3.2.71 \(\int \frac {(d-c^2 d x^2)^2 (a+b \arcsin (c x))^2}{x^2} \, dx\) [171]

3.2.71.1 Optimal result

Integrand size = 27, antiderivative size = 249 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x^2} \, dx=\frac {32}{9} b^2 c^2 d^2 x-\frac {2}{27} b^2 c^4 d^2 x^3-\frac {10}{3} b c d^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {2}{9} b c d^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {8}{3} c^2 d^2 x (a+b \arcsin (c x))^2-\frac {4}{3} c^2 d^2 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}-4 b c d^2 (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )+2 i b^2 c d^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-2 i b^2 c d^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) \]

output

32/9*b^2*c^2*d^2*x-2/27*b^2*c^4*d^2*x^3-2/9*b*c*d^2*(-c^2*x^2+1)^(3/2)*(a+ 
b*arcsin(c*x))-8/3*c^2*d^2*x*(a+b*arcsin(c*x))^2-4/3*c^2*d^2*x*(-c^2*x^2+1 
)*(a+b*arcsin(c*x))^2-d^2*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))^2/x-4*b*c*d^2*( 
a+b*arcsin(c*x))*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))+2*I*b^2*c*d^2*polylog(2 
,-I*c*x-(-c^2*x^2+1)^(1/2))-2*I*b^2*c*d^2*polylog(2,I*c*x+(-c^2*x^2+1)^(1/ 
2))-10/3*b*c*d^2*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)
 

3.2.71.2 Mathematica [A] (verified)

Time = 1.11 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.29 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x^2} \, dx=\frac {1}{54} d^2 \left (-\frac {54 a^2}{x}-108 a^2 c^2 x+18 a^2 c^4 x^3+12 a b c \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )+36 a b c^4 x^3 \arcsin (c x)-189 b^2 c \sqrt {1-c^2 x^2} \arcsin (c x)-216 a b c \left (\sqrt {1-c^2 x^2}+c x \arcsin (c x)\right )-108 b^2 c^2 x \left (-2+\arcsin (c x)^2\right )+2 b^2 c^2 x \left (-2 \left (6+c^2 x^2\right )+9 c^2 x^2 \arcsin (c x)^2\right )-\frac {108 a b \left (\arcsin (c x)+c x \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{x}-3 b^2 c \arcsin (c x) \cos (3 \arcsin (c x))-\frac {54 b^2 \arcsin (c x) \left (\arcsin (c x)+2 c x \left (-\log \left (1-e^{i \arcsin (c x)}\right )+\log \left (1+e^{i \arcsin (c x)}\right )\right )\right )}{x}+108 i b^2 c \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-108 i b^2 c \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right ) \]

input

Integrate[((d - c^2*d*x^2)^2*(a + b*ArcSin[c*x])^2)/x^2,x]
 

output

(d^2*((-54*a^2)/x - 108*a^2*c^2*x + 18*a^2*c^4*x^3 + 12*a*b*c*Sqrt[1 - c^2 
*x^2]*(2 + c^2*x^2) + 36*a*b*c^4*x^3*ArcSin[c*x] - 189*b^2*c*Sqrt[1 - c^2* 
x^2]*ArcSin[c*x] - 216*a*b*c*(Sqrt[1 - c^2*x^2] + c*x*ArcSin[c*x]) - 108*b 
^2*c^2*x*(-2 + ArcSin[c*x]^2) + 2*b^2*c^2*x*(-2*(6 + c^2*x^2) + 9*c^2*x^2* 
ArcSin[c*x]^2) - (108*a*b*(ArcSin[c*x] + c*x*ArcTanh[Sqrt[1 - c^2*x^2]]))/ 
x - 3*b^2*c*ArcSin[c*x]*Cos[3*ArcSin[c*x]] - (54*b^2*ArcSin[c*x]*(ArcSin[c 
*x] + 2*c*x*(-Log[1 - E^(I*ArcSin[c*x])] + Log[1 + E^(I*ArcSin[c*x])])))/x 
 + (108*I)*b^2*c*PolyLog[2, -E^(I*ArcSin[c*x])] - (108*I)*b^2*c*PolyLog[2, 
 E^(I*ArcSin[c*x])]))/54
 

3.2.71.3 Rubi [A] (verified)

Time = 1.88 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.26, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used = {5200, 27, 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.

\(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x^2} \, dx\)

\(\Big \downarrow \) 5200

\(\displaystyle 2 b c d^2 \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx-4 c^2 d \int d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}\)

\(\Big \downarrow \) 27

\(\displaystyle 2 b c d^2 \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx-4 c^2 d^2 \int \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}\)

\(\Big \downarrow \) 5158

\(\displaystyle 2 b c d^2 \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx-4 c^2 d^2 \left (-\frac {2}{3} b c \int x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {2}{3} \int (a+b \arcsin (c x))^2dx+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}\)

\(\Big \downarrow \) 5130

\(\displaystyle -4 c^2 d^2 \left (\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )-\frac {2}{3} b c \int x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )+2 b c d^2 \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}\)

\(\Big \downarrow \) 5182

\(\displaystyle -4 c^2 d^2 \left (\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b \int 1dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \int \left (1-c^2 x^2\right )dx}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )+2 b c d^2 \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}\)

\(\Big \downarrow \) 24

\(\displaystyle -4 c^2 d^2 \left (-\frac {2}{3} b c \left (\frac {b \int \left (1-c^2 x^2\right )dx}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )\right )+2 b c d^2 \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 b c d^2 \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}-4 c^2 d^2 \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )\)

\(\Big \downarrow \) 5202

\(\displaystyle 2 b c d^2 \left (\int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}dx-\frac {1}{3} b c \int \left (1-c^2 x^2\right )dx+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}-4 c^2 d^2 \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 b c d^2 \left (\int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}dx+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}-4 c^2 d^2 \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )\)

\(\Big \downarrow \) 5198

\(\displaystyle 2 b c d^2 \left (\int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx-b c \int 1dx+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}-4 c^2 d^2 \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )\)

\(\Big \downarrow \) 24

\(\displaystyle 2 b c d^2 \left (\int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )-b c x\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}-4 c^2 d^2 \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )\)

\(\Big \downarrow \) 5218

\(\displaystyle 2 b c d^2 \left (\int \frac {a+b \arcsin (c x)}{c x}d\arcsin (c x)+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )-b c x\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}-4 c^2 d^2 \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 b c d^2 \left (\int (a+b \arcsin (c x)) \csc (\arcsin (c x))d\arcsin (c x)+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )-b c x\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}-4 c^2 d^2 \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )\)

\(\Big \downarrow \) 4671

\(\displaystyle 2 b c d^2 \left (-b \int \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )-b c x\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}-4 c^2 d^2 \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )\)

\(\Big \downarrow \) 2715

\(\displaystyle 2 b c d^2 \left (i b \int e^{-i \arcsin (c x)} \log \left (1-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )-b c x\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}-4 c^2 d^2 \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle 2 b c d^2 \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{3} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-\frac {1}{3} b c \left (x-\frac {c^2 x^3}{3}\right )-b c x\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}-4 c^2 d^2 \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\right )\)

input

Int[((d - c^2*d*x^2)^2*(a + b*ArcSin[c*x])^2)/x^2,x]
 

output

-((d^2*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/x) - 4*c^2*d^2*((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) 
 + 2*b*c*d^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])])
 

3.2.71.3.1 Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2715

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]
 

rule 2838

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4671

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]
 

rule 5130

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]
 

rule 5158

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]
 

rule 5182

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]
 

rule 5198

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])
 

rule 5200

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]
 

rule 5202

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]
 

rule 5218

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]
 

3.2.71.4 Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.49

method	result	size
derivativedivides	\(c \left (d^{2} a^{2} \left (\frac {c^{3} x^{3}}{3}-2 c x -\frac {1}{c x}\right )-\frac {7 d^{2} b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{2}-\frac {7 d^{2} b^{2} \arcsin \left (c x \right )^{2} c x}{4}+\frac {7 d^{2} b^{2} c x}{2}-\frac {d^{2} b^{2} \arcsin \left (c x \right )^{2}}{c x}-2 d^{2} b^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 d^{2} b^{2} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i d^{2} b^{2} \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i d^{2} b^{2} \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {d^{2} b^{2} \arcsin \left (c x \right ) \cos \left (3 \arcsin \left (c x \right )\right )}{18}-\frac {d^{2} b^{2} \arcsin \left (c x \right )^{2} \sin \left (3 \arcsin \left (c x \right )\right )}{12}+\frac {d^{2} b^{2} \sin \left (3 \arcsin \left (c x \right )\right )}{54}+2 d^{2} a b \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}-2 c x \arcsin \left (c x \right )-\frac {\arcsin \left (c x \right )}{c x}+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{9}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\)	\(372\)
default	\(c \left (d^{2} a^{2} \left (\frac {c^{3} x^{3}}{3}-2 c x -\frac {1}{c x}\right )-\frac {7 d^{2} b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{2}-\frac {7 d^{2} b^{2} \arcsin \left (c x \right )^{2} c x}{4}+\frac {7 d^{2} b^{2} c x}{2}-\frac {d^{2} b^{2} \arcsin \left (c x \right )^{2}}{c x}-2 d^{2} b^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 d^{2} b^{2} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i d^{2} b^{2} \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i d^{2} b^{2} \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {d^{2} b^{2} \arcsin \left (c x \right ) \cos \left (3 \arcsin \left (c x \right )\right )}{18}-\frac {d^{2} b^{2} \arcsin \left (c x \right )^{2} \sin \left (3 \arcsin \left (c x \right )\right )}{12}+\frac {d^{2} b^{2} \sin \left (3 \arcsin \left (c x \right )\right )}{54}+2 d^{2} a b \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}-2 c x \arcsin \left (c x \right )-\frac {\arcsin \left (c x \right )}{c x}+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{9}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\)	\(372\)
parts	\(d^{2} a^{2} \left (\frac {c^{4} x^{3}}{3}-2 c^{2} x -\frac {1}{x}\right )-\frac {7 d^{2} b^{2} c \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )}{2}-\frac {7 d^{2} b^{2} c^{2} \arcsin \left (c x \right )^{2} x}{4}+\frac {7 b^{2} c^{2} d^{2} x}{2}-\frac {d^{2} b^{2} \arcsin \left (c x \right )^{2}}{x}-2 d^{2} b^{2} c \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 d^{2} b^{2} c \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i b^{2} c \,d^{2} \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i b^{2} c \,d^{2} \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {d^{2} b^{2} c \arcsin \left (c x \right ) \cos \left (3 \arcsin \left (c x \right )\right )}{18}-\frac {d^{2} b^{2} c \arcsin \left (c x \right )^{2} \sin \left (3 \arcsin \left (c x \right )\right )}{12}+\frac {d^{2} b^{2} c \sin \left (3 \arcsin \left (c x \right )\right )}{54}+2 d^{2} a b c \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}-2 c x \arcsin \left (c x \right )-\frac {\arcsin \left (c x \right )}{c x}+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{9}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\)	\(379\)

input

int((-c^2*d*x^2+d)^2*(a+b*arcsin(c*x))^2/x^2,x,method=_RETURNVERBOSE)
 

output

c*(d^2*a^2*(1/3*c^3*x^3-2*c*x-1/c/x)-7/2*d^2*b^2*arcsin(c*x)*(-c^2*x^2+1)^ 
(1/2)-7/4*d^2*b^2*arcsin(c*x)^2*c*x+7/2*d^2*b^2*c*x-d^2*b^2/c/x*arcsin(c*x 
)^2-2*d^2*b^2*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+2*d^2*b^2*arcsin( 
c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+2*I*d^2*b^2*polylog(2,-I*c*x-(-c^2*x^2 
+1)^(1/2))-2*I*d^2*b^2*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-1/18*d^2*b^2*ar 
csin(c*x)*cos(3*arcsin(c*x))-1/12*d^2*b^2*arcsin(c*x)^2*sin(3*arcsin(c*x)) 
+1/54*d^2*b^2*sin(3*arcsin(c*x))+2*d^2*a*b*(1/3*c^3*x^3*arcsin(c*x)-2*c*x* 
arcsin(c*x)-1/c/x*arcsin(c*x)+1/9*c^2*x^2*(-c^2*x^2+1)^(1/2)-16/9*(-c^2*x^ 
2+1)^(1/2)-arctanh(1/(-c^2*x^2+1)^(1/2))))
 

3.2.71.5 Fricas [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x^2} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]

input

integrate((-c^2*d*x^2+d)^2*(a+b*arcsin(c*x))^2/x^2,x, algorithm="fricas")
 

output

integral((a^2*c^4*d^2*x^4 - 2*a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^4*d^2*x^4 
 - 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arcsin(c*x)^2 + 2*(a*b*c^4*d^2*x^4 - 2*a*b 
*c^2*d^2*x^2 + a*b*d^2)*arcsin(c*x))/x^2, x)
 

3.2.71.6 Sympy [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x^2} \, dx=d^{2} \left (\int \left (- 2 a^{2} c^{2}\right )\, dx + \int \frac {a^{2}}{x^{2}}\, dx + \int a^{2} c^{4} x^{2}\, dx + \int \left (- 2 b^{2} c^{2} \operatorname {asin}^{2}{\left (c x \right )}\right )\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int \left (- 4 a b c^{2} \operatorname {asin}{\left (c x \right )}\right )\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{x^{2}}\, dx + \int b^{2} c^{4} x^{2} \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int 2 a b c^{4} x^{2} \operatorname {asin}{\left (c x \right )}\, dx\right ) \]

input

integrate((-c**2*d*x**2+d)**2*(a+b*asin(c*x))**2/x**2,x)
 

output

d**2*(Integral(-2*a**2*c**2, x) + Integral(a**2/x**2, x) + Integral(a**2*c 
**4*x**2, x) + Integral(-2*b**2*c**2*asin(c*x)**2, x) + Integral(b**2*asin 
(c*x)**2/x**2, x) + Integral(-4*a*b*c**2*asin(c*x), x) + Integral(2*a*b*as 
in(c*x)/x**2, x) + Integral(b**2*c**4*x**2*asin(c*x)**2, x) + Integral(2*a 
*b*c**4*x**2*asin(c*x), x))
 

3.2.71.7 Maxima [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x^2} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]

input

integrate((-c^2*d*x^2+d)^2*(a+b*arcsin(c*x))^2/x^2,x, algorithm="maxima")
 

output

1/3*a^2*c^4*d^2*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^4*d^2 - 2*b^2*c^2*d^2*x*arcsin(c*x)^ 
2 + 4*b^2*c^2*d^2*(x - sqrt(-c^2*x^2 + 1)*arcsin(c*x)/c) - 2*a^2*c^2*d^2*x 
 - 4*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*a*b*c*d^2 - 2*(c*log(2*sqrt(-c 
^2*x^2 + 1)/abs(x) + 2/abs(x)) + arcsin(c*x)/x)*a*b*d^2 - a^2*d^2/x + 1/3* 
((b^2*c^4*d^2*x^4 - 3*b^2*d^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^ 
2 + 3*x*integrate(2/3*(b^2*c^5*d^2*x^4 - 3*b^2*c*d^2)*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
 

3.2.71.8 Giac [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x^2} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]

input

integrate((-c^2*d*x^2+d)^2*(a+b*arcsin(c*x))^2/x^2,x, algorithm="giac")
 

output

integrate((c^2*d*x^2 - d)^2*(b*arcsin(c*x) + a)^2/x^2, x)
 

3.2.71.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^2}{x^2} \,d x \]

input

int(((a + b*asin(c*x))^2*(d - c^2*d*x^2)^2)/x^2,x)
 

output

int(((a + b*asin(c*x))^2*(d - c^2*d*x^2)^2)/x^2, x)