Integrand size = 27, antiderivative size = 396 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2}{x^3} \, dx=-\frac {35}{32} b^2 c^4 d^3 x^2+\frac {1}{4} b^2 c^6 d^3 x^4-\frac {7}{32} b^2 c^2 d^3 \left (1-c^2 x^2\right )^2+\frac {3}{16} b c^3 d^3 x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {7}{8} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}+\frac {3}{32} c^2 d^3 (a+b \arccos (c x))^2-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}+\frac {i c^2 d^3 (a+b \arccos (c x))^3}{b}-3 c^2 d^3 (a+b \arccos (c x))^2 \log \left (1-e^{2 i \arccos (c x)}\right )+b^2 c^2 d^3 \log (x)+3 i b c^2 d^3 (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )-\frac {3}{2} b^2 c^2 d^3 \operatorname {PolyLog}\left (3,e^{2 i \arccos (c x)}\right ) \] 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*arccos(c*x))-7/8*b*c^3*d^3*x*(-c ^2*x^2+1)^(3/2)*(a+b*arccos(c*x))-b*c*d^3*(-c^2*x^2+1)^(5/2)*(a+b*arccos(c *x))/x+3/32*c^2*d^3*(a+b*arccos(c*x))^2-3/2*c^2*d^3*(-c^2*x^2+1)*(a+b*arcc os(c*x))^2-3/4*c^2*d^3*(-c^2*x^2+1)^2*(a+b*arccos(c*x))^2-1/2*d^3*(-c^2*x^ 2+1)^3*(a+b*arccos(c*x))^2/x^2+I*c^2*d^3*(a+b*arccos(c*x))^3/b-3*c^2*d^3*( a+b*arccos(c*x))^2*ln(1-(c*x+I*(-c^2*x^2+1)^(1/2))^2)+b^2*c^2*d^3*ln(x)+3* I*b*c^2*d^3*(a+b*arccos(c*x))*polylog(2,(c*x+I*(-c^2*x^2+1)^(1/2))^2)-3/2* b^2*c^2*d^3*polylog(3,(c*x+I*(-c^2*x^2+1)^(1/2))^2)
Time = 0.85 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.31 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2}{x^3} \, dx=\frac {d^3 \left (-128 a^2+384 a^2 c^4 x^4-64 a^2 c^6 x^6+256 a b c x \sqrt {1-c^2 x^2}-336 a b c^3 x^3 \sqrt {1-c^2 x^2}+32 a b c^5 x^5 \sqrt {1-c^2 x^2}-256 a b \arccos (c x)+768 a b c^4 x^4 \arccos (c x)-128 a b c^6 x^6 \arccos (c x)+256 b^2 c x \sqrt {1-c^2 x^2} \arccos (c x)-128 b^2 \arccos (c x)^2+768 i a b c^2 x^2 \arccos (c x)^2+256 i b^2 c^2 x^2 \arccos (c x)^3+672 a b c^2 x^2 \arctan \left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )-80 b^2 c^2 x^2 \cos (2 \arccos (c x))+160 b^2 c^2 x^2 \arccos (c x)^2 \cos (2 \arccos (c x))+b^2 c^2 x^2 \cos (4 \arccos (c x))-8 b^2 c^2 x^2 \arccos (c x)^2 \cos (4 \arccos (c x))-1536 a b c^2 x^2 \arccos (c x) \log \left (1+e^{2 i \arccos (c x)}\right )-768 b^2 c^2 x^2 \arccos (c x)^2 \log \left (1+e^{2 i \arccos (c x)}\right )-768 a^2 c^2 x^2 \log (x)+256 b^2 c^2 x^2 \log (c x)+768 i b c^2 x^2 (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )-384 b^2 c^2 x^2 \operatorname {PolyLog}\left (3,-e^{2 i \arccos (c x)}\right )-160 b^2 c^2 x^2 \arccos (c x) \sin (2 \arccos (c x))+4 b^2 c^2 x^2 \arccos (c x) \sin (4 \arccos (c x))\right )}{256 x^2} \] Input:
Integrate[((d - c^2*d*x^2)^3*(a + b*ArcCos[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*ArcCos[c*x] + 768*a*b*c^4*x^4*ArcCos[c*x] - 128*a*b*c^6*x^6 *ArcCos[c*x] + 256*b^2*c*x*Sqrt[1 - c^2*x^2]*ArcCos[c*x] - 128*b^2*ArcCos[ c*x]^2 + (768*I)*a*b*c^2*x^2*ArcCos[c*x]^2 + (256*I)*b^2*c^2*x^2*ArcCos[c* x]^3 + 672*a*b*c^2*x^2*ArcTan[(c*x)/(-1 + Sqrt[1 - c^2*x^2])] - 80*b^2*c^2 *x^2*Cos[2*ArcCos[c*x]] + 160*b^2*c^2*x^2*ArcCos[c*x]^2*Cos[2*ArcCos[c*x]] + b^2*c^2*x^2*Cos[4*ArcCos[c*x]] - 8*b^2*c^2*x^2*ArcCos[c*x]^2*Cos[4*ArcC os[c*x]] - 1536*a*b*c^2*x^2*ArcCos[c*x]*Log[1 + E^((2*I)*ArcCos[c*x])] - 7 68*b^2*c^2*x^2*ArcCos[c*x]^2*Log[1 + E^((2*I)*ArcCos[c*x])] - 768*a^2*c^2* x^2*Log[x] + 256*b^2*c^2*x^2*Log[c*x] + (768*I)*b*c^2*x^2*(a + b*ArcCos[c* x])*PolyLog[2, -E^((2*I)*ArcCos[c*x])] - 384*b^2*c^2*x^2*PolyLog[3, -E^((2 *I)*ArcCos[c*x])] - 160*b^2*c^2*x^2*ArcCos[c*x]*Sin[2*ArcCos[c*x]] + 4*b^2 *c^2*x^2*ArcCos[c*x]*Sin[4*ArcCos[c*x]]))/(256*x^2)
Time = 3.97 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.41, number of steps used = 31, number of rules used = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.111, Rules used = {5201, 27, 5201, 243, 49, 2009, 5159, 244, 2009, 5157, 15, 5153, 5203, 5159, 244, 2009, 5157, 15, 5153, 5203, 5137, 3042, 4202, 2620, 3011, 2720, 5157, 15, 5153, 7143}
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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2}{x^3} \, dx\) |
| \(\Big \downarrow \) 5201 |
| \(\displaystyle -b c d^3 \int \frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x^2}dx-3 c^2 d \int \frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2}{x}dx-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -b c d^3 \int \frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x^2}dx-3 c^2 d^3 \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2}{x}dx-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 5201 |
| \(\displaystyle -b c d^3 \left (-5 c^2 \int \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx-b c \int \frac {\left (1-c^2 x^2\right )^2}{x}dx-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}\right )-3 c^2 d^3 \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2}{x}dx-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -b c d^3 \left (-5 c^2 \int \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx-\frac {1}{2} b c \int \frac {\left (1-c^2 x^2\right )^2}{x^2}dx^2-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}\right )-3 c^2 d^3 \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2}{x}dx-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle -3 c^2 d^3 \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2}{x}dx-b c d^3 \left (-5 c^2 \int \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx-\frac {1}{2} b c \int \left (x^2 c^4-2 c^2+\frac {1}{x^2}\right )dx^2-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 c^2 d^3 \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2}{x}dx-b c d^3 \left (-5 c^2 \int \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}-\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 5159 |
| \(\displaystyle -3 c^2 d^3 \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2}{x}dx-b c d^3 \left (-5 c^2 \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {1}{4} b c \int x \left (1-c^2 x^2\right )dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))\right )-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}-\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle -3 c^2 d^3 \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2}{x}dx-b c d^3 \left (-5 c^2 \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {1}{4} b c \int \left (x-c^2 x^3\right )dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))\right )-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}-\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 c^2 d^3 \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2}{x}dx-b c d^3 \left (-5 c^2 \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}-\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 5157 |
| \(\displaystyle -3 c^2 d^3 \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2}{x}dx-b c d^3 \left (-5 c^2 \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}-\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -3 c^2 d^3 \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2}{x}dx-b c d^3 \left (-5 c^2 \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {1}{4} b c x^2\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}-\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 5153 |
| \(\displaystyle -3 c^2 d^3 \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2}{x}dx-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}-b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 5203 |
| \(\displaystyle -3 c^2 d^3 \left (\frac {1}{2} b c \int \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx+\int \frac {\left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{x}dx+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}-b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 5159 |
| \(\displaystyle -3 c^2 d^3 \left (\frac {1}{2} b c \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {1}{4} b c \int x \left (1-c^2 x^2\right )dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))\right )+\int \frac {\left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{x}dx+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}-b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle -3 c^2 d^3 \left (\int \frac {\left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{x}dx+\frac {1}{2} b c \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {1}{4} b c \int \left (x-c^2 x^3\right )dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}-b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 c^2 d^3 \left (\int \frac {\left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{x}dx+\frac {1}{2} b c \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}-b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 5157 |
| \(\displaystyle -3 c^2 d^3 \left (\int \frac {\left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{x}dx+\frac {1}{2} b c \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}-b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -3 c^2 d^3 \left (\int \frac {\left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{x}dx+\frac {1}{2} b c \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {1}{4} b c x^2\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}-b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 5153 |
| \(\displaystyle -3 c^2 d^3 \left (\int \frac {\left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{x}dx+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2+\frac {1}{2} b c \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}-b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 5203 |
| \(\displaystyle -3 c^2 d^3 \left (b c \int \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\int \frac {(a+b \arccos (c x))^2}{x}dx+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {1}{2} b c \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}-b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 5137 |
| \(\displaystyle -3 c^2 d^3 \left (b c \int \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx-\int \frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c x}d\arccos (c x)+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {1}{2} b c \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}-b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -3 c^2 d^3 \left (b c \int \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx-\int (a+b \arccos (c x))^2 \tan (\arccos (c x))d\arccos (c x)+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {1}{2} b c \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}-b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 4202 |
| \(\displaystyle -3 c^2 d^3 \left (b c \int \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+2 i \int \frac {e^{2 i \arccos (c x)} (a+b \arccos (c x))^2}{1+e^{2 i \arccos (c x)}}d\arccos (c x)+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {1}{2} b c \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {i (a+b \arccos (c x))^3}{3 b}\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}-b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -3 c^2 d^3 \left (b c \int \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+2 i \left (i b \int (a+b \arccos (c x)) \log \left (1+e^{2 i \arccos (c x)}\right )d\arccos (c x)-\frac {1}{2} i \log \left (1+e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))^2\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {1}{2} b c \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {i (a+b \arccos (c x))^3}{3 b}\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}-b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -3 c^2 d^3 \left (b c \int \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+2 i \left (i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{2} i b \int \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )d\arccos (c x)\right )-\frac {1}{2} i \log \left (1+e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))^2\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {1}{2} b c \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {i (a+b \arccos (c x))^3}{3 b}\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}-b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -3 c^2 d^3 \left (b c \int \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+2 i \left (i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{4} b \int e^{-2 i \arccos (c x)} \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )de^{2 i \arccos (c x)}\right )-\frac {1}{2} i \log \left (1+e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))^2\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {1}{2} b c \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {i (a+b \arccos (c x))^3}{3 b}\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}-b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 5157 |
| \(\displaystyle -3 c^2 d^3 \left (b c \left (\frac {1}{2} \int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))\right )+2 i \left (i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{4} b \int e^{-2 i \arccos (c x)} \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )de^{2 i \arccos (c x)}\right )-\frac {1}{2} i \log \left (1+e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))^2\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {1}{2} b c \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {i (a+b \arccos (c x))^3}{3 b}\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}-b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -3 c^2 d^3 \left (b c \left (\frac {1}{2} \int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {1}{4} b c x^2\right )+2 i \left (i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{4} b \int e^{-2 i \arccos (c x)} \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )de^{2 i \arccos (c x)}\right )-\frac {1}{2} i \log \left (1+e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))^2\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {1}{2} b c \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {i (a+b \arccos (c x))^3}{3 b}\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}-b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 5153 |
| \(\displaystyle -3 c^2 d^3 \left (2 i \left (i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{4} b \int e^{-2 i \arccos (c x)} \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )de^{2 i \arccos (c x)}\right )-\frac {1}{2} i \log \left (1+e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))^2\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+b c \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{2} b c \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {i (a+b \arccos (c x))^3}{3 b}\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}-b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -3 c^2 d^3 \left (\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+b c \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{2} b c \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+2 i \left (i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (3,-e^{2 i \arccos (c x)}\right )\right )-\frac {1}{2} i \log \left (1+e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))^2\right )-\frac {i (a+b \arccos (c x))^3}{3 b}\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{2 x^2}-b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
Input:
Int[((d - c^2*d*x^2)^3*(a + b*ArcCos[c*x])^2)/x^3,x]
Output:
-1/2*(d^3*(1 - c^2*x^2)^3*(a + b*ArcCos[c*x])^2)/x^2 - b*c*d^3*(-(((1 - c^ 2*x^2)^(5/2)*(a + b*ArcCos[c*x]))/x) - 5*c^2*((b*c*(x^2/2 - (c^2*x^4)/4))/ 4 + (x*(1 - c^2*x^2)^(3/2)*(a + b*ArcCos[c*x]))/4 + (3*((b*c*x^2)/4 + (x*S qrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/2 - (a + b*ArcCos[c*x])^2/(4*b*c)))/ 4) - (b*c*(-2*c^2*x^2 + (c^4*x^4)/2 + Log[x^2]))/2) - 3*c^2*d^3*(((1 - c^2 *x^2)*(a + b*ArcCos[c*x])^2)/2 + ((1 - c^2*x^2)^2*(a + b*ArcCos[c*x])^2)/4 - ((I/3)*(a + b*ArcCos[c*x])^3)/b + b*c*((b*c*x^2)/4 + (x*Sqrt[1 - c^2*x^ 2]*(a + b*ArcCos[c*x]))/2 - (a + b*ArcCos[c*x])^2/(4*b*c)) + (b*c*((b*c*(x ^2/2 - (c^2*x^4)/4))/4 + (x*(1 - c^2*x^2)^(3/2)*(a + b*ArcCos[c*x]))/4 + ( 3*((b*c*x^2)/4 + (x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/2 - (a + b*ArcC os[c*x])^2/(4*b*c)))/4))/2 + (2*I)*((-1/2*I)*(a + b*ArcCos[c*x])^2*Log[1 + E^((2*I)*ArcCos[c*x])] + I*b*((I/2)*(a + b*ArcCos[c*x])*PolyLog[2, -E^((2 *I)*ArcCos[c*x])] - (b*PolyLog[3, -E^((2*I)*ArcCos[c*x])])/4)))
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[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I *((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I Int[(c + d*x)^m*(E^(2*I*( e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt Q[m, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> -Subst[Int[ (a + b*x)^n*Tan[x], x], x, ArcCos[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0 ]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[(-(b*c*(n + 1))^(-1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2] ]*(a + b*ArcCos[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^ 2*d + e, 0] && NeQ[n, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcCos[c*x])^n/2), x] + (Simp[(1/2 )*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[(a + b*ArcCos[c*x])^n/Sqrt[ 1 - c^2*x^2], x], x] + Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2 ]] Int[x*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x ] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcCos[c*x])^n/(2*p + 1)), x] + (S imp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x], x] + Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c , d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]
Int[((a_.) + ArcCos[(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*ArcC os[c*x])^n/(f*(m + 1))), x] + (-Simp[2*e*(p/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x], x] + Simp[b*c*(n/(f*(m + 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*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f} , x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
Int[((a_.) + ArcCos[(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*ArcC os[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*ArcCos[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*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -1]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Time = 0.75 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.46
| method | result | size |
| parts | \(-d^{3} a^{2} \left (\frac {c^{6} x^{4}}{4}-\frac {3 c^{4} x^{2}}{2}+3 c^{2} \ln \left (x \right )+\frac {1}{2 x^{2}}\right )-d^{3} b^{2} c^{2} \left (-i \arccos \left (c x \right )^{3}-\frac {5 \left (2 \arccos \left (c x \right )^{2}-1+2 i \arccos \left (c x \right )\right ) \left (2 c^{2} x^{2}-1+2 i \sqrt {-c^{2} x^{2}+1}\, x c \right )}{32}-\frac {5 \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x c +2 c^{2} x^{2}-1\right ) \left (2 \arccos \left (c x \right )^{2}-1-2 i \arccos \left (c x \right )\right )}{32}+\frac {\arccos \left (c x \right ) \left (-2 i c^{2} x^{2}-2 c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )\right )}{2 c^{2} x^{2}}-\ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 \ln \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )+3 \arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-3 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {3 \operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {\left (8 \arccos \left (c x \right )^{2}-1\right ) \cos \left (4 \arccos \left (c x \right )\right )}{256}-\frac {\arccos \left (c x \right ) \sin \left (4 \arccos \left (c x \right )\right )}{64}\right )-2 d^{3} a b \,c^{2} \left (-\frac {3 i \arccos \left (c x \right )^{2}}{2}-\frac {5 \left (i+2 \arccos \left (c x \right )\right ) \left (2 c^{2} x^{2}-1+2 i \sqrt {-c^{2} x^{2}+1}\, x c \right )}{32}-\frac {5 \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x c +2 c^{2} x^{2}-1\right ) \left (-i+2 \arccos \left (c x \right )\right )}{32}+\frac {-i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )}{2 c^{2} x^{2}}+3 \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {3 i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {\arccos \left (c x \right ) \cos \left (4 \arccos \left (c x \right )\right )}{32}-\frac {\sin \left (4 \arccos \left (c x \right )\right )}{128}\right )\) | \(579\) |
| derivativedivides | \(c^{2} \left (-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 (c x \right )\right )-d^{3} b^{2} \left (-i \arccos \left (c x \right )^{3}-\frac {5 \left (2 \arccos \left (c x \right )^{2}-1+2 i \arccos \left (c x \right )\right ) \left (2 c^{2} x^{2}-1+2 i \sqrt {-c^{2} x^{2}+1}\, x c \right )}{32}-\frac {5 \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x c +2 c^{2} x^{2}-1\right ) \left (2 \arccos \left (c x \right )^{2}-1-2 i \arccos \left (c x \right )\right )}{32}+\frac {\arccos \left (c x \right ) \left (-2 i c^{2} x^{2}-2 c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )\right )}{2 c^{2} x^{2}}-\ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 \ln \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )+3 \arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-3 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {3 \operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {\left (8 \arccos \left (c x \right )^{2}-1\right ) \cos \left (4 \arccos \left (c x \right )\right )}{256}-\frac {\arccos \left (c x \right ) \sin \left (4 \arccos \left (c x \right )\right )}{64}\right )+3 i d^{3} a b \arccos \left (c x \right )^{2}-\frac {5 d^{3} a b \sqrt {-c^{2} x^{2}+1}\, x c}{4}+\frac {5 d^{3} a b \arccos \left (c x \right ) x^{2} c^{2}}{2}-\frac {5 d^{3} a b \arccos \left (c x \right )}{4}+i d^{3} a b +\frac {d^{3} a b \sqrt {-c^{2} x^{2}+1}}{c x}-\frac {d^{3} a b \arccos \left (c x \right )}{c^{2} x^{2}}-6 d^{3} a b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+3 i d^{3} a b \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {d^{3} a b \arccos \left (c x \right ) \cos \left (4 \arccos \left (c x \right )\right )}{16}+\frac {d^{3} a b \sin \left (4 \arccos \left (c x \right )\right )}{64}\right )\) | \(580\) |
| default | \(c^{2} \left (-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 (c x \right )\right )-d^{3} b^{2} \left (-i \arccos \left (c x \right )^{3}-\frac {5 \left (2 \arccos \left (c x \right )^{2}-1+2 i \arccos \left (c x \right )\right ) \left (2 c^{2} x^{2}-1+2 i \sqrt {-c^{2} x^{2}+1}\, x c \right )}{32}-\frac {5 \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x c +2 c^{2} x^{2}-1\right ) \left (2 \arccos \left (c x \right )^{2}-1-2 i \arccos \left (c x \right )\right )}{32}+\frac {\arccos \left (c x \right ) \left (-2 i c^{2} x^{2}-2 c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )\right )}{2 c^{2} x^{2}}-\ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 \ln \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )+3 \arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-3 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {3 \operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {\left (8 \arccos \left (c x \right )^{2}-1\right ) \cos \left (4 \arccos \left (c x \right )\right )}{256}-\frac {\arccos \left (c x \right ) \sin \left (4 \arccos \left (c x \right )\right )}{64}\right )+3 i d^{3} a b \arccos \left (c x \right )^{2}-\frac {5 d^{3} a b \sqrt {-c^{2} x^{2}+1}\, x c}{4}+\frac {5 d^{3} a b \arccos \left (c x \right ) x^{2} c^{2}}{2}-\frac {5 d^{3} a b \arccos \left (c x \right )}{4}+i d^{3} a b +\frac {d^{3} a b \sqrt {-c^{2} x^{2}+1}}{c x}-\frac {d^{3} a b \arccos \left (c x \right )}{c^{2} x^{2}}-6 d^{3} a b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+3 i d^{3} a b \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {d^{3} a b \arccos \left (c x \right ) \cos \left (4 \arccos \left (c x \right )\right )}{16}+\frac {d^{3} a b \sin \left (4 \arccos \left (c x \right )\right )}{64}\right )\) | \(580\) |
Input:
int((-c^2*d*x^2+d)^3*(a+b*arccos(c*x))^2/x^3,x,method=_RETURNVERBOSE)
Output:
-d^3*a^2*(1/4*c^6*x^4-3/2*c^4*x^2+3*c^2*ln(x)+1/2/x^2)-d^3*b^2*c^2*(-I*arc cos(c*x)^3-5/32*(2*arccos(c*x)^2-1+2*I*arccos(c*x))*(2*c^2*x^2-1+2*I*(-c^2 *x^2+1)^(1/2)*c*x)-5/32*(-2*I*(-c^2*x^2+1)^(1/2)*c*x+2*c^2*x^2-1)*(2*arcco s(c*x)^2-1-2*I*arccos(c*x))+1/2*arccos(c*x)*(-2*I*c^2*x^2-2*c*x*(-c^2*x^2+ 1)^(1/2)+arccos(c*x))/c^2/x^2-ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))^2)+2*ln(c*x+ I*(-c^2*x^2+1)^(1/2))+3*arccos(c*x)^2*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))^2)-3 *I*arccos(c*x)*polylog(2,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)+3/2*polylog(3,-(c* x+I*(-c^2*x^2+1)^(1/2))^2)+1/256*(8*arccos(c*x)^2-1)*cos(4*arccos(c*x))-1/ 64*arccos(c*x)*sin(4*arccos(c*x)))-2*d^3*a*b*c^2*(-3/2*I*arccos(c*x)^2-5/3 2*(I+2*arccos(c*x))*(2*c^2*x^2-1+2*I*(-c^2*x^2+1)^(1/2)*c*x)-5/32*(-2*I*(- c^2*x^2+1)^(1/2)*c*x+2*c^2*x^2-1)*(-I+2*arccos(c*x))+1/2*(-I*c^2*x^2-c*x*( -c^2*x^2+1)^(1/2)+arccos(c*x))/c^2/x^2+3*arccos(c*x)*ln(1+(c*x+I*(-c^2*x^2 +1)^(1/2))^2)-3/2*I*polylog(2,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)+1/32*arccos(c *x)*cos(4*arccos(c*x))-1/128*sin(4*arccos(c*x)))
\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2}{x^3} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \arccos \left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \] Input:
integrate((-c^2*d*x^2+d)^3*(a+b*arccos(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)*a rccos(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)*arccos(c*x))/x^3, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2}{x^3} \, dx=- d^{3} \left (\int \left (- \frac {a^{2}}{x^{3}}\right )\, dx + \int \frac {3 a^{2} c^{2}}{x}\, dx + \int \left (- 3 a^{2} c^{4} x\right )\, dx + \int a^{2} c^{6} x^{3}\, dx + \int \left (- \frac {b^{2} \operatorname {acos}^{2}{\left (c x \right )}}{x^{3}}\right )\, dx + \int \left (- \frac {2 a b \operatorname {acos}{\left (c x \right )}}{x^{3}}\right )\, dx + \int \frac {3 b^{2} c^{2} \operatorname {acos}^{2}{\left (c x \right )}}{x}\, dx + \int \left (- 3 b^{2} c^{4} x \operatorname {acos}^{2}{\left (c x \right )}\right )\, dx + \int b^{2} c^{6} x^{3} \operatorname {acos}^{2}{\left (c x \right )}\, dx + \int \frac {6 a b c^{2} \operatorname {acos}{\left (c x \right )}}{x}\, dx + \int \left (- 6 a b c^{4} x \operatorname {acos}{\left (c x \right )}\right )\, dx + \int 2 a b c^{6} x^{3} \operatorname {acos}{\left (c x \right )}\, dx\right ) \] Input:
integrate((-c**2*d*x**2+d)**3*(a+b*acos(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*acos(c*x)** 2/x**3, x) + Integral(-2*a*b*acos(c*x)/x**3, x) + Integral(3*b**2*c**2*aco s(c*x)**2/x, x) + Integral(-3*b**2*c**4*x*acos(c*x)**2, x) + Integral(b**2 *c**6*x**3*acos(c*x)**2, x) + Integral(6*a*b*c**2*acos(c*x)/x, x) + Integr al(-6*a*b*c**4*x*acos(c*x), x) + Integral(2*a*b*c**6*x**3*acos(c*x), x))
\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2}{x^3} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \arccos \left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \] Input:
integrate((-c^2*d*x^2+d)^3*(a+b*arccos(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 - arccos(c*x)/x^2) - 1/2*a^2*d^3/x^2 - integrate (((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)*arct an2(sqrt(c*x + 1)*sqrt(-c*x + 1), 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)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x))/x ^3, x)
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2}{x^3} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((-c^2*d*x^2+d)^3*(a+b*arccos(c*x))^2/x^3,x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^3}{x^3} \,d x \] Input:
int(((a + b*acos(c*x))^2*(d - c^2*d*x^2)^3)/x^3,x)
Output:
int(((a + b*acos(c*x))^2*(d - c^2*d*x^2)^3)/x^3, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2}{x^3} \, dx=\frac {d^{3} \left (24 \mathit {acos} \left (c x \right )^{2} b^{2} c^{4} x^{4}-12 \mathit {acos} \left (c x \right )^{2} b^{2} c^{2} x^{2}-24 \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) b^{2} c^{3} x^{3}-8 \mathit {acos} \left (c x \right ) a b \,c^{6} x^{6}+48 \mathit {acos} \left (c x \right ) a b \,c^{4} x^{4}-16 \mathit {acos} \left (c x \right ) a b +21 \mathit {asin} \left (c x \right ) a b \,c^{2} x^{2}+2 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{5} x^{5}-21 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{3} x^{3}+16 \sqrt {-c^{2} x^{2}+1}\, a b c x -96 \left (\int \frac {\mathit {acos} \left (c x \right )}{x}d x \right ) a b \,c^{2} x^{2}+16 \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{x^{3}}d x \right ) b^{2} x^{2}-48 \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{x}d x \right ) b^{2} c^{2} x^{2}-16 \left (\int \mathit {acos} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{6} x^{2}-48 \,\mathrm {log}\left (x \right ) a^{2} c^{2} x^{2}-4 a^{2} c^{6} x^{6}+24 a^{2} c^{4} x^{4}-8 a^{2}-12 b^{2} c^{4} x^{4}\right )}{16 x^{2}} \] Input:
int((-c^2*d*x^2+d)^3*(a+b*acos(c*x))^2/x^3,x)
Output:
(d**3*(24*acos(c*x)**2*b**2*c**4*x**4 - 12*acos(c*x)**2*b**2*c**2*x**2 - 2 4*sqrt( - c**2*x**2 + 1)*acos(c*x)*b**2*c**3*x**3 - 8*acos(c*x)*a*b*c**6*x **6 + 48*acos(c*x)*a*b*c**4*x**4 - 16*acos(c*x)*a*b + 21*asin(c*x)*a*b*c** 2*x**2 + 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(acos(c*x)/x,x )*a*b*c**2*x**2 + 16*int(acos(c*x)**2/x**3,x)*b**2*x**2 - 48*int(acos(c*x) **2/x,x)*b**2*c**2*x**2 - 16*int(acos(c*x)**2*x**3,x)*b**2*c**6*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 - 12 *b**2*c**4*x**4))/(16*x**2)