\(\int \frac {(d-c^2 d x^2)^{3/2} (a+b \arccos (c x))^2}{x^2} \, dx\) [225]

Optimal result

Integrand size = 29, antiderivative size = 424 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x^2} \, dx=\frac {1}{4} b^2 c^2 d x \sqrt {d-c^2 d x^2}-\frac {5 b^2 c d \sqrt {d-c^2 d x^2} \arccos (c x)}{4 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d x^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{2 \sqrt {1-c^2 x^2}}+b c d \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2-\frac {i c d \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}-\frac {c d \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^3}{2 b \sqrt {1-c^2 x^2}}+\frac {2 b c d \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \log \left (1-e^{2 i \arccos (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {i b^2 c d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{\sqrt {1-c^2 x^2}} \] Output:

1/4*b^2*c^2*d*x*(-c^2*d*x^2+d)^(1/2)-5/4*b^2*c*d*(-c^2*d*x^2+d)^(1/2)*arcc 
os(c*x)/(-c^2*x^2+1)^(1/2)+3/2*b*c^3*d*x^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arcco 
s(c*x))/(-c^2*x^2+1)^(1/2)+b*c*d*(-c^2*x^2+1)^(1/2)*(-c^2*d*x^2+d)^(1/2)*( 
a+b*arccos(c*x))-3/2*c^2*d*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2-I*c* 
d*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2/(-c^2*x^2+1)^(1/2)-(-c^2*d*x^2+ 
d)^(3/2)*(a+b*arccos(c*x))^2/x-1/2*c*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c* 
x))^3/b/(-c^2*x^2+1)^(1/2)+2*b*c*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))* 
ln(1-(c*x+I*(-c^2*x^2+1)^(1/2))^2)/(-c^2*x^2+1)^(1/2)-I*b^2*c*d*(-c^2*d*x^ 
2+d)^(1/2)*polylog(2,(c*x+I*(-c^2*x^2+1)^(1/2))^2)/(-c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 3.87 (sec) , antiderivative size = 401, normalized size of antiderivative = 0.95 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x^2} \, dx=\frac {-12 a^2 d \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right ) \sqrt {d-c^2 d x^2}+36 a^2 c d^{3/2} x \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+24 a b d \sqrt {d-c^2 d x^2} \left (-2 \sqrt {1-c^2 x^2} \arccos (c x)+c x \arccos (c x)^2-2 c x \log (c x)\right )+8 i b^2 d \sqrt {d-c^2 d x^2} \left (i \arccos (c x) \left (3 \sqrt {1-c^2 x^2} \arccos (c x)-c x \arccos (c x) (3 i+\arccos (c x))+6 c x \log \left (1+e^{2 i \arccos (c x)}\right )\right )+3 c x \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )\right )+b^2 c d x \sqrt {d-c^2 d x^2} \left (4 \arccos (c x)^3-6 \arccos (c x) \cos (2 \arccos (c x))+\left (3-6 \arccos (c x)^2\right ) \sin (2 \arccos (c x))\right )-6 a b c d x \sqrt {d-c^2 d x^2} (\cos (2 \arccos (c x))+2 \arccos (c x) (-\arccos (c x)+\sin (2 \arccos (c x))))}{24 x \sqrt {1-c^2 x^2}} \] Input:

Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x])^2)/x^2,x]
 

Output:

(-12*a^2*d*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2)*Sqrt[d - c^2*d*x^2] + 36*a^2*c* 
d^(3/2)*x*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 
+ c^2*x^2))] + 24*a*b*d*Sqrt[d - c^2*d*x^2]*(-2*Sqrt[1 - c^2*x^2]*ArcCos[c 
*x] + c*x*ArcCos[c*x]^2 - 2*c*x*Log[c*x]) + (8*I)*b^2*d*Sqrt[d - c^2*d*x^2 
]*(I*ArcCos[c*x]*(3*Sqrt[1 - c^2*x^2]*ArcCos[c*x] - c*x*ArcCos[c*x]*(3*I + 
 ArcCos[c*x]) + 6*c*x*Log[1 + E^((2*I)*ArcCos[c*x])]) + 3*c*x*PolyLog[2, - 
E^((2*I)*ArcCos[c*x])]) + b^2*c*d*x*Sqrt[d - c^2*d*x^2]*(4*ArcCos[c*x]^3 - 
 6*ArcCos[c*x]*Cos[2*ArcCos[c*x]] + (3 - 6*ArcCos[c*x]^2)*Sin[2*ArcCos[c*x 
]]) - 6*a*b*c*d*x*Sqrt[d - c^2*d*x^2]*(Cos[2*ArcCos[c*x]] + 2*ArcCos[c*x]* 
(-ArcCos[c*x] + Sin[2*ArcCos[c*x]])))/(24*x*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 1.83 (sec) , antiderivative size = 369, normalized size of antiderivative = 0.87, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {5201, 5157, 5139, 262, 223, 5153, 5189, 211, 223, 5137, 3042, 4202, 2620, 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.

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

\(\Big \downarrow \) 5201

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

\(\Big \downarrow \) 5157

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5153

\(\displaystyle -\frac {2 b c d \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right ) (a+b \arccos (c x))}{x}dx}{\sqrt {1-c^2 x^2}}-3 c^2 d \left (\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}\)

\(\Big \downarrow \) 5189

\(\displaystyle -\frac {2 b c d \sqrt {d-c^2 d x^2} \left (\int \frac {a+b \arccos (c x)}{x}dx+\frac {1}{2} b c \int \sqrt {1-c^2 x^2}dx+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))\right )}{\sqrt {1-c^2 x^2}}-3 c^2 d \left (\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}\)

\(\Big \downarrow \) 211

\(\displaystyle -\frac {2 b c d \sqrt {d-c^2 d x^2} \left (\int \frac {a+b \arccos (c x)}{x}dx+\frac {1}{2} b c \left (\frac {1}{2} \int \frac {1}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))\right )}{\sqrt {1-c^2 x^2}}-3 c^2 d \left (\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}\)

\(\Big \downarrow \) 223

\(\displaystyle -\frac {2 b c d \sqrt {d-c^2 d x^2} \left (\int \frac {a+b \arccos (c x)}{x}dx+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )}{\sqrt {1-c^2 x^2}}-3 c^2 d \left (\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}\)

\(\Big \downarrow \) 5137

\(\displaystyle -\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\int \frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c x}d\arccos (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )}{\sqrt {1-c^2 x^2}}-3 c^2 d \left (\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\int (a+b \arccos (c x)) \tan (\arccos (c x))d\arccos (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )}{\sqrt {1-c^2 x^2}}-3 c^2 d \left (\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}\)

\(\Big \downarrow \) 4202

\(\displaystyle -\frac {2 b c d \sqrt {d-c^2 d x^2} \left (2 i \int \frac {e^{2 i \arccos (c x)} (a+b \arccos (c x))}{1+e^{2 i \arccos (c x)}}d\arccos (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))-\frac {i (a+b \arccos (c x))^2}{2 b}+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )}{\sqrt {1-c^2 x^2}}-3 c^2 d \left (\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}\)

\(\Big \downarrow \) 2620

\(\displaystyle -\frac {2 b c d \sqrt {d-c^2 d x^2} \left (2 i \left (\frac {1}{2} i b \int \log \left (1+e^{2 i \arccos (c x)}\right )d\arccos (c x)-\frac {1}{2} i \log \left (1+e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))-\frac {i (a+b \arccos (c x))^2}{2 b}+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )}{\sqrt {1-c^2 x^2}}-3 c^2 d \left (\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {2 b c d \sqrt {d-c^2 d x^2} \left (2 i \left (\frac {1}{4} b \int e^{-2 i \arccos (c x)} \log \left (1+e^{2 i \arccos (c x)}\right )de^{2 i \arccos (c x)}-\frac {1}{2} i \log \left (1+e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))-\frac {i (a+b \arccos (c x))^2}{2 b}+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )}{\sqrt {1-c^2 x^2}}-3 c^2 d \left (\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {2 b c d \sqrt {d-c^2 d x^2} \left (\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))+2 i \left (-\frac {1}{2} i \log \left (1+e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )\right )-\frac {i (a+b \arccos (c x))^2}{2 b}+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )}{\sqrt {1-c^2 x^2}}-3 c^2 d \left (\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}\)

Input:

Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x])^2)/x^2,x]
 

Output:

-(((d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x])^2)/x) - 3*c^2*d*((x*Sqrt[d - 
c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/2 - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c 
*x])^3)/(6*b*c*Sqrt[1 - c^2*x^2]) + (b*c*Sqrt[d - c^2*d*x^2]*((x^2*(a + b* 
ArcCos[c*x]))/2 + (b*c*(-1/2*(x*Sqrt[1 - c^2*x^2])/c^2 + ArcSin[c*x]/(2*c^ 
3)))/2))/Sqrt[1 - c^2*x^2]) - (2*b*c*d*Sqrt[d - c^2*d*x^2]*(((1 - c^2*x^2) 
*(a + b*ArcCos[c*x]))/2 - ((I/2)*(a + b*ArcCos[c*x])^2)/b + (b*c*((x*Sqrt[ 
1 - c^2*x^2])/2 + ArcSin[c*x]/(2*c)))/2 + (2*I)*((-1/2*I)*(a + b*ArcCos[c* 
x])*Log[1 + E^((2*I)*ArcCos[c*x])] - (b*PolyLog[2, -E^((2*I)*ArcCos[c*x])] 
)/4)))/Sqrt[1 - c^2*x^2]
 

Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

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 4202

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I 
*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I   Int[(c + d*x)^m*(E^(2*I*( 
e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt 
Q[m, 0]
 

rule 5137

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> -Subst[Int[ 
(a + b*x)^n*Tan[x], x], x, ArcCos[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0 
]
 

rule 5139

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
:> Simp[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^n/(d*(m + 1))), x] + Simp[b*c*(n 
/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2 
*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
 

rule 5153

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[(-(b*c*(n + 1))^(-1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2] 
]*(a + b*ArcCos[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^ 
2*d + e, 0] && NeQ[n, -1]
 

rule 5157

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcCos[c*x])^n/2), x] + (Simp[(1/2 
)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[(a + b*ArcCos[c*x])^n/Sqrt[ 
1 - c^2*x^2], x], x] + Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2 
]]   Int[x*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x 
] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
 

rule 5189

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.))/(x_), 
x_Symbol] :> Simp[(d + e*x^2)^p*((a + b*ArcCos[c*x])/(2*p)), x] + (Simp[d 
 Int[(d + e*x^2)^(p - 1)*((a + b*ArcCos[c*x])/x), x], x] + Simp[b*c*(d^p/(2 
*p))   Int[(1 - c^2*x^2)^(p - 1/2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && 
 EqQ[c^2*d + e, 0] && IGtQ[p, 0]
 

rule 5201

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcC 
os[c*x])^n/(f*(m + 1))), x] + (-Simp[2*e*(p/(f^2*(m + 1)))   Int[(f*x)^(m + 
 2)*(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x], x] + Simp[b*c*(n/(f*(m + 
 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[(f*x)^(m + 1)*(1 - c^2*x^2) 
^(p - 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f} 
, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
 

Maple [A] (verified)

Time = 0.66 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.11

method	result	size
default	\(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{d x}-a^{2} c^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}-\frac {3 a^{2} c^{2} d x \sqrt {-c^{2} d \,x^{2}+d}}{2}-\frac {3 a^{2} c^{2} d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (-2 \arccos \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-2 c^{3} x^{3} \arccos \left (c x \right )+2 \arccos \left (c x \right )^{3} c x +4 i \arccos \left (c x \right )^{2} x c +c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}-8 \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) x c +4 i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) x c -4 \arccos \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}+c x \arccos \left (c x \right )\right ) d}{4 \left (c^{2} x^{2}-1\right ) x}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (-4 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{2} x^{2}-2 c^{3} x^{3}+6 \arccos \left (c x \right )^{2} c x +8 i \arccos \left (c x \right ) x c -8 \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) x c -8 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+c x \right ) d}{4 \left (c^{2} x^{2}-1\right ) x}\)	\(472\)
parts	\(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{d x}-a^{2} c^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}-\frac {3 a^{2} c^{2} d x \sqrt {-c^{2} d \,x^{2}+d}}{2}-\frac {3 a^{2} c^{2} d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (-2 \arccos \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-2 c^{3} x^{3} \arccos \left (c x \right )+2 \arccos \left (c x \right )^{3} c x +4 i \arccos \left (c x \right )^{2} x c +c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}-8 \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) x c +4 i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) x c -4 \arccos \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}+c x \arccos \left (c x \right )\right ) d}{4 \left (c^{2} x^{2}-1\right ) x}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (-4 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{2} x^{2}-2 c^{3} x^{3}+6 \arccos \left (c x \right )^{2} c x +8 i \arccos \left (c x \right ) x c -8 \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) x c -8 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+c x \right ) d}{4 \left (c^{2} x^{2}-1\right ) x}\)	\(472\)

Input:

int((-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))^2/x^2,x,method=_RETURNVERBOSE)
 

Output:

-a^2/d/x*(-c^2*d*x^2+d)^(5/2)-a^2*c^2*x*(-c^2*d*x^2+d)^(3/2)-3/2*a^2*c^2*d 
*x*(-c^2*d*x^2+d)^(1/2)-3/2*a^2*c^2*d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2) 
*x/(-c^2*d*x^2+d)^(1/2))-1/4*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2) 
/(c^2*x^2-1)/x*(-2*arccos(c*x)^2*(-c^2*x^2+1)^(1/2)*x^2*c^2-2*c^3*x^3*arcc 
os(c*x)+2*arccos(c*x)^3*c*x+4*I*arccos(c*x)^2*x*c+c^2*x^2*(-c^2*x^2+1)^(1/ 
2)-8*arccos(c*x)*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))^2)*x*c+4*I*polylog(2,-(c* 
x+I*(-c^2*x^2+1)^(1/2))^2)*x*c-4*arccos(c*x)^2*(-c^2*x^2+1)^(1/2)+c*x*arcc 
os(c*x))*d-1/4*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)/x 
*(-4*(-c^2*x^2+1)^(1/2)*arccos(c*x)*c^2*x^2-2*c^3*x^3+6*arccos(c*x)^2*c*x+ 
8*I*arccos(c*x)*x*c-8*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))^2)*x*c-8*arccos(c*x) 
*(-c^2*x^2+1)^(1/2)+c*x)*d
 

Fricas [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x^2} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \] Input:

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))^2/x^2,x, algorithm="frica 
s")
 

Output:

integral(-(a^2*c^2*d*x^2 - a^2*d + (b^2*c^2*d*x^2 - b^2*d)*arccos(c*x)^2 + 
 2*(a*b*c^2*d*x^2 - a*b*d)*arccos(c*x))*sqrt(-c^2*d*x^2 + d)/x^2, x)
 

Sympy [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x^2} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}{x^{2}}\, dx \] Input:

integrate((-c**2*d*x**2+d)**(3/2)*(a+b*acos(c*x))**2/x**2,x)
 

Output:

Integral((-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*acos(c*x))**2/x**2, x)
 

Maxima [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x^2} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \] Input:

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))^2/x^2,x, algorithm="maxim 
a")
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))^2/x^2,x, algorithm="giac" 
)
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^2} \,d x \] Input:

int(((a + b*acos(c*x))^2*(d - c^2*d*x^2)^(3/2))/x^2,x)
 

Output:

int(((a + b*acos(c*x))^2*(d - c^2*d*x^2)^(3/2))/x^2, x)
 

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x^2} \, dx=\frac {\sqrt {d}\, d \left (2 \mathit {acos} \left (c x \right )^{3} b^{2} c x +6 \mathit {acos} \left (c x \right )^{2} a b c x -9 \mathit {asin} \left (c x \right ) a^{2} c x -3 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-6 \sqrt {-c^{2} x^{2}+1}\, a^{2}+12 \left (\int \frac {\mathit {acos} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, x^{2}}d x \right ) a b x +6 \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, x^{2}}d x \right ) b^{2} x -12 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )d x \right ) a b \,c^{2} x -6 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )^{2}d x \right ) b^{2} c^{2} x \right )}{6 x} \] Input:

int((-c^2*d*x^2+d)^(3/2)*(a+b*acos(c*x))^2/x^2,x)
 

Output:

(sqrt(d)*d*(2*acos(c*x)**3*b**2*c*x + 6*acos(c*x)**2*a*b*c*x - 9*asin(c*x) 
*a**2*c*x - 3*sqrt( - c**2*x**2 + 1)*a**2*c**2*x**2 - 6*sqrt( - c**2*x**2 
+ 1)*a**2 + 12*int(acos(c*x)/(sqrt( - c**2*x**2 + 1)*x**2),x)*a*b*x + 6*in 
t(acos(c*x)**2/(sqrt( - c**2*x**2 + 1)*x**2),x)*b**2*x - 12*int(sqrt( - c* 
*2*x**2 + 1)*acos(c*x),x)*a*b*c**2*x - 6*int(sqrt( - c**2*x**2 + 1)*acos(c 
*x)**2,x)*b**2*c**2*x))/(6*x)