Integrand size = 27, antiderivative size = 348 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2}{x^4} \, dx=-\frac {b^2 c^2 d^3}{3 x}-\frac {50}{9} b^2 c^4 d^3 x+\frac {2}{27} b^2 c^6 d^3 x^3+5 b c^3 d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{9} b c^3 d^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{3 x^2}+\frac {16}{3} c^4 d^3 x (a+b \arcsin (c x))^2+\frac {8}{3} c^4 d^3 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2 c^2 d^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{3 x^3}+\frac {34}{3} b c^3 d^3 (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )-\frac {17}{3} i b^2 c^3 d^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )+\frac {17}{3} i b^2 c^3 d^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) \] Output:
-1/3*b^2*c^2*d^3/x-50/9*b^2*c^4*d^3*x+2/27*b^2*c^6*d^3*x^3+5*b*c^3*d^3*(-c ^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))-1/9*b*c^3*d^3*(-c^2*x^2+1)^(3/2)*(a+b*ar csin(c*x))-1/3*b*c*d^3*(-c^2*x^2+1)^(5/2)*(a+b*arcsin(c*x))/x^2+16/3*c^4*d ^3*x*(a+b*arcsin(c*x))^2+8/3*c^4*d^3*x*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2+2* c^2*d^3*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))^2/x-1/3*d^3*(-c^2*x^2+1)^3*(a+b*a rcsin(c*x))^2/x^3+34/3*b*c^3*d^3*(a+b*arcsin(c*x))*arctanh(I*c*x+(-c^2*x^2 +1)^(1/2))-17/3*I*b^2*c^3*d^3*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))+17/3*I* b^2*c^3*d^3*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))
Time = 0.80 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.38 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2}{x^4} \, dx=-\frac {d^3 \left (9 a^2-81 a^2 c^2 x^2+9 b^2 c^2 x^2-81 a^2 c^4 x^4+150 b^2 c^4 x^4+9 a^2 c^6 x^6-2 b^2 c^6 x^6+9 a b c x \sqrt {1-c^2 x^2}-150 a b c^3 x^3 \sqrt {1-c^2 x^2}+6 a b c^5 x^5 \sqrt {1-c^2 x^2}+18 a b \arcsin (c x)-162 a b c^2 x^2 \arcsin (c x)-162 a b c^4 x^4 \arcsin (c x)+18 a b c^6 x^6 \arcsin (c x)+9 b^2 c x \sqrt {1-c^2 x^2} \arcsin (c x)-150 b^2 c^3 x^3 \sqrt {1-c^2 x^2} \arcsin (c x)+6 b^2 c^5 x^5 \sqrt {1-c^2 x^2} \arcsin (c x)+9 b^2 \arcsin (c x)^2-81 b^2 c^2 x^2 \arcsin (c x)^2-81 b^2 c^4 x^4 \arcsin (c x)^2+9 b^2 c^6 x^6 \arcsin (c x)^2-153 a b c^3 x^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+153 b^2 c^3 x^3 \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )-153 b^2 c^3 x^3 \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )+153 i b^2 c^3 x^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-153 i b^2 c^3 x^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )}{27 x^3} \] Input:
Integrate[((d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2)/x^4,x]
Output:
-1/27*(d^3*(9*a^2 - 81*a^2*c^2*x^2 + 9*b^2*c^2*x^2 - 81*a^2*c^4*x^4 + 150* b^2*c^4*x^4 + 9*a^2*c^6*x^6 - 2*b^2*c^6*x^6 + 9*a*b*c*x*Sqrt[1 - c^2*x^2] - 150*a*b*c^3*x^3*Sqrt[1 - c^2*x^2] + 6*a*b*c^5*x^5*Sqrt[1 - c^2*x^2] + 18 *a*b*ArcSin[c*x] - 162*a*b*c^2*x^2*ArcSin[c*x] - 162*a*b*c^4*x^4*ArcSin[c* x] + 18*a*b*c^6*x^6*ArcSin[c*x] + 9*b^2*c*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x] - 150*b^2*c^3*x^3*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + 6*b^2*c^5*x^5*Sqrt[1 - c ^2*x^2]*ArcSin[c*x] + 9*b^2*ArcSin[c*x]^2 - 81*b^2*c^2*x^2*ArcSin[c*x]^2 - 81*b^2*c^4*x^4*ArcSin[c*x]^2 + 9*b^2*c^6*x^6*ArcSin[c*x]^2 - 153*a*b*c^3* x^3*ArcTanh[Sqrt[1 - c^2*x^2]] + 153*b^2*c^3*x^3*ArcSin[c*x]*Log[1 - E^(I* ArcSin[c*x])] - 153*b^2*c^3*x^3*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + ( 153*I)*b^2*c^3*x^3*PolyLog[2, -E^(I*ArcSin[c*x])] - (153*I)*b^2*c^3*x^3*Po lyLog[2, E^(I*ArcSin[c*x])]))/x^3
Time = 3.39 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.58, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.704, Rules used = {5200, 27, 5200, 244, 2009, 5158, 5130, 5182, 24, 2009, 5202, 2009, 5198, 24, 5218, 3042, 4671, 2715, 2838}
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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2}{x^4} \, dx\) |
| \(\Big \downarrow \) 5200 |
| \(\displaystyle \frac {2}{3} b c d^3 \int \frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x^3}dx-2 c^2 d \int \frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x^2}dx-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 c^2 d^3 \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x^2}dx+\frac {2}{3} b c d^3 \int \frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x^3}dx-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 5200 |
| \(\displaystyle \frac {2}{3} b c d^3 \left (-\frac {5}{2} c^2 \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx+\frac {1}{2} b c \int \frac {\left (1-c^2 x^2\right )^2}{x^2}dx-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}\right )-2 c^2 d^3 \left (-4 c^2 \int \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx+2 b c \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle -2 c^2 d^3 \left (-4 c^2 \int \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx+2 b c \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}\right )+\frac {2}{3} b c d^3 \left (-\frac {5}{2} c^2 \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx+\frac {1}{2} b c \int \left (x^2 c^4-2 c^2+\frac {1}{x^2}\right )dx-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 c^2 d^3 \left (-4 c^2 \int \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx+2 b c \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}\right )+\frac {2}{3} b c d^3 \left (-\frac {5}{2} c^2 \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}-2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 5158 |
| \(\displaystyle -2 c^2 d^3 \left (-4 c^2 \left (-\frac {2}{3} b c \int x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {2}{3} \int (a+b \arcsin (c x))^2dx+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )+2 b c \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}\right )+\frac {2}{3} b c d^3 \left (-\frac {5}{2} c^2 \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}-2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 5130 |
| \(\displaystyle -2 c^2 d^3 \left (-4 c^2 \left (\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )-\frac {2}{3} b c \int x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )+2 b c \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}\right )+\frac {2}{3} b c d^3 \left (-\frac {5}{2} c^2 \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}-2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle -2 c^2 d^3 \left (-4 c^2 \left (\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b \int 1dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \int \left (1-c^2 x^2\right )dx}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )+2 b c \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}\right )+\frac {2}{3} b c d^3 \left (-\frac {5}{2} c^2 \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}-2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -2 c^2 d^3 \left (-4 c^2 \left (-\frac {2}{3} b c \left (\frac {b \int \left (1-c^2 x^2\right )dx}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )\right )+2 b c \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}\right )+\frac {2}{3} b c d^3 \left (-\frac {5}{2} c^2 \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}-2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 c^2 d^3 \left (2 b c \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}-4 c^2 \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )\right )+\frac {2}{3} b c d^3 \left (-\frac {5}{2} c^2 \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}-2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle -2 c^2 d^3 \left (2 b c \left (\int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}dx-\frac {1}{3} b c \int \left (1-c^2 x^2\right )dx+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}-4 c^2 \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )\right )+\frac {2}{3} b c d^3 \left (-\frac {5}{2} c^2 \left (\int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}dx-\frac {1}{3} b c \int \left (1-c^2 x^2\right )dx+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))\right )-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}-2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 c^2 d^3 \left (2 b c \left (\int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}dx+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}-4 c^2 \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )\right )+\frac {2}{3} b c d^3 \left (-\frac {5}{2} c^2 \left (\int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}dx+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )\right )-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}-2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 5198 |
| \(\displaystyle -2 c^2 d^3 \left (2 b c \left (\int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx-b c \int 1dx+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}-4 c^2 \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )\right )+\frac {2}{3} b c d^3 \left (-\frac {5}{2} c^2 \left (\int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx-b c \int 1dx+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )\right )-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}-2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -2 c^2 d^3 \left (2 b c \left (\int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )-b c x\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}-4 c^2 \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )\right )+\frac {2}{3} b c d^3 \left (-\frac {5}{2} c^2 \left (\int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )-b c x\right )-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}-2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 5218 |
| \(\displaystyle -2 c^2 d^3 \left (2 b c \left (\int \frac {a+b \arcsin (c x)}{c x}d\arcsin (c x)+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )-b c x\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}-4 c^2 \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )\right )+\frac {2}{3} b c d^3 \left (-\frac {5}{2} c^2 \left (\int \frac {a+b \arcsin (c x)}{c x}d\arcsin (c x)+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )-b c x\right )-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}-2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -2 c^2 d^3 \left (2 b c \left (\int (a+b \arcsin (c x)) \csc (\arcsin (c x))d\arcsin (c x)+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )-b c x\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}-4 c^2 \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )\right )+\frac {2}{3} b c d^3 \left (-\frac {5}{2} c^2 \left (\int (a+b \arcsin (c x)) \csc (\arcsin (c x))d\arcsin (c x)+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )-b c x\right )-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}-2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle -2 c^2 d^3 \left (2 b c \left (-b \int \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )-b c x\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}-4 c^2 \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )\right )+\frac {2}{3} b c d^3 \left (-\frac {5}{2} c^2 \left (-b \int \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )-b c x\right )-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}-2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -2 c^2 d^3 \left (2 b c \left (i b \int e^{-i \arcsin (c x)} \log \left (1-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )-b c x\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}-4 c^2 \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )\right )+\frac {2}{3} b c d^3 \left (-\frac {5}{2} c^2 \left (i b \int e^{-i \arcsin (c x)} \log \left (1-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )-b c x\right )-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}-2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -2 c^2 d^3 \left (2 b c \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )-b c x\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}-4 c^2 \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )\right )+\frac {2}{3} b c d^3 \left (-\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )-b c x\right )-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}-2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{3 x^3}\) |
Input:
Int[((d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2)/x^4,x]
Output:
-1/3*(d^3*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/x^3 - 2*c^2*d^3*(-(((1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/x) - 4*c^2*((x*(1 - c^2*x^2)*(a + b*ArcS in[c*x])^2)/3 - (2*b*c*((b*(x - (c^2*x^3)/3))/(3*c) - ((1 - c^2*x^2)^(3/2) *(a + b*ArcSin[c*x]))/(3*c^2)))/3 + (2*(x*(a + b*ArcSin[c*x])^2 - 2*b*c*(( b*x)/c - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c^2)))/3) + 2*b*c*(-(b*c* x) - (b*c*(x - (c^2*x^3)/3))/3 + Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) + ( (1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/3 - 2*(a + b*ArcSin[c*x])*ArcTanh [E^(I*ArcSin[c*x])] + I*b*PolyLog[2, -E^(I*ArcSin[c*x])] - I*b*PolyLog[2, E^(I*ArcSin[c*x])])) + (2*b*c*d^3*((b*c*(-x^(-1) - 2*c^2*x + (c^4*x^3)/3)) /2 - ((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(2*x^2) - (5*c^2*(-(b*c*x) - (b*c*(x - (c^2*x^3)/3))/3 + Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) + ((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/3 - 2*(a + b*ArcSin[c*x])*ArcTanh[E^ (I*ArcSin[c*x])] + I*b*PolyLog[2, -E^(I*ArcSin[c*x])] - I*b*PolyLog[2, E^( I*ArcSin[c*x])]))/2))/3
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[m, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Ar cSin[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSin[c*x])^n/(2*p + 1)), x] + (S imp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[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*ArcSin[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_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcS in[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[(f*x)^m*((a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2]), x], x ] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[ (f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
Int[((a_.) + ArcSin[(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*ArcS in[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*ArcSin[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*ArcSin[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_.) + ArcSin[(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*ArcS in[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*ArcSin[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*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -1]
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)* (x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e* x^2]] Subst[Int[(a + b*x)^n*Sin[x]^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a , b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]
Time = 0.79 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.40
| method | result | size |
| derivativedivides | \(c^{3} \left (-d^{3} a^{2} \left (\frac {c^{3} x^{3}}{3}-3 c x +\frac {1}{3 c^{3} x^{3}}-\frac {3}{c x}\right )-\frac {50 d^{3} b^{2} c x}{9}+\frac {2 d^{3} b^{2} c^{3} x^{3}}{27}-\frac {d^{3} b^{2}}{3 c x}+\frac {17 d^{3} b^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {17 d^{3} b^{2} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {2 d^{3} b^{2} \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{2} x^{2}}{9}+\frac {3 d^{3} b^{2} \arcsin \left (c x \right )^{2}}{c x}-\frac {d^{3} b^{2} \arcsin \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {d^{3} b^{2} \arcsin \left (c x \right )^{2} c^{3} x^{3}}{3}+3 d^{3} b^{2} \arcsin \left (c x \right )^{2} c x +\frac {17 i d^{3} b^{2} \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {17 i d^{3} b^{2} \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {d^{3} b^{2} \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )}{3 c^{2} x^{2}}+\frac {50 d^{3} b^{2} \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )}{9}-2 d^{3} a b \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}-3 c x \arcsin \left (c x \right )+\frac {\arcsin \left (c x \right )}{3 c^{3} x^{3}}-\frac {3 \arcsin \left (c x \right )}{c x}+\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {17 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {25 \sqrt {-c^{2} x^{2}+1}}{9}\right )\right )\) | \(488\) |
| default | \(c^{3} \left (-d^{3} a^{2} \left (\frac {c^{3} x^{3}}{3}-3 c x +\frac {1}{3 c^{3} x^{3}}-\frac {3}{c x}\right )-\frac {50 d^{3} b^{2} c x}{9}+\frac {2 d^{3} b^{2} c^{3} x^{3}}{27}-\frac {d^{3} b^{2}}{3 c x}+\frac {17 d^{3} b^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {17 d^{3} b^{2} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {2 d^{3} b^{2} \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{2} x^{2}}{9}+\frac {3 d^{3} b^{2} \arcsin \left (c x \right )^{2}}{c x}-\frac {d^{3} b^{2} \arcsin \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {d^{3} b^{2} \arcsin \left (c x \right )^{2} c^{3} x^{3}}{3}+3 d^{3} b^{2} \arcsin \left (c x \right )^{2} c x +\frac {17 i d^{3} b^{2} \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {17 i d^{3} b^{2} \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {d^{3} b^{2} \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )}{3 c^{2} x^{2}}+\frac {50 d^{3} b^{2} \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )}{9}-2 d^{3} a b \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}-3 c x \arcsin \left (c x \right )+\frac {\arcsin \left (c x \right )}{3 c^{3} x^{3}}-\frac {3 \arcsin \left (c x \right )}{c x}+\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {17 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {25 \sqrt {-c^{2} x^{2}+1}}{9}\right )\right )\) | \(488\) |
| parts | \(-d^{3} a^{2} \left (\frac {c^{6} x^{3}}{3}-3 c^{4} x +\frac {1}{3 x^{3}}-\frac {3 c^{2}}{x}\right )-\frac {50 b^{2} c^{4} d^{3} x}{9}-\frac {2 d^{3} b^{2} c^{5} \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2}}{9}+\frac {2 b^{2} c^{6} d^{3} x^{3}}{27}-\frac {d^{3} b^{2} c \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )}{3 x^{2}}-\frac {b^{2} c^{2} d^{3}}{3 x}-\frac {17 i b^{2} c^{3} d^{3} \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {17 i b^{2} c^{3} d^{3} \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {17 d^{3} b^{2} c^{3} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {17 d^{3} b^{2} c^{3} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {3 d^{3} b^{2} c^{2} \arcsin \left (c x \right )^{2}}{x}-\frac {d^{3} b^{2} \arcsin \left (c x \right )^{2}}{3 x^{3}}-\frac {d^{3} b^{2} c^{6} \arcsin \left (c x \right )^{2} x^{3}}{3}+\frac {50 d^{3} b^{2} c^{3} \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )}{9}+3 d^{3} b^{2} c^{4} \arcsin \left (c x \right )^{2} x -2 d^{3} a b \,c^{3} \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}-3 c x \arcsin \left (c x \right )+\frac {\arcsin \left (c x \right )}{3 c^{3} x^{3}}-\frac {3 \arcsin \left (c x \right )}{c x}+\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {17 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {25 \sqrt {-c^{2} x^{2}+1}}{9}\right )\) | \(500\) |
Input:
int((-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))^2/x^4,x,method=_RETURNVERBOSE)
Output:
c^3*(-d^3*a^2*(1/3*c^3*x^3-3*c*x+1/3/c^3/x^3-3/c/x)-50/9*d^3*b^2*c*x+2/27* d^3*b^2*c^3*x^3-1/3*d^3*b^2/c/x+17/3*d^3*b^2*arcsin(c*x)*ln(1+I*c*x+(-c^2* x^2+1)^(1/2))-17/3*d^3*b^2*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-2/9* d^3*b^2*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^2*x^2+3*d^3*b^2/c/x*arcsin(c*x)^2 -1/3*d^3*b^2/c^3/x^3*arcsin(c*x)^2-1/3*d^3*b^2*arcsin(c*x)^2*c^3*x^3+3*d^3 *b^2*arcsin(c*x)^2*c*x+17/3*I*d^3*b^2*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))- 17/3*I*d^3*b^2*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-1/3*d^3*b^2/c^2/x^2*(- c^2*x^2+1)^(1/2)*arcsin(c*x)+50/9*d^3*b^2*(-c^2*x^2+1)^(1/2)*arcsin(c*x)-2 *d^3*a*b*(1/3*c^3*x^3*arcsin(c*x)-3*c*x*arcsin(c*x)+1/3*arcsin(c*x)/c^3/x^ 3-3*arcsin(c*x)/c/x+1/6/c^2/x^2*(-c^2*x^2+1)^(1/2)-17/6*arctanh(1/(-c^2*x^ 2+1)^(1/2))+1/9*c^2*x^2*(-c^2*x^2+1)^(1/2)-25/9*(-c^2*x^2+1)^(1/2)))
\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2}{x^4} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \] Input:
integrate((-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))^2/x^4,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 rcsin(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)*arcsin(c*x))/x^4, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2}{x^4} \, dx=- d^{3} \left (\int \left (- 3 a^{2} c^{4}\right )\, dx + \int \left (- \frac {a^{2}}{x^{4}}\right )\, dx + \int \frac {3 a^{2} c^{2}}{x^{2}}\, dx + \int a^{2} c^{6} x^{2}\, dx + \int \left (- 3 b^{2} c^{4} \operatorname {asin}^{2}{\left (c x \right )}\right )\, dx + \int \left (- \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x^{4}}\right )\, dx + \int \left (- 6 a b c^{4} \operatorname {asin}{\left (c x \right )}\right )\, dx + \int \left (- \frac {2 a b \operatorname {asin}{\left (c x \right )}}{x^{4}}\right )\, dx + \int \frac {3 b^{2} c^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int b^{2} c^{6} x^{2} \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int \frac {6 a b c^{2} \operatorname {asin}{\left (c x \right )}}{x^{2}}\, dx + \int 2 a b c^{6} x^{2} \operatorname {asin}{\left (c x \right )}\, dx\right ) \] Input:
integrate((-c**2*d*x**2+d)**3*(a+b*asin(c*x))**2/x**4,x)
Output:
-d**3*(Integral(-3*a**2*c**4, x) + Integral(-a**2/x**4, x) + Integral(3*a* *2*c**2/x**2, x) + Integral(a**2*c**6*x**2, x) + Integral(-3*b**2*c**4*asi n(c*x)**2, x) + Integral(-b**2*asin(c*x)**2/x**4, x) + Integral(-6*a*b*c** 4*asin(c*x), x) + Integral(-2*a*b*asin(c*x)/x**4, x) + Integral(3*b**2*c** 2*asin(c*x)**2/x**2, x) + Integral(b**2*c**6*x**2*asin(c*x)**2, x) + Integ ral(6*a*b*c**2*asin(c*x)/x**2, x) + Integral(2*a*b*c**6*x**2*asin(c*x), x) )
\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2}{x^4} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \] Input:
integrate((-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))^2/x^4,x, algorithm="maxima")
Output:
-1/3*a^2*c^6*d^3*x^3 - 2/9*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/ c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*a*b*c^6*d^3 + 3*b^2*c^4*d^3*x*arcsin(c*x) ^2 - 6*b^2*c^4*d^3*(x - sqrt(-c^2*x^2 + 1)*arcsin(c*x)/c) + 3*a^2*c^4*d^3* x + 6*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*a*b*c^3*d^3 + 6*(c*log(2*sqrt (-c^2*x^2 + 1)/abs(x) + 2/abs(x)) + arcsin(c*x)/x)*a*b*c^2*d^3 - 1/3*((c^2 *log(2*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) + sqrt(-c^2*x^2 + 1)/x^2)*c + 2*arcsin(c*x)/x^3)*a*b*d^3 + 3*a^2*c^2*d^3/x - 1/3*a^2*d^3/x^3 - 1/3*(3*x ^3*integrate(2/3*(b^2*c^7*d^3*x^6 - 9*b^2*c^3*d^3*x^2 + b^2*c*d^3)*sqrt(c* x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^2*x^5 - x^3), x) + (b^2*c^6*d^3*x^6 - 9*b^2*c^2*d^3*x^2 + b^2*d^3)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2)/x^3
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2}{x^4} \, dx=\text {Timed out} \] Input:
integrate((-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))^2/x^4,x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^3}{x^4} \,d x \] Input:
int(((a + b*asin(c*x))^2*(d - c^2*d*x^2)^3)/x^4,x)
Output:
int(((a + b*asin(c*x))^2*(d - c^2*d*x^2)^3)/x^4, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2}{x^4} \, dx=\frac {d^{3} \left (27 \mathit {asin} \left (c x \right )^{2} b^{2} c^{4} x^{4}+54 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) b^{2} c^{3} x^{3}-6 \mathit {asin} \left (c x \right ) a b \,c^{6} x^{6}+54 \mathit {asin} \left (c x \right ) a b \,c^{4} x^{4}+54 \mathit {asin} \left (c x \right ) a b \,c^{2} x^{2}-6 \mathit {asin} \left (c x \right ) a b -2 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{5} x^{5}+50 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{3} x^{3}-3 \sqrt {-c^{2} x^{2}+1}\, a b c x +9 \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{x^{4}}d x \right ) b^{2} x^{3}-27 \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{x^{2}}d x \right ) b^{2} c^{2} x^{3}-9 \left (\int \mathit {asin} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{6} x^{3}-51 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right )\right ) a b \,c^{3} x^{3}-3 a^{2} c^{6} x^{6}+27 a^{2} c^{4} x^{4}+27 a^{2} c^{2} x^{2}-3 a^{2}-54 b^{2} c^{4} x^{4}\right )}{9 x^{3}} \] Input:
int((-c^2*d*x^2+d)^3*(a+b*asin(c*x))^2/x^4,x)
Output:
(d**3*(27*asin(c*x)**2*b**2*c**4*x**4 + 54*sqrt( - c**2*x**2 + 1)*asin(c*x )*b**2*c**3*x**3 - 6*asin(c*x)*a*b*c**6*x**6 + 54*asin(c*x)*a*b*c**4*x**4 + 54*asin(c*x)*a*b*c**2*x**2 - 6*asin(c*x)*a*b - 2*sqrt( - c**2*x**2 + 1)* a*b*c**5*x**5 + 50*sqrt( - c**2*x**2 + 1)*a*b*c**3*x**3 - 3*sqrt( - c**2*x **2 + 1)*a*b*c*x + 9*int(asin(c*x)**2/x**4,x)*b**2*x**3 - 27*int(asin(c*x) **2/x**2,x)*b**2*c**2*x**3 - 9*int(asin(c*x)**2*x**2,x)*b**2*c**6*x**3 - 5 1*log(tan(asin(c*x)/2))*a*b*c**3*x**3 - 3*a**2*c**6*x**6 + 27*a**2*c**4*x* *4 + 27*a**2*c**2*x**2 - 3*a**2 - 54*b**2*c**4*x**4))/(9*x**3)