456*(c^2*x^2+1)^3- 55/9216*(c^2*x^2+1)^2-55/3072*c^2*x^2-55/3072)+2*d^2*a*b*(1/8*arcsinh(x*c) *x^8*c^8+1/3*arcsinh(x*c)*x^6*c^6+1/4*arcsinh(x*c)*c^4*x^4-73/3072*arcsinh (x*c)+11/1152*x*c*(c^2*x^2+1)^(5/2)+55/4608*x*c*(c^2*x^2+1)^(3/2)+55/3072* (c^2*x^2+1)^(1/2)*x*c-1/64*x*c*(c^2*x^2+1)^(7/2)))
Time = 0.11 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.18 \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {108 \, {\left (32 \, a^{2} + b^{2}\right )} c^{8} d^{2} x^{8} + 8 \, {\left (1152 \, a^{2} + 43 \, b^{2}\right )} c^{6} d^{2} x^{6} + 3 \, {\left (2304 \, a^{2} + 73 \, b^{2}\right )} c^{4} d^{2} x^{4} - 657 \, b^{2} c^{2} d^{2} x^{2} + 9 \, {\left (384 \, b^{2} c^{8} d^{2} x^{8} + 1024 \, b^{2} c^{6} d^{2} x^{6} + 768 \, b^{2} c^{4} d^{2} x^{4} - 73 \, b^{2} d^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 6 \, {\left (1152 \, a b c^{8} d^{2} x^{8} + 3072 \, a b c^{6} d^{2} x^{6} + 2304 \, a b c^{4} d^{2} x^{4} - 219 \, a b d^{2} - {\left (144 \, b^{2} c^{7} d^{2} x^{7} + 344 \, b^{2} c^{5} d^{2} x^{5} + 146 \, b^{2} c^{3} d^{2} x^{3} - 219 \, b^{2} c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 6 \, {\left (144 \, a b c^{7} d^{2} x^{7} + 344 \, a b c^{5} d^{2} x^{5} + 146 \, a b c^{3} d^{2} x^{3} - 219 \, a b c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}}{27648 \, c^{4}} \] Input:
integrate(x^3*(c^2*d*x^2+d)^2*(a+b*arcsinh(c*x))^2,x, algorithm="fricas")
Output:
1/27648*(108*(32*a^2 + b^2)*c^8*d^2*x^8 + 8*(1152*a^2 + 43*b^2)*c^6*d^2*x^ 6 + 3*(2304*a^2 + 73*b^2)*c^4*d^2*x^4 - 657*b^2*c^2*d^2*x^2 + 9*(384*b^2*c ^8*d^2*x^8 + 1024*b^2*c^6*d^2*x^6 + 768*b^2*c^4*d^2*x^4 - 73*b^2*d^2)*log( c*x + sqrt(c^2*x^2 + 1))^2 + 6*(1152*a*b*c^8*d^2*x^8 + 3072*a*b*c^6*d^2*x^ 6 + 2304*a*b*c^4*d^2*x^4 - 219*a*b*d^2 - (144*b^2*c^7*d^2*x^7 + 344*b^2*c^ 5*d^2*x^5 + 146*b^2*c^3*d^2*x^3 - 219*b^2*c*d^2*x)*sqrt(c^2*x^2 + 1))*log( c*x + sqrt(c^2*x^2 + 1)) - 6*(144*a*b*c^7*d^2*x^7 + 344*a*b*c^5*d^2*x^5 + 146*a*b*c^3*d^2*x^3 - 219*a*b*c*d^2*x)*sqrt(c^2*x^2 + 1))/c^4
Time = 1.35 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.74 \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} \frac {a^{2} c^{4} d^{2} x^{8}}{8} + \frac {a^{2} c^{2} d^{2} x^{6}}{3} + \frac {a^{2} d^{2} x^{4}}{4} + \frac {a b c^{4} d^{2} x^{8} \operatorname {asinh}{\left (c x \right )}}{4} - \frac {a b c^{3} d^{2} x^{7} \sqrt {c^{2} x^{2} + 1}}{32} + \frac {2 a b c^{2} d^{2} x^{6} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {43 a b c d^{2} x^{5} \sqrt {c^{2} x^{2} + 1}}{576} + \frac {a b d^{2} x^{4} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {73 a b d^{2} x^{3} \sqrt {c^{2} x^{2} + 1}}{2304 c} + \frac {73 a b d^{2} x \sqrt {c^{2} x^{2} + 1}}{1536 c^{3}} - \frac {73 a b d^{2} \operatorname {asinh}{\left (c x \right )}}{1536 c^{4}} + \frac {b^{2} c^{4} d^{2} x^{8} \operatorname {asinh}^{2}{\left (c x \right )}}{8} + \frac {b^{2} c^{4} d^{2} x^{8}}{256} - \frac {b^{2} c^{3} d^{2} x^{7} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{32} + \frac {b^{2} c^{2} d^{2} x^{6} \operatorname {asinh}^{2}{\left (c x \right )}}{3} + \frac {43 b^{2} c^{2} d^{2} x^{6}}{3456} - \frac {43 b^{2} c d^{2} x^{5} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{576} + \frac {b^{2} d^{2} x^{4} \operatorname {asinh}^{2}{\left (c x \right )}}{4} + \frac {73 b^{2} d^{2} x^{4}}{9216} - \frac {73 b^{2} d^{2} x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{2304 c} - \frac {73 b^{2} d^{2} x^{2}}{3072 c^{2}} + \frac {73 b^{2} d^{2} x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{1536 c^{3}} - \frac {73 b^{2} d^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{3072 c^{4}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{2} x^{4}}{4} & \text {otherwise} \end {cases} \] Input:
integrate(x**3*(c**2*d*x**2+d)**2*(a+b*asinh(c*x))**2,x)
Output:
Piecewise((a**2*c**4*d**2*x**8/8 + a**2*c**2*d**2*x**6/3 + a**2*d**2*x**4/ 4 + a*b*c**4*d**2*x**8*asinh(c*x)/4 - a*b*c**3*d**2*x**7*sqrt(c**2*x**2 + 1)/32 + 2*a*b*c**2*d**2*x**6*asinh(c*x)/3 - 43*a*b*c*d**2*x**5*sqrt(c**2*x **2 + 1)/576 + a*b*d**2*x**4*asinh(c*x)/2 - 73*a*b*d**2*x**3*sqrt(c**2*x** 2 + 1)/(2304*c) + 73*a*b*d**2*x*sqrt(c**2*x**2 + 1)/(1536*c**3) - 73*a*b*d **2*asinh(c*x)/(1536*c**4) + b**2*c**4*d**2*x**8*asinh(c*x)**2/8 + b**2*c* *4*d**2*x**8/256 - b**2*c**3*d**2*x**7*sqrt(c**2*x**2 + 1)*asinh(c*x)/32 + b**2*c**2*d**2*x**6*asinh(c*x)**2/3 + 43*b**2*c**2*d**2*x**6/3456 - 43*b* *2*c*d**2*x**5*sqrt(c**2*x**2 + 1)*asinh(c*x)/576 + b**2*d**2*x**4*asinh(c *x)**2/4 + 73*b**2*d**2*x**4/9216 - 73*b**2*d**2*x**3*sqrt(c**2*x**2 + 1)* asinh(c*x)/(2304*c) - 73*b**2*d**2*x**2/(3072*c**2) + 73*b**2*d**2*x*sqrt( c**2*x**2 + 1)*asinh(c*x)/(1536*c**3) - 73*b**2*d**2*asinh(c*x)**2/(3072*c **4), Ne(c, 0)), (a**2*d**2*x**4/4, True))
Leaf count of result is larger than twice the leaf count of optimal. 762 vs. \(2 (264) = 528\).
Time = 0.07 (sec) , antiderivative size = 762, normalized size of antiderivative = 2.57 \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx =\text {Too large to display} \] Input:
integrate(x^3*(c^2*d*x^2+d)^2*(a+b*arcsinh(c*x))^2,x, algorithm="maxima")
Output:
1/8*b^2*c^4*d^2*x^8*arcsinh(c*x)^2 + 1/8*a^2*c^4*d^2*x^8 + 1/3*b^2*c^2*d^2 *x^6*arcsinh(c*x)^2 + 1/3*a^2*c^2*d^2*x^6 + 1/4*b^2*d^2*x^4*arcsinh(c*x)^2 + 1/1536*(384*x^8*arcsinh(c*x) - (48*sqrt(c^2*x^2 + 1)*x^7/c^2 - 56*sqrt( c^2*x^2 + 1)*x^5/c^4 + 70*sqrt(c^2*x^2 + 1)*x^3/c^6 - 105*sqrt(c^2*x^2 + 1 )*x/c^8 + 105*arcsinh(c*x)/c^9)*c)*a*b*c^4*d^2 + 1/9216*((36*x^8/c^2 - 56* x^6/c^4 + 105*x^4/c^6 - 315*x^2/c^8 + 315*log(c*x + sqrt(c^2*x^2 + 1))^2/c ^10)*c^2 - 6*(48*sqrt(c^2*x^2 + 1)*x^7/c^2 - 56*sqrt(c^2*x^2 + 1)*x^5/c^4 + 70*sqrt(c^2*x^2 + 1)*x^3/c^6 - 105*sqrt(c^2*x^2 + 1)*x/c^8 + 105*arcsinh (c*x)/c^9)*c*arcsinh(c*x))*b^2*c^4*d^2 + 1/4*a^2*d^2*x^4 + 1/72*(48*x^6*ar csinh(c*x) - (8*sqrt(c^2*x^2 + 1)*x^5/c^2 - 10*sqrt(c^2*x^2 + 1)*x^3/c^4 + 15*sqrt(c^2*x^2 + 1)*x/c^6 - 15*arcsinh(c*x)/c^7)*c)*a*b*c^2*d^2 + 1/432* ((8*x^6/c^2 - 15*x^4/c^4 + 45*x^2/c^6 - 45*log(c*x + sqrt(c^2*x^2 + 1))^2/ c^8)*c^2 - 6*(8*sqrt(c^2*x^2 + 1)*x^5/c^2 - 10*sqrt(c^2*x^2 + 1)*x^3/c^4 + 15*sqrt(c^2*x^2 + 1)*x/c^6 - 15*arcsinh(c*x)/c^7)*c*arcsinh(c*x))*b^2*c^2 *d^2 + 1/16*(8*x^4*arcsinh(c*x) - (2*sqrt(c^2*x^2 + 1)*x^3/c^2 - 3*sqrt(c^ 2*x^2 + 1)*x/c^4 + 3*arcsinh(c*x)/c^5)*c)*a*b*d^2 + 1/32*((x^4/c^2 - 3*x^2 /c^4 + 3*log(c*x + sqrt(c^2*x^2 + 1))^2/c^6)*c^2 - 2*(2*sqrt(c^2*x^2 + 1)* x^3/c^2 - 3*sqrt(c^2*x^2 + 1)*x/c^4 + 3*arcsinh(c*x)/c^5)*c*arcsinh(c*x))* b^2*d^2
Exception generated. \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3*(c^2*d*x^2+d)^2*(a+b*arcsinh(c*x))^2,x, algorithm="giac")
Output:
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. \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\int x^3\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^2 \,d x \] Input:
int(x^3*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^2,x)
Output:
int(x^3*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^2, x)
\[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^{2} \left (1152 \mathit {asinh} \left (c x \right ) a b \,c^{8} x^{8}+3072 \mathit {asinh} \left (c x \right ) a b \,c^{6} x^{6}+2304 \mathit {asinh} \left (c x \right ) a b \,c^{4} x^{4}-144 \sqrt {c^{2} x^{2}+1}\, a b \,c^{7} x^{7}-344 \sqrt {c^{2} x^{2}+1}\, a b \,c^{5} x^{5}-146 \sqrt {c^{2} x^{2}+1}\, a b \,c^{3} x^{3}+219 \sqrt {c^{2} x^{2}+1}\, a b c x +4608 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{7}d x \right ) b^{2} c^{8}+9216 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{5}d x \right ) b^{2} c^{6}+4608 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{4}-219 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a b +576 a^{2} c^{8} x^{8}+1536 a^{2} c^{6} x^{6}+1152 a^{2} c^{4} x^{4}\right )}{4608 c^{4}} \] Input:
int(x^3*(c^2*d*x^2+d)^2*(a+b*asinh(c*x))^2,x)
Output:
(d**2*(1152*asinh(c*x)*a*b*c**8*x**8 + 3072*asinh(c*x)*a*b*c**6*x**6 + 230 4*asinh(c*x)*a*b*c**4*x**4 - 144*sqrt(c**2*x**2 + 1)*a*b*c**7*x**7 - 344*s qrt(c**2*x**2 + 1)*a*b*c**5*x**5 - 146*sqrt(c**2*x**2 + 1)*a*b*c**3*x**3 + 219*sqrt(c**2*x**2 + 1)*a*b*c*x + 4608*int(asinh(c*x)**2*x**7,x)*b**2*c** 8 + 9216*int(asinh(c*x)**2*x**5,x)*b**2*c**6 + 4608*int(asinh(c*x)**2*x**3 ,x)*b**2*c**4 - 219*log(sqrt(c**2*x**2 + 1) + c*x)*a*b + 576*a**2*c**8*x** 8 + 1536*a**2*c**6*x**6 + 1152*a**2*c**4*x**4))/(4608*c**4)