Integrand size = 27, antiderivative size = 329 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2}{x^2} \, dx=\frac {122}{25} b^2 c^2 d^3 x-\frac {14}{75} b^2 c^4 d^3 x^3+\frac {2}{125} b^2 c^6 d^3 x^5-\frac {22}{5} b c d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {2}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {2}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {16}{5} c^2 d^3 x (a+b \arcsin (c x))^2-\frac {8}{5} c^2 d^3 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{x}-4 b c d^3 (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )+2 i b^2 c d^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-2 i b^2 c d^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) \]
122/25*b^2*c^2*d^3*x-14/75*b^2*c^4*d^3*x^3+2/125*b^2*c^6*d^3*x^5-2/5*b*c*d ^3*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))-2/25*b*c*d^3*(-c^2*x^2+1)^(5/2)*(a +b*arcsin(c*x))-16/5*c^2*d^3*x*(a+b*arcsin(c*x))^2-8/5*c^2*d^3*x*(-c^2*x^2 +1)*(a+b*arcsin(c*x))^2-6/5*c^2*d^3*x*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))^2-d ^3*(-c^2*x^2+1)^3*(a+b*arcsin(c*x))^2/x-4*b*c*d^3*(a+b*arcsin(c*x))*arctan h(I*c*x+(-c^2*x^2+1)^(1/2))+2*I*b^2*c*d^3*polylog(2,-I*c*x-(-c^2*x^2+1)^(1 /2))-2*I*b^2*c*d^3*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-22/5*b*c*d^3*(a+b*a rcsin(c*x))*(-c^2*x^2+1)^(1/2)
Time = 1.36 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.47 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2}{x^2} \, dx=\frac {1}{720} d^3 \left (-\frac {720 a^2}{x}-2160 a^2 c^2 x+3460 b^2 c^2 x+720 a^2 c^4 x^3-144 a^2 c^6 x^5-\frac {17568}{5} a b c \sqrt {1-c^2 x^2}+\frac {2016}{5} a b c^3 x^2 \sqrt {1-c^2 x^2}-\frac {288}{5} a b c^5 x^4 \sqrt {1-c^2 x^2}-\frac {1440 a b \arcsin (c x)}{x}-4320 a b c^2 x \arcsin (c x)+1440 a b c^4 x^3 \arcsin (c x)-288 a b c^6 x^5 \arcsin (c x)-3420 b^2 c \sqrt {1-c^2 x^2} \arcsin (c x)-\frac {720 b^2 \arcsin (c x)^2}{x}-1890 b^2 c^2 x \arcsin (c x)^2-1440 a b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+80 b^2 c^2 x \cos (2 \arcsin (c x))-360 b^2 c^2 x \arcsin (c x)^2 \cos (2 \arcsin (c x))-90 b^2 c \arcsin (c x) \cos (3 \arcsin (c x))-\frac {18}{5} b^2 c \arcsin (c x) \cos (5 \arcsin (c x))+1440 b^2 c \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )-1440 b^2 c \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )+1440 i b^2 c \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-1440 i b^2 c \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-10 b^2 c \sin (3 \arcsin (c x))+45 b^2 c \arcsin (c x)^2 \sin (3 \arcsin (c x))+\frac {18}{25} b^2 c \sin (5 \arcsin (c x))-9 b^2 c \arcsin (c x)^2 \sin (5 \arcsin (c x))\right ) \]
(d^3*((-720*a^2)/x - 2160*a^2*c^2*x + 3460*b^2*c^2*x + 720*a^2*c^4*x^3 - 1 44*a^2*c^6*x^5 - (17568*a*b*c*Sqrt[1 - c^2*x^2])/5 + (2016*a*b*c^3*x^2*Sqr t[1 - c^2*x^2])/5 - (288*a*b*c^5*x^4*Sqrt[1 - c^2*x^2])/5 - (1440*a*b*ArcS in[c*x])/x - 4320*a*b*c^2*x*ArcSin[c*x] + 1440*a*b*c^4*x^3*ArcSin[c*x] - 2 88*a*b*c^6*x^5*ArcSin[c*x] - 3420*b^2*c*Sqrt[1 - c^2*x^2]*ArcSin[c*x] - (7 20*b^2*ArcSin[c*x]^2)/x - 1890*b^2*c^2*x*ArcSin[c*x]^2 - 1440*a*b*c*ArcTan h[Sqrt[1 - c^2*x^2]] + 80*b^2*c^2*x*Cos[2*ArcSin[c*x]] - 360*b^2*c^2*x*Arc Sin[c*x]^2*Cos[2*ArcSin[c*x]] - 90*b^2*c*ArcSin[c*x]*Cos[3*ArcSin[c*x]] - (18*b^2*c*ArcSin[c*x]*Cos[5*ArcSin[c*x]])/5 + 1440*b^2*c*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] - 1440*b^2*c*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + (1440*I)*b^2*c*PolyLog[2, -E^(I*ArcSin[c*x])] - (1440*I)*b^2*c*PolyLog[2 , E^(I*ArcSin[c*x])] - 10*b^2*c*Sin[3*ArcSin[c*x]] + 45*b^2*c*ArcSin[c*x]^ 2*Sin[3*ArcSin[c*x]] + (18*b^2*c*Sin[5*ArcSin[c*x]])/25 - 9*b^2*c*ArcSin[c *x]^2*Sin[5*ArcSin[c*x]]))/720
Time = 2.82 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.41, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {5200, 27, 5158, 5158, 5130, 5182, 24, 210, 2009, 5202, 210, 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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2}{x^2} \, dx\) |
| \(\Big \downarrow \) 5200 |
| \(\displaystyle 2 b c d^3 \int \frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}dx-6 c^2 d \int d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2dx-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 b c d^3 \int \frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}dx-6 c^2 d^3 \int \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2dx-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{x}\) |
| \(\Big \downarrow \) 5158 |
| \(\displaystyle 2 b c d^3 \int \frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}dx-6 c^2 d^3 \left (-\frac {2}{5} b c \int x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx+\frac {4}{5} \int \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{x}\) |
| \(\Big \downarrow \) 5158 |
| \(\displaystyle 2 b c d^3 \int \frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}dx-6 c^2 d^3 \left (-\frac {2}{5} b c \int x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx+\frac {4}{5} \left (-\frac {2}{3} b c \int x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {2}{3} \int (a+b \arcsin (c x))^2dx+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{x}\) |
| \(\Big \downarrow \) 5130 |
| \(\displaystyle -6 c^2 d^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )-\frac {2}{3} b c \int x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )-\frac {2}{5} b c \int x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )+2 b c d^3 \int \frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}dx-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{x}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle -6 c^2 d^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b \int 1dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \int \left (1-c^2 x^2\right )dx}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )-\frac {2}{5} b c \left (\frac {b \int \left (1-c^2 x^2\right )^2dx}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^2}\right )+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )+2 b c d^3 \int \frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}dx-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{x}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -6 c^2 d^3 \left (\frac {4}{5} \left (-\frac {2}{3} b c \left (\frac {b \int \left (1-c^2 x^2\right )dx}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )\right )-\frac {2}{5} b c \left (\frac {b \int \left (1-c^2 x^2\right )^2dx}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^2}\right )+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )+2 b c d^3 \int \frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}dx-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{x}\) |
| \(\Big \downarrow \) 210 |
| \(\displaystyle 2 b c d^3 \int \frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}dx-6 c^2 d^3 \left (\frac {4}{5} \left (-\frac {2}{3} b c \left (\frac {b \int \left (1-c^2 x^2\right )dx}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )\right )-\frac {2}{5} b c \left (\frac {b \int \left (c^4 x^4-2 c^2 x^2+1\right )dx}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^2}\right )+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 b c d^3 \int \frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}dx-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{x}-6 c^2 d^3 \left (\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )-\frac {2}{5} b c \left (\frac {b \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^2}\right )\right )\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle 2 b c d^3 \left (\int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx-\frac {1}{5} b c \int \left (1-c^2 x^2\right )^2dx+\frac {1}{5} \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{x}-6 c^2 d^3 \left (\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )-\frac {2}{5} b c \left (\frac {b \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^2}\right )\right )\) |
| \(\Big \downarrow \) 210 |
| \(\displaystyle 2 b c d^3 \left (\int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx-\frac {1}{5} b c \int \left (c^4 x^4-2 c^2 x^2+1\right )dx+\frac {1}{5} \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{x}-6 c^2 d^3 \left (\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )-\frac {2}{5} b c \left (\frac {b \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^2}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 b c d^3 \left (\int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx+\frac {1}{5} \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {1}{5} b c \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{x}-6 c^2 d^3 \left (\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )-\frac {2}{5} b c \left (\frac {b \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^2}\right )\right )\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle 2 b c d^3 \left (\int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}dx-\frac {1}{3} b c \int \left (1-c^2 x^2\right )dx+\frac {1}{5} \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{5} b c \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{x}-6 c^2 d^3 \left (\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )-\frac {2}{5} b c \left (\frac {b \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^2}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 b c d^3 \left (\int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}dx+\frac {1}{5} \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )-\frac {1}{5} b c \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{x}-6 c^2 d^3 \left (\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )-\frac {2}{5} b c \left (\frac {b \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^2}\right )\right )\) |
| \(\Big \downarrow \) 5198 |
| \(\displaystyle 2 b c d^3 \left (\int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx-b c \int 1dx+\frac {1}{5} \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )-\frac {1}{5} b c \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{x}-6 c^2 d^3 \left (\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )-\frac {2}{5} b c \left (\frac {b \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^2}\right )\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle 2 b c d^3 \left (\int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx+\frac {1}{5} \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )-\frac {1}{5} b c \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right )-b c x\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{x}-6 c^2 d^3 \left (\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )-\frac {2}{5} b c \left (\frac {b \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^2}\right )\right )\) |
| \(\Big \downarrow \) 5218 |
| \(\displaystyle 2 b c d^3 \left (\int \frac {a+b \arcsin (c x)}{c x}d\arcsin (c x)+\frac {1}{5} \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )-\frac {1}{5} b c \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right )-b c x\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{x}-6 c^2 d^3 \left (\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )-\frac {2}{5} b c \left (\frac {b \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^2}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 b c d^3 \left (\int (a+b \arcsin (c x)) \csc (\arcsin (c x))d\arcsin (c x)+\frac {1}{5} \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )-\frac {1}{5} b c \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right )-b c x\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{x}-6 c^2 d^3 \left (\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )-\frac {2}{5} b c \left (\frac {b \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^2}\right )\right )\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle 2 b c d^3 \left (-b \int \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{5} \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )-\frac {1}{5} b c \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right )-b c x\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{x}-6 c^2 d^3 \left (\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )-\frac {2}{5} b c \left (\frac {b \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^2}\right )\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle 2 b c d^3 \left (i b \int e^{-i \arcsin (c x)} \log \left (1-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{5} \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )-\frac {1}{5} b c \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right )-b c x\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{x}-6 c^2 d^3 \left (\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )-\frac {2}{5} b c \left (\frac {b \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^2}\right )\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle 2 b c d^3 \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{5} \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )-\frac {1}{5} b c \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right )-b c x\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{x}-6 c^2 d^3 \left (\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )-\frac {2}{5} b c \left (\frac {b \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^2}\right )\right )\) |
-((d^3*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/x) - 6*c^2*d^3*((x*(1 - c^2* x^2)^2*(a + b*ArcSin[c*x])^2)/5 - (2*b*c*((b*(x - (2*c^2*x^3)/3 + (c^4*x^5 )/5))/(5*c) - ((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(5*c^2)))/5 + (4*( (x*(1 - c^2*x^2)*(a + b*ArcSin[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))/5) + 2*b*c*d^3*(-(b*c*x) - (b*c*(x - (c^2*x^3)/3))/3 - (b*c* (x - (2*c^2*x^3)/3 + (c^4*x^5)/5))/5 + Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x ]) + ((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/3 + ((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/5 - 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])])
3.2.80.3.1 Defintions of rubi rules used
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_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 )^p, x], x] /; FreeQ[{a, b}, 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 = 0.19 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.26
| method | result | size |
| parts | \(-d^{3} a^{2} \left (\frac {c^{6} x^{5}}{5}-c^{4} x^{3}+3 c^{2} x +\frac {1}{x}\right )-d^{3} b^{2} c \left (\frac {19 \left (-i \sqrt {-c^{2} x^{2}+1}+c x \right ) \left (\arcsin \left (c x \right )^{2}-2+2 i \arcsin \left (c x \right )\right )}{16}+\frac {19 \left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) \left (\arcsin \left (c x \right )^{2}-2-2 i \arcsin \left (c x \right )\right )}{16}+\frac {\arcsin \left (c x \right )^{2}}{c x}+2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {\arcsin \left (c x \right ) \cos \left (5 \arcsin \left (c x \right )\right )}{200}+\frac {\left (25 \arcsin \left (c x \right )^{2}-2\right ) \sin \left (5 \arcsin \left (c x \right )\right )}{2000}+\frac {\arcsin \left (c x \right ) \cos \left (3 \arcsin \left (c x \right )\right )}{8}+\frac {\left (9 \arcsin \left (c x \right )^{2}-2\right ) \sin \left (3 \arcsin \left (c x \right )\right )}{48}\right )-2 d^{3} a b c \left (\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}-c^{3} x^{3} \arcsin \left (c x \right )+3 c x \arcsin \left (c x \right )+\frac {\arcsin \left (c x \right )}{c x}+\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {7 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{25}+\frac {61 \sqrt {-c^{2} x^{2}+1}}{25}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\) | \(416\) |
| derivativedivides | \(c \left (-d^{3} a^{2} \left (\frac {c^{5} x^{5}}{5}-c^{3} x^{3}+3 c x +\frac {1}{c x}\right )-d^{3} b^{2} \left (\frac {19 \left (-i \sqrt {-c^{2} x^{2}+1}+c x \right ) \left (\arcsin \left (c x \right )^{2}-2+2 i \arcsin \left (c x \right )\right )}{16}+\frac {19 \left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) \left (\arcsin \left (c x \right )^{2}-2-2 i \arcsin \left (c x \right )\right )}{16}+\frac {\arcsin \left (c x \right )^{2}}{c x}+2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {\arcsin \left (c x \right ) \cos \left (5 \arcsin \left (c x \right )\right )}{200}+\frac {\left (25 \arcsin \left (c x \right )^{2}-2\right ) \sin \left (5 \arcsin \left (c x \right )\right )}{2000}+\frac {\arcsin \left (c x \right ) \cos \left (3 \arcsin \left (c x \right )\right )}{8}+\frac {\left (9 \arcsin \left (c x \right )^{2}-2\right ) \sin \left (3 \arcsin \left (c x \right )\right )}{48}\right )-2 d^{3} a b \left (\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}-c^{3} x^{3} \arcsin \left (c x \right )+3 c x \arcsin \left (c x \right )+\frac {\arcsin \left (c x \right )}{c x}+\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {7 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{25}+\frac {61 \sqrt {-c^{2} x^{2}+1}}{25}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\) | \(418\) |
| default | \(c \left (-d^{3} a^{2} \left (\frac {c^{5} x^{5}}{5}-c^{3} x^{3}+3 c x +\frac {1}{c x}\right )-d^{3} b^{2} \left (\frac {19 \left (-i \sqrt {-c^{2} x^{2}+1}+c x \right ) \left (\arcsin \left (c x \right )^{2}-2+2 i \arcsin \left (c x \right )\right )}{16}+\frac {19 \left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) \left (\arcsin \left (c x \right )^{2}-2-2 i \arcsin \left (c x \right )\right )}{16}+\frac {\arcsin \left (c x \right )^{2}}{c x}+2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {\arcsin \left (c x \right ) \cos \left (5 \arcsin \left (c x \right )\right )}{200}+\frac {\left (25 \arcsin \left (c x \right )^{2}-2\right ) \sin \left (5 \arcsin \left (c x \right )\right )}{2000}+\frac {\arcsin \left (c x \right ) \cos \left (3 \arcsin \left (c x \right )\right )}{8}+\frac {\left (9 \arcsin \left (c x \right )^{2}-2\right ) \sin \left (3 \arcsin \left (c x \right )\right )}{48}\right )-2 d^{3} a b \left (\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}-c^{3} x^{3} \arcsin \left (c x \right )+3 c x \arcsin \left (c x \right )+\frac {\arcsin \left (c x \right )}{c x}+\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {7 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{25}+\frac {61 \sqrt {-c^{2} x^{2}+1}}{25}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\) | \(418\) |
-d^3*a^2*(1/5*c^6*x^5-c^4*x^3+3*c^2*x+1/x)-d^3*b^2*c*(19/16*(-I*(-c^2*x^2+ 1)^(1/2)+c*x)*(arcsin(c*x)^2-2+2*I*arcsin(c*x))+19/16*(c*x+I*(-c^2*x^2+1)^ (1/2))*(arcsin(c*x)^2-2-2*I*arcsin(c*x))+1/c/x*arcsin(c*x)^2+2*arcsin(c*x) *ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-2*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2 ))-2*I*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))+2*I*polylog(2,I*c*x+(-c^2*x^2+ 1)^(1/2))+1/200*arcsin(c*x)*cos(5*arcsin(c*x))+1/2000*(25*arcsin(c*x)^2-2) *sin(5*arcsin(c*x))+1/8*arcsin(c*x)*cos(3*arcsin(c*x))+1/48*(9*arcsin(c*x) ^2-2)*sin(3*arcsin(c*x)))-2*d^3*a*b*c*(1/5*arcsin(c*x)*c^5*x^5-c^3*x^3*arc sin(c*x)+3*c*x*arcsin(c*x)+1/c/x*arcsin(c*x)+1/25*c^4*x^4*(-c^2*x^2+1)^(1/ 2)-7/25*c^2*x^2*(-c^2*x^2+1)^(1/2)+61/25*(-c^2*x^2+1)^(1/2)+arctanh(1/(-c^ 2*x^2+1)^(1/2)))
\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2}{x^2} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
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^2, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2}{x^2} \, dx=- d^{3} \left (\int 3 a^{2} c^{2}\, dx + \int \left (- \frac {a^{2}}{x^{2}}\right )\, dx + \int \left (- 3 a^{2} c^{4} x^{2}\right )\, dx + \int a^{2} c^{6} x^{4}\, dx + \int 3 b^{2} c^{2} \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int \left (- \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x^{2}}\right )\, dx + \int 6 a b c^{2} \operatorname {asin}{\left (c x \right )}\, dx + \int \left (- \frac {2 a b \operatorname {asin}{\left (c x \right )}}{x^{2}}\right )\, dx + \int \left (- 3 b^{2} c^{4} x^{2} \operatorname {asin}^{2}{\left (c x \right )}\right )\, dx + \int b^{2} c^{6} x^{4} \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int \left (- 6 a b c^{4} x^{2} \operatorname {asin}{\left (c x \right )}\right )\, dx + \int 2 a b c^{6} x^{4} \operatorname {asin}{\left (c x \right )}\, dx\right ) \]
-d**3*(Integral(3*a**2*c**2, x) + Integral(-a**2/x**2, x) + Integral(-3*a* *2*c**4*x**2, x) + Integral(a**2*c**6*x**4, x) + Integral(3*b**2*c**2*asin (c*x)**2, x) + Integral(-b**2*asin(c*x)**2/x**2, x) + Integral(6*a*b*c**2* asin(c*x), x) + Integral(-2*a*b*asin(c*x)/x**2, x) + Integral(-3*b**2*c**4 *x**2*asin(c*x)**2, x) + Integral(b**2*c**6*x**4*asin(c*x)**2, x) + Integr al(-6*a*b*c**4*x**2*asin(c*x), x) + Integral(2*a*b*c**6*x**4*asin(c*x), x) )
\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2}{x^2} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
-1/5*a^2*c^6*d^3*x^5 - 2/75*(15*x^5*arcsin(c*x) + (3*sqrt(-c^2*x^2 + 1)*x^ 4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c)*a*b*c^ 6*d^3 + a^2*c^4*d^3*x^3 + 2/3*(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^4*d^3 - 3*b^2*c^2*d^3*x*arcsin(c *x)^2 + 6*b^2*c^2*d^3*(x - sqrt(-c^2*x^2 + 1)*arcsin(c*x)/c) - 3*a^2*c^2*d ^3*x - 6*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*a*b*c*d^3 - 2*(c*log(2*sqr t(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) + arcsin(c*x)/x)*a*b*d^3 - a^2*d^3/x - 1/5*((b^2*c^6*d^3*x^6 - 5*b^2*c^4*d^3*x^4 + 5*b^2*d^3)*arctan2(c*x, sqrt(c *x + 1)*sqrt(-c*x + 1))^2 + 5*x*integrate(2/5*(b^2*c^7*d^3*x^6 - 5*b^2*c^5 *d^3*x^4 + 5*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^3 - x), x))/x
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2}{x^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^3}{x^2} \,d x \]