Integrand size = 26, antiderivative size = 379 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\frac {35}{32} b^2 c^4 d^3 x^2+\frac {1}{4} b^2 c^6 d^3 x^4-\frac {7}{32} b^2 c^2 d^3 \left (1+c^2 x^2\right )^2-\frac {3}{16} b c^3 d^3 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+\frac {7}{8} b c^3 d^3 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}-\frac {3}{32} c^2 d^3 (a+b \text {arcsinh}(c x))^2+\frac {3}{2} c^2 d^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {3}{4} c^2 d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}-\frac {c^2 d^3 (a+b \text {arcsinh}(c x))^3}{b}+3 c^2 d^3 (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+b^2 c^2 d^3 \log (x)+3 b c^2 d^3 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )-\frac {3}{2} b^2 c^2 d^3 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right ) \] Output:
35/32*b^2*c^4*d^3*x^2+1/4*b^2*c^6*d^3*x^4-7/32*b^2*c^2*d^3*(c^2*x^2+1)^2-3 /16*b*c^3*d^3*x*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))+7/8*b*c^3*d^3*x*(c^2* x^2+1)^(3/2)*(a+b*arcsinh(c*x))-b*c*d^3*(c^2*x^2+1)^(5/2)*(a+b*arcsinh(c*x ))/x-3/32*c^2*d^3*(a+b*arcsinh(c*x))^2+3/2*c^2*d^3*(c^2*x^2+1)*(a+b*arcsin h(c*x))^2+3/4*c^2*d^3*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2-1/2*d^3*(c^2*x^2+ 1)^3*(a+b*arcsinh(c*x))^2/x^2-c^2*d^3*(a+b*arcsinh(c*x))^3/b+3*c^2*d^3*(a+ b*arcsinh(c*x))^2*ln(1-(c*x+(c^2*x^2+1)^(1/2))^2)+b^2*c^2*d^3*ln(x)+3*b*c^ 2*d^3*(a+b*arcsinh(c*x))*polylog(2,(c*x+(c^2*x^2+1)^(1/2))^2)-3/2*b^2*c^2* d^3*polylog(3,(c*x+(c^2*x^2+1)^(1/2))^2)
Time = 0.58 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.31 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\frac {d^3 \left (-128 a^2+384 a^2 c^4 x^4+64 a^2 c^6 x^6-256 a b c x \sqrt {1+c^2 x^2}-336 a b c^3 x^3 \sqrt {1+c^2 x^2}-32 a b c^5 x^5 \sqrt {1+c^2 x^2}-256 a b \text {arcsinh}(c x)+768 a b c^4 x^4 \text {arcsinh}(c x)+128 a b c^6 x^6 \text {arcsinh}(c x)-256 b^2 c x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-128 b^2 \text {arcsinh}(c x)^2-768 a b c^2 x^2 \text {arcsinh}(c x)^2-256 b^2 c^2 x^2 \text {arcsinh}(c x)^3+80 b^2 c^2 x^2 \cosh (2 \text {arcsinh}(c x))+160 b^2 c^2 x^2 \text {arcsinh}(c x)^2 \cosh (2 \text {arcsinh}(c x))+b^2 c^2 x^2 \cosh (4 \text {arcsinh}(c x))+8 b^2 c^2 x^2 \text {arcsinh}(c x)^2 \cosh (4 \text {arcsinh}(c x))+1536 a b c^2 x^2 \text {arcsinh}(c x) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+768 b^2 c^2 x^2 \text {arcsinh}(c x)^2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+768 a^2 c^2 x^2 \log (x)+256 b^2 c^2 x^2 \log (c x)-336 a b c^2 x^2 \log \left (-c x+\sqrt {1+c^2 x^2}\right )+768 b c^2 x^2 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )-384 b^2 c^2 x^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )-160 b^2 c^2 x^2 \text {arcsinh}(c x) \sinh (2 \text {arcsinh}(c x))-4 b^2 c^2 x^2 \text {arcsinh}(c x) \sinh (4 \text {arcsinh}(c x))\right )}{256 x^2} \] Input:
Integrate[((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2)/x^3,x]
Output:
(d^3*(-128*a^2 + 384*a^2*c^4*x^4 + 64*a^2*c^6*x^6 - 256*a*b*c*x*Sqrt[1 + c ^2*x^2] - 336*a*b*c^3*x^3*Sqrt[1 + c^2*x^2] - 32*a*b*c^5*x^5*Sqrt[1 + c^2* x^2] - 256*a*b*ArcSinh[c*x] + 768*a*b*c^4*x^4*ArcSinh[c*x] + 128*a*b*c^6*x ^6*ArcSinh[c*x] - 256*b^2*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - 128*b^2*Arc Sinh[c*x]^2 - 768*a*b*c^2*x^2*ArcSinh[c*x]^2 - 256*b^2*c^2*x^2*ArcSinh[c*x ]^3 + 80*b^2*c^2*x^2*Cosh[2*ArcSinh[c*x]] + 160*b^2*c^2*x^2*ArcSinh[c*x]^2 *Cosh[2*ArcSinh[c*x]] + b^2*c^2*x^2*Cosh[4*ArcSinh[c*x]] + 8*b^2*c^2*x^2*A rcSinh[c*x]^2*Cosh[4*ArcSinh[c*x]] + 1536*a*b*c^2*x^2*ArcSinh[c*x]*Log[1 - E^(2*ArcSinh[c*x])] + 768*b^2*c^2*x^2*ArcSinh[c*x]^2*Log[1 - E^(2*ArcSinh [c*x])] + 768*a^2*c^2*x^2*Log[x] + 256*b^2*c^2*x^2*Log[c*x] - 336*a*b*c^2* x^2*Log[-(c*x) + Sqrt[1 + c^2*x^2]] + 768*b*c^2*x^2*(a + b*ArcSinh[c*x])*P olyLog[2, E^(2*ArcSinh[c*x])] - 384*b^2*c^2*x^2*PolyLog[3, E^(2*ArcSinh[c* x])] - 160*b^2*c^2*x^2*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]] - 4*b^2*c^2*x^2*A rcSinh[c*x]*Sinh[4*ArcSinh[c*x]]))/(256*x^2)
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.
| \(\displaystyle \int \frac {\left (c^2 d x^2+d\right )^3 (a+b \text {arcsinh}(c x))^2}{x^3} \, dx\) |
| \(\Big \downarrow \) 6222 |
| \(\displaystyle b c d^3 \int \frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^2}dx+3 c^2 d \int \frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle b c d^3 \int \frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^2}dx+3 c^2 d^3 \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 6222 |
| \(\displaystyle b c d^3 \left (5 c^2 \int \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+b c \int \frac {\left (c^2 x^2+1\right )^2}{x}dx-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}\right )+3 c^2 d^3 \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle b c d^3 \left (5 c^2 \int \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {1}{2} b c \int \frac {\left (c^2 x^2+1\right )^2}{x^2}dx^2-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}\right )+3 c^2 d^3 \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle 3 c^2 d^3 \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}dx+b c d^3 \left (5 c^2 \int \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {1}{2} b c \int \left (x^2 c^4+2 c^2+\frac {1}{x^2}\right )dx^2-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 c^2 d^3 \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}dx+b c d^3 \left (5 c^2 \int \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}+2 c^2 x^2+\log \left (x^2\right )\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle 3 c^2 d^3 \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}dx+b c d^3 \left (5 c^2 \left (\frac {3}{4} \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {1}{4} b c \int x \left (c^2 x^2+1\right )dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))\right )-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}+2 c^2 x^2+\log \left (x^2\right )\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle 3 c^2 d^3 \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}dx+b c d^3 \left (5 c^2 \left (\frac {3}{4} \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {1}{4} b c \int \left (c^2 x^3+x\right )dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))\right )-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}+2 c^2 x^2+\log \left (x^2\right )\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 c^2 d^3 \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}dx+b c d^3 \left (5 c^2 \left (\frac {3}{4} \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}+2 c^2 x^2+\log \left (x^2\right )\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 6200 |
| \(\displaystyle 3 c^2 d^3 \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}dx+b c d^3 \left (5 c^2 \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx-\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}+2 c^2 x^2+\log \left (x^2\right )\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle 3 c^2 d^3 \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}dx+b c d^3 \left (5 c^2 \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c x^2\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}+2 c^2 x^2+\log \left (x^2\right )\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle 3 c^2 d^3 \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+5 c^2 \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}+2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle 3 c^2 d^3 \left (-\frac {1}{2} b c \int \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{x}dx+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+5 c^2 \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}+2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle 3 c^2 d^3 \left (-\frac {1}{2} b c \left (\frac {3}{4} \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {1}{4} b c \int x \left (c^2 x^2+1\right )dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))\right )+\int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{x}dx+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+5 c^2 \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}+2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle 3 c^2 d^3 \left (\int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {1}{2} b c \left (\frac {3}{4} \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {1}{4} b c \int \left (c^2 x^3+x\right )dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))\right )+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+5 c^2 \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}+2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 c^2 d^3 \left (\int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {1}{2} b c \left (\frac {3}{4} \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+5 c^2 \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}+2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 6200 |
| \(\displaystyle 3 c^2 d^3 \left (\int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {1}{2} b c \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx-\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+5 c^2 \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}+2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle 3 c^2 d^3 \left (\int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {1}{2} b c \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c x^2\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+5 c^2 \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}+2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle 3 c^2 d^3 \left (\int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{x}dx+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+5 c^2 \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}+2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle 3 c^2 d^3 \left (-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\int \frac {(a+b \text {arcsinh}(c x))^2}{x}dx+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+5 c^2 \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}+2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 6190 |
| \(\displaystyle 3 c^2 d^3 \left (-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {\int -(a+b \text {arcsinh}(c x))^2 \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+5 c^2 \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}+2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 3 c^2 d^3 \left (-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {\int (a+b \text {arcsinh}(c x))^2 \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+5 c^2 \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}+2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 c^2 d^3 \left (-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {\int -i (a+b \text {arcsinh}(c x))^2 \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}+\frac {\pi }{2}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+5 c^2 \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}+2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle 3 c^2 d^3 \left (-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {i \int (a+b \text {arcsinh}(c x))^2 \tan \left (\frac {1}{2} \left (\frac {2 i a}{b}+\pi \right )-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+5 c^2 \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}+2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle 3 c^2 d^3 \left (-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {i \left (2 i \int \frac {e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi } (a+b \text {arcsinh}(c x))^2}{1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }}d(a+b \text {arcsinh}(c x))-\frac {1}{3} i (a+b \text {arcsinh}(c x))^3\right )}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+5 c^2 \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}+2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle 3 c^2 d^3 \left (-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {i \left (2 i \left (b \int (a+b \text {arcsinh}(c x)) \log \left (1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )d(a+b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x))^2 \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c x))^3\right )}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+5 c^2 \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}+2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle 3 c^2 d^3 \left (-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {i \left (2 i \left (b \left (\frac {1}{2} b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )-\frac {1}{2} b \int \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )d(a+b \text {arcsinh}(c x))\right )-\frac {1}{2} b (a+b \text {arcsinh}(c x))^2 \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c x))^3\right )}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+5 c^2 \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}+2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle 3 c^2 d^3 \left (\frac {i \left (2 i \left (b \left (\frac {1}{4} b^2 \int e^{-\frac {2 a}{b}+\frac {2 (a+b \text {arcsinh}(c x))}{b}+i \pi } \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))de^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }+\frac {1}{2} b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )\right )-\frac {1}{2} b (a+b \text {arcsinh}(c x))^2 \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c x))^3\right )}{b}-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+5 c^2 \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}+2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 6200 |
| \(\displaystyle 3 c^2 d^3 \left (\frac {i \left (2 i \left (b \left (\frac {1}{4} b^2 \int e^{-\frac {2 a}{b}+\frac {2 (a+b \text {arcsinh}(c x))}{b}+i \pi } \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))de^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }+\frac {1}{2} b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )\right )-\frac {1}{2} b (a+b \text {arcsinh}(c x))^2 \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c x))^3\right )}{b}-b c \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx-\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))\right )+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+5 c^2 \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}+2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle 3 c^2 d^3 \left (\frac {i \left (2 i \left (b \left (\frac {1}{4} b^2 \int e^{-\frac {2 a}{b}+\frac {2 (a+b \text {arcsinh}(c x))}{b}+i \pi } \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))de^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }+\frac {1}{2} b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )\right )-\frac {1}{2} b (a+b \text {arcsinh}(c x))^2 \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c x))^3\right )}{b}-b c \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c x^2\right )+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}+5 c^2 \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}+2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
Input:
Int[((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2)/x^3,x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(771\) vs. \(2(380)=760\).
Time = 1.52 (sec) , antiderivative size = 772, normalized size of antiderivative = 2.04
| method | result | size |
| derivativedivides | \(c^{2} \left (-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}}{8}-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{x c}-\frac {21 d^{3} b^{2} \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c}{16}+\frac {81 d^{3} b^{2}}{256}-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (x c \right )^{2}}{2 x^{2} c^{2}}+\frac {21 d^{3} a b \,\operatorname {arcsinh}\left (x c \right )}{16}-3 d^{3} a b \operatorname {arcsinh}\left (x c \right )^{2}+6 d^{3} a b \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )+6 d^{3} a b \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )+6 d^{3} b^{2} \operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )+6 d^{3} b^{2} \operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )+3 d^{3} b^{2} \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+3 d^{3} b^{2} \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\frac {d^{3} b^{2} x^{4} c^{4}}{32}-\frac {d^{3} a b \,\operatorname {arcsinh}\left (x c \right )}{x^{2} c^{2}}-\frac {d^{3} a b \sqrt {c^{2} x^{2}+1}}{x c}+d^{3} a^{2} \left (\frac {c^{4} x^{4}}{4}+\frac {3 c^{2} x^{2}}{2}-\frac {1}{2 c^{2} x^{2}}+3 \ln \left (x c \right )\right )+6 d^{3} a b \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+6 d^{3} a b \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\frac {d^{3} b^{2} \operatorname {arcsinh}\left (x c \right )^{2} x^{4} c^{4}}{4}+\frac {3 d^{3} b^{2} \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}}{2}+d^{3} a b +\frac {d^{3} a b \,\operatorname {arcsinh}\left (x c \right ) x^{4} c^{4}}{2}+3 d^{3} a b \,\operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}-\frac {d^{3} a b \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}}{8}-\frac {21 d^{3} a b x c \sqrt {c^{2} x^{2}+1}}{16}-6 d^{3} b^{2} \operatorname {polylog}\left (3, x c +\sqrt {c^{2} x^{2}+1}\right )+\frac {21 d^{3} b^{2} \operatorname {arcsinh}\left (x c \right )^{2}}{32}-d^{3} b^{2} \operatorname {arcsinh}\left (x c \right )^{3}-6 d^{3} b^{2} \operatorname {polylog}\left (3, -x c -\sqrt {c^{2} x^{2}+1}\right )-2 d^{3} b^{2} \ln \left (x c +\sqrt {c^{2} x^{2}+1}\right )+d^{3} b^{2} \operatorname {arcsinh}\left (x c \right )+d^{3} b^{2} \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+d^{3} b^{2} \ln \left (x c +\sqrt {c^{2} x^{2}+1}-1\right )+\frac {21 b^{2} c^{2} d^{3} x^{2}}{32}\right )\) | \(772\) |
| default | \(c^{2} \left (-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}}{8}-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{x c}-\frac {21 d^{3} b^{2} \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c}{16}+\frac {81 d^{3} b^{2}}{256}-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (x c \right )^{2}}{2 x^{2} c^{2}}+\frac {21 d^{3} a b \,\operatorname {arcsinh}\left (x c \right )}{16}-3 d^{3} a b \operatorname {arcsinh}\left (x c \right )^{2}+6 d^{3} a b \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )+6 d^{3} a b \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )+6 d^{3} b^{2} \operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )+6 d^{3} b^{2} \operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )+3 d^{3} b^{2} \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+3 d^{3} b^{2} \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\frac {d^{3} b^{2} x^{4} c^{4}}{32}-\frac {d^{3} a b \,\operatorname {arcsinh}\left (x c \right )}{x^{2} c^{2}}-\frac {d^{3} a b \sqrt {c^{2} x^{2}+1}}{x c}+d^{3} a^{2} \left (\frac {c^{4} x^{4}}{4}+\frac {3 c^{2} x^{2}}{2}-\frac {1}{2 c^{2} x^{2}}+3 \ln \left (x c \right )\right )+6 d^{3} a b \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+6 d^{3} a b \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\frac {d^{3} b^{2} \operatorname {arcsinh}\left (x c \right )^{2} x^{4} c^{4}}{4}+\frac {3 d^{3} b^{2} \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}}{2}+d^{3} a b +\frac {d^{3} a b \,\operatorname {arcsinh}\left (x c \right ) x^{4} c^{4}}{2}+3 d^{3} a b \,\operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}-\frac {d^{3} a b \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}}{8}-\frac {21 d^{3} a b x c \sqrt {c^{2} x^{2}+1}}{16}-6 d^{3} b^{2} \operatorname {polylog}\left (3, x c +\sqrt {c^{2} x^{2}+1}\right )+\frac {21 d^{3} b^{2} \operatorname {arcsinh}\left (x c \right )^{2}}{32}-d^{3} b^{2} \operatorname {arcsinh}\left (x c \right )^{3}-6 d^{3} b^{2} \operatorname {polylog}\left (3, -x c -\sqrt {c^{2} x^{2}+1}\right )-2 d^{3} b^{2} \ln \left (x c +\sqrt {c^{2} x^{2}+1}\right )+d^{3} b^{2} \operatorname {arcsinh}\left (x c \right )+d^{3} b^{2} \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+d^{3} b^{2} \ln \left (x c +\sqrt {c^{2} x^{2}+1}-1\right )+\frac {21 b^{2} c^{2} d^{3} x^{2}}{32}\right )\) | \(772\) |
| parts | \(-\frac {d^{3} a b c \sqrt {c^{2} x^{2}+1}}{x}+\frac {3 d^{3} b^{2} c^{4} \operatorname {arcsinh}\left (x c \right )^{2} x^{2}}{2}+3 d^{3} b^{2} c^{2} \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+6 d^{3} b^{2} c^{2} \operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )+3 d^{3} b^{2} c^{2} \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+6 d^{3} b^{2} c^{2} \operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-6 d^{3} b^{2} c^{2} \operatorname {polylog}\left (3, -x c -\sqrt {c^{2} x^{2}+1}\right )-6 d^{3} b^{2} c^{2} \operatorname {polylog}\left (3, x c +\sqrt {c^{2} x^{2}+1}\right )+d^{3} b^{2} c^{2} \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+d^{3} b^{2} c^{2} \ln \left (x c +\sqrt {c^{2} x^{2}+1}-1\right )+d^{3} b^{2} c^{2} \operatorname {arcsinh}\left (x c \right )-d^{3} b^{2} c^{2} \operatorname {arcsinh}\left (x c \right )^{3}-2 d^{3} b^{2} c^{2} \ln \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\frac {21 d^{3} b^{2} c^{2} \operatorname {arcsinh}\left (x c \right )^{2}}{32}-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (x c \right )^{2}}{2 x^{2}}+\frac {81 d^{3} b^{2} c^{2}}{256}+6 d^{3} a b \,c^{2} \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )+\frac {21 d^{3} a b \,c^{2} \operatorname {arcsinh}\left (x c \right )}{16}-3 d^{3} a b \,c^{2} \operatorname {arcsinh}\left (x c \right )^{2}-\frac {d^{3} a b \,\operatorname {arcsinh}\left (x c \right )}{x^{2}}+6 d^{3} a b \,c^{2} \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )+\frac {d^{3} b^{2} c^{6} \operatorname {arcsinh}\left (x c \right )^{2} x^{4}}{4}-\frac {21 d^{3} b^{2} c^{3} \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x}{16}-\frac {d^{3} b^{2} c \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{x}-\frac {d^{3} b^{2} c^{5} \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{3}}{8}+d^{3} a b \,c^{2}+6 d^{3} a b \,c^{2} \operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+6 d^{3} a b \,c^{2} \operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{3} a b \,c^{5} \sqrt {c^{2} x^{2}+1}\, x^{3}}{8}-\frac {21 d^{3} a b \,c^{3} \sqrt {c^{2} x^{2}+1}\, x}{16}+\frac {d^{3} a b \,c^{6} \operatorname {arcsinh}\left (x c \right ) x^{4}}{2}+3 d^{3} a b \,c^{4} \operatorname {arcsinh}\left (x c \right ) x^{2}+\frac {21 b^{2} c^{4} d^{3} x^{2}}{32}+\frac {b^{2} c^{6} d^{3} x^{4}}{32}+d^{3} a^{2} \left (\frac {x^{4} c^{6}}{4}+\frac {3 c^{4} x^{2}}{2}+3 c^{2} \ln \left (x \right )-\frac {1}{2 x^{2}}\right )\) | \(820\) |
Input:
int((c^2*d*x^2+d)^3*(a+b*arcsinh(x*c))^2/x^3,x,method=_RETURNVERBOSE)
Output:
c^2*(-1/8*d^3*b^2*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*x^3*c^3-d^3*b^2*arcsinh(x *c)/x/c*(c^2*x^2+1)^(1/2)-21/16*d^3*b^2*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*x*c +81/256*d^3*b^2-1/2*d^3*b^2*arcsinh(x*c)^2/x^2/c^2+21/16*d^3*a*b*arcsinh(x *c)-3*d^3*a*b*arcsinh(x*c)^2+6*d^3*a*b*polylog(2,-x*c-(c^2*x^2+1)^(1/2))+6 *d^3*a*b*polylog(2,x*c+(c^2*x^2+1)^(1/2))+6*d^3*b^2*arcsinh(x*c)*polylog(2 ,-x*c-(c^2*x^2+1)^(1/2))+6*d^3*b^2*arcsinh(x*c)*polylog(2,x*c+(c^2*x^2+1)^ (1/2))+3*d^3*b^2*arcsinh(x*c)^2*ln(1-x*c-(c^2*x^2+1)^(1/2))+3*d^3*b^2*arcs inh(x*c)^2*ln(1+x*c+(c^2*x^2+1)^(1/2))+1/32*d^3*b^2*x^4*c^4-d^3*a*b*arcsin h(x*c)/x^2/c^2-d^3*a*b/x/c*(c^2*x^2+1)^(1/2)+d^3*a^2*(1/4*c^4*x^4+3/2*c^2* x^2-1/2/c^2/x^2+3*ln(x*c))+6*d^3*a*b*arcsinh(x*c)*ln(1-x*c-(c^2*x^2+1)^(1/ 2))+6*d^3*a*b*arcsinh(x*c)*ln(1+x*c+(c^2*x^2+1)^(1/2))+1/4*d^3*b^2*arcsinh (x*c)^2*x^4*c^4+3/2*d^3*b^2*arcsinh(x*c)^2*x^2*c^2+d^3*a*b+1/2*d^3*a*b*arc sinh(x*c)*x^4*c^4+3*d^3*a*b*arcsinh(x*c)*x^2*c^2-1/8*d^3*a*b*(c^2*x^2+1)^( 1/2)*x^3*c^3-21/16*d^3*a*b*x*c*(c^2*x^2+1)^(1/2)-6*d^3*b^2*polylog(3,x*c+( c^2*x^2+1)^(1/2))+21/32*d^3*b^2*arcsinh(x*c)^2-d^3*b^2*arcsinh(x*c)^3-6*d^ 3*b^2*polylog(3,-x*c-(c^2*x^2+1)^(1/2))-2*d^3*b^2*ln(x*c+(c^2*x^2+1)^(1/2) )+d^3*b^2*arcsinh(x*c)+d^3*b^2*ln(1+x*c+(c^2*x^2+1)^(1/2))+d^3*b^2*ln(x*c+ (c^2*x^2+1)^(1/2)-1)+21/32*b^2*c^2*d^3*x^2)
\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{3} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \] Input:
integrate((c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2/x^3,x, algorithm="fricas")
Output:
integral((a^2*c^6*d^3*x^6 + 3*a^2*c^4*d^3*x^4 + 3*a^2*c^2*d^3*x^2 + a^2*d^ 3 + (b^2*c^6*d^3*x^6 + 3*b^2*c^4*d^3*x^4 + 3*b^2*c^2*d^3*x^2 + b^2*d^3)*ar csinh(c*x)^2 + 2*(a*b*c^6*d^3*x^6 + 3*a*b*c^4*d^3*x^4 + 3*a*b*c^2*d^3*x^2 + a*b*d^3)*arcsinh(c*x))/x^3, x)
\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=d^{3} \left (\int \frac {a^{2}}{x^{3}}\, dx + \int \frac {3 a^{2} c^{2}}{x}\, dx + \int 3 a^{2} c^{4} x\, dx + \int a^{2} c^{6} x^{3}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {3 b^{2} c^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x}\, dx + \int 3 b^{2} c^{4} x \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{6} x^{3} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int \frac {6 a b c^{2} \operatorname {asinh}{\left (c x \right )}}{x}\, dx + \int 6 a b c^{4} x \operatorname {asinh}{\left (c x \right )}\, dx + \int 2 a b c^{6} x^{3} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \] Input:
integrate((c**2*d*x**2+d)**3*(a+b*asinh(c*x))**2/x**3,x)
Output:
d**3*(Integral(a**2/x**3, x) + Integral(3*a**2*c**2/x, x) + Integral(3*a** 2*c**4*x, x) + Integral(a**2*c**6*x**3, x) + Integral(b**2*asinh(c*x)**2/x **3, x) + Integral(2*a*b*asinh(c*x)/x**3, x) + Integral(3*b**2*c**2*asinh( c*x)**2/x, x) + Integral(3*b**2*c**4*x*asinh(c*x)**2, x) + Integral(b**2*c **6*x**3*asinh(c*x)**2, x) + Integral(6*a*b*c**2*asinh(c*x)/x, x) + Integr al(6*a*b*c**4*x*asinh(c*x), x) + Integral(2*a*b*c**6*x**3*asinh(c*x), x))
\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{3} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \] Input:
integrate((c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2/x^3,x, algorithm="maxima")
Output:
1/4*a^2*c^6*d^3*x^4 + 3/2*a^2*c^4*d^3*x^2 + 3*a^2*c^2*d^3*log(x) - a*b*d^3 *(sqrt(c^2*x^2 + 1)*c/x + arcsinh(c*x)/x^2) - 1/2*a^2*d^3/x^2 + integrate( b^2*c^6*d^3*x^3*log(c*x + sqrt(c^2*x^2 + 1))^2 + 2*a*b*c^6*d^3*x^3*log(c*x + sqrt(c^2*x^2 + 1)) + 3*b^2*c^4*d^3*x*log(c*x + sqrt(c^2*x^2 + 1))^2 + 6 *a*b*c^4*d^3*x*log(c*x + sqrt(c^2*x^2 + 1)) + 3*b^2*c^2*d^3*log(c*x + sqrt (c^2*x^2 + 1))^2/x + 6*a*b*c^2*d^3*log(c*x + sqrt(c^2*x^2 + 1))/x + b^2*d^ 3*log(c*x + sqrt(c^2*x^2 + 1))^2/x^3, x)
Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2/x^3,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. \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^3}{x^3} \,d x \] Input:
int(((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^3)/x^3,x)
Output:
int(((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^3)/x^3, x)
\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\frac {d^{3} \left (24 \mathit {asinh} \left (c x \right )^{2} b^{2} c^{4} x^{4}+12 \mathit {asinh} \left (c x \right )^{2} b^{2} c^{2} x^{2}-24 \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) b^{2} c^{3} x^{3}+8 \mathit {asinh} \left (c x \right ) a b \,c^{6} x^{6}+48 \mathit {asinh} \left (c x \right ) a b \,c^{4} x^{4}-16 \mathit {asinh} \left (c x \right ) a b -2 \sqrt {c^{2} x^{2}+1}\, a b \,c^{5} x^{5}-21 \sqrt {c^{2} x^{2}+1}\, a b \,c^{3} x^{3}-16 \sqrt {c^{2} x^{2}+1}\, a b c x +96 \left (\int \frac {\mathit {asinh} \left (c x \right )}{x}d x \right ) a b \,c^{2} x^{2}+16 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{x^{3}}d x \right ) b^{2} x^{2}+48 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{x}d x \right ) b^{2} c^{2} x^{2}+16 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{6} x^{2}+21 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a b \,c^{2} x^{2}+48 \,\mathrm {log}\left (x \right ) a^{2} c^{2} x^{2}+4 a^{2} c^{6} x^{6}+24 a^{2} c^{4} x^{4}-8 a^{2}-16 a b \,c^{2} x^{2}+12 b^{2} c^{4} x^{4}\right )}{16 x^{2}} \] Input:
int((c^2*d*x^2+d)^3*(a+b*asinh(c*x))^2/x^3,x)
Output:
(d**3*(24*asinh(c*x)**2*b**2*c**4*x**4 + 12*asinh(c*x)**2*b**2*c**2*x**2 - 24*sqrt(c**2*x**2 + 1)*asinh(c*x)*b**2*c**3*x**3 + 8*asinh(c*x)*a*b*c**6* x**6 + 48*asinh(c*x)*a*b*c**4*x**4 - 16*asinh(c*x)*a*b - 2*sqrt(c**2*x**2 + 1)*a*b*c**5*x**5 - 21*sqrt(c**2*x**2 + 1)*a*b*c**3*x**3 - 16*sqrt(c**2*x **2 + 1)*a*b*c*x + 96*int(asinh(c*x)/x,x)*a*b*c**2*x**2 + 16*int(asinh(c*x )**2/x**3,x)*b**2*x**2 + 48*int(asinh(c*x)**2/x,x)*b**2*c**2*x**2 + 16*int (asinh(c*x)**2*x**3,x)*b**2*c**6*x**2 + 21*log(sqrt(c**2*x**2 + 1) + c*x)* a*b*c**2*x**2 + 48*log(x)*a**2*c**2*x**2 + 4*a**2*c**6*x**6 + 24*a**2*c**4 *x**4 - 8*a**2 - 16*a*b*c**2*x**2 + 12*b**2*c**4*x**4))/(16*x**2)