\(\int \frac {(d-c^2 d x^2)^{3/2} (a+b \arccos (c x))^2}{x^4} \, dx\) [227]

Optimal result

Integrand size = 29, antiderivative size = 400 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x^4} \, dx=-\frac {b^2 c^2 d \sqrt {d-c^2 d x^2}}{3 x}-\frac {b^2 c^3 d \sqrt {d-c^2 d x^2} \arccos (c x)}{3 \sqrt {1-c^2 x^2}}-\frac {b c d \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{3 x^2}+\frac {c^2 d \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{x}+\frac {4 i c^3 d \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{3 x^3}+\frac {c^3 d \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^3}{3 b \sqrt {1-c^2 x^2}}-\frac {8 b c^3 d \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \log \left (1-e^{2 i \arccos (c x)}\right )}{3 \sqrt {1-c^2 x^2}}+\frac {4 i b^2 c^3 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{3 \sqrt {1-c^2 x^2}} \] Output:

-1/3*b^2*c^2*d*(-c^2*d*x^2+d)^(1/2)/x-1/3*b^2*c^3*d*(-c^2*d*x^2+d)^(1/2)*a 
rccos(c*x)/(-c^2*x^2+1)^(1/2)-1/3*b*c*d*(-c^2*x^2+1)^(1/2)*(-c^2*d*x^2+d)^ 
(1/2)*(a+b*arccos(c*x))/x^2+c^2*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2 
/x+4/3*I*c^3*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2/(-c^2*x^2+1)^(1/2) 
-1/3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))^2/x^3+1/3*c^3*d*(-c^2*d*x^2+d) 
^(1/2)*(a+b*arccos(c*x))^3/b/(-c^2*x^2+1)^(1/2)-8/3*b*c^3*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)+4/3*I*b^2*c^3*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 = 1.63 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.23 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x^4} \, dx=\frac {a b c d x \sqrt {d-c^2 d x^2}-a^2 d \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}+4 a^2 c^2 d x^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}-b^2 c^2 d x^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}+b d \sqrt {d-c^2 d x^2} \left (-3 a c^3 x^3+b \left (-4 i c^3 x^3-\sqrt {1-c^2 x^2}+4 c^2 x^2 \sqrt {1-c^2 x^2}\right )\right ) \arccos (c x)^2-b^2 c^3 d x^3 \sqrt {d-c^2 d x^2} \arccos (c x)^3-3 a^2 c^3 d^{3/2} x^3 \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+b d \sqrt {d-c^2 d x^2} \arccos (c x) \left (b c x+2 a \sqrt {1-c^2 x^2} \left (-1+4 c^2 x^2\right )+8 b c^3 x^3 \log \left (1+e^{2 i \arccos (c x)}\right )\right )+8 a b c^3 d x^3 \sqrt {d-c^2 d x^2} \log (c x)-4 i b^2 c^3 d x^3 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{3 x^3 \sqrt {1-c^2 x^2}} \] Input:

Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x])^2)/x^4,x]
 

Output:

(a*b*c*d*x*Sqrt[d - c^2*d*x^2] - a^2*d*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^ 
2] + 4*a^2*c^2*d*x^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2] - b^2*c^2*d*x^2 
*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2] + b*d*Sqrt[d - c^2*d*x^2]*(-3*a*c^3 
*x^3 + b*((-4*I)*c^3*x^3 - Sqrt[1 - c^2*x^2] + 4*c^2*x^2*Sqrt[1 - c^2*x^2] 
))*ArcCos[c*x]^2 - b^2*c^3*d*x^3*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^3 - 3*a^2 
*c^3*d^(3/2)*x^3*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[ 
d]*(-1 + c^2*x^2))] + b*d*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*(b*c*x + 2*a*Sqr 
t[1 - c^2*x^2]*(-1 + 4*c^2*x^2) + 8*b*c^3*x^3*Log[1 + E^((2*I)*ArcCos[c*x] 
)]) + 8*a*b*c^3*d*x^3*Sqrt[d - c^2*d*x^2]*Log[c*x] - (4*I)*b^2*c^3*d*x^3*S 
qrt[d - c^2*d*x^2]*PolyLog[2, -E^((2*I)*ArcCos[c*x])])/(3*x^3*Sqrt[1 - c^2 
*x^2])
 

Rubi [A] (verified)

Time = 3.19 (sec) , antiderivative size = 391, normalized size of antiderivative = 0.98, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.621, Rules used = {5201, 5191, 247, 223, 5137, 3042, 4202, 2620, 2715, 2838, 5197, 5137, 3042, 4202, 2620, 2715, 2838, 5153}

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.

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

\(\Big \downarrow \) 5201

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

\(\Big \downarrow \) 5191

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

\(\Big \downarrow \) 247

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5137

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 5197

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

\(\Big \downarrow \) 5137

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 5153

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

Input:

Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x])^2)/x^4,x]
 

Output:

-1/3*((d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x])^2)/x^3 - (2*b*c*d*Sqrt[d - 
 c^2*d*x^2]*(-1/2*((1 - c^2*x^2)*(a + b*ArcCos[c*x]))/x^2 - (b*c*(-(Sqrt[1 
 - c^2*x^2]/x) - c*ArcSin[c*x]))/2 + c^2*(((I/2)*(a + b*ArcCos[c*x])^2)/b 
- (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))))/(3*Sqrt[1 - c^2*x^2]) - c^2*d*(- 
((Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/x) + (c*Sqrt[d - c^2*d*x^2]*( 
a + b*ArcCos[c*x])^3)/(3*b*Sqrt[1 - c^2*x^2]) + (2*b*c*Sqrt[d - c^2*d*x^2] 
*(((I/2)*(a + b*ArcCos[c*x])^2)/b - (2*I)*((-1/2*I)*(a + b*ArcCos[c*x])*Lo 
g[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 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 247

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1)))   Int[ 
(c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 
0] && LtQ[m, -1] &&  !ILtQ[(m + 2*p + 3)/2, 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 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 5191

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

rule 5197

Int[((a_.) + ArcCos[(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*ArcC 
os[c*x])^n/(f*(m + 1))), x] + (Simp[b*c*(n/(f*(m + 1)))*Simp[Sqrt[d + e*x^2 
]/Sqrt[1 - c^2*x^2]]   Int[(f*x)^(m + 1)*(a + b*ArcCos[c*x])^(n - 1), x], x 
] + Simp[(c^2/(f^2*(m + 1)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[ 
(f*x)^(m + 2)*((a + b*ArcCos[c*x])^n/Sqrt[1 - c^2*x^2]), x], x]) /; FreeQ[{ 
a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1]
 

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 [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2116 vs. \(2 (374 ) = 748\).

Time = 0.71 (sec) , antiderivative size = 2117, normalized size of antiderivative = 5.29

method	result	size
default	\(\text {Expression too large to display}\)	\(2117\)
parts	\(\text {Expression too large to display}\)	\(2117\)

Input:

int((-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))^2/x^4,x,method=_RETURNVERBOSE)
 

Output:

73/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x/(c^2*x^2-1)*a 
rccos(c*x)^2*c^4-14/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1 
)/x/(c^2*x^2-1)*arccos(c*x)^2*c^2+4/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4 
*x^4-9*c^2*x^2+1)*x^3/(c^2*x^2-1)*(-c^2*x^2+1)*c^6-8*b^2*(-d*(c^2*x^2-1))^ 
(1/2)*(-c^2*x^2+1)^(1/2)*d*c^3/(3*c^2*x^2-3)*arccos(c*x)*ln(1+(c*x+I*(-c^2 
*x^2+1)^(1/2))^2)+8*I*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*d*c^3/ 
(3*c^2*x^2-3)*arccos(c*x)^2+32*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9* 
c^2*x^2+1)*x^5/(c^2*x^2-1)*arccos(c*x)^2*c^8+3*b^2*(-d*(c^2*x^2-1))^(1/2)* 
d/(24*c^4*x^4-9*c^2*x^2+1)/(c^2*x^2-1)*arccos(c*x)*(-c^2*x^2+1)^(1/2)*c^3- 
52*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^3/(c^2*x^2-1)*a 
rccos(c*x)^2*c^6-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1) 
/x^2/(c^2*x^2-1)*arccos(c*x)*(-c^2*x^2+1)^(1/2)*c-16/3*I*b^2*(-d*(c^2*x^2- 
1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^5/(c^2*x^2-1)*arccos(c*x)*c^8+8*I*b 
^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^4/(c^2*x^2-1)*(-c^2 
*x^2+1)^(1/2)*c^7+20/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^ 
2+1)*x^3/(c^2*x^2-1)*arccos(c*x)*c^6-3*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24* 
c^4*x^4-9*c^2*x^2+1)*x^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^5-4/3*I*b^2*(-d* 
(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x/(c^2*x^2-1)*arccos(c*x)*c^ 
4-4/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)/(c^2*x^2-1)* 
arccos(c*x)^2*(-c^2*x^2+1)^(1/2)*c^3+2/3*a^2*c^2/d/x*(-c^2*d*x^2+d)^(5/...
 

Fricas [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x^4} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \] Input:

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))^2/x^4,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^4, x)
 

Sympy [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x^4} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}{x^{4}}\, dx \] Input:

integrate((-c**2*d*x**2+d)**(3/2)*(a+b*acos(c*x))**2/x**4,x)
 

Output:

Integral((-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*acos(c*x))**2/x**4, x)
 

Maxima [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x^4} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \] Input:

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))^2/x^4,x, algorithm="maxim 
a")
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x^4} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))^2/x^4,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. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^4} \,d x \] Input:

int(((a + b*acos(c*x))^2*(d - c^2*d*x^2)^(3/2))/x^4,x)
 

Output:

int(((a + b*acos(c*x))^2*(d - c^2*d*x^2)^(3/2))/x^4, x)
 

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x^4} \, dx=\frac {\sqrt {d}\, d \left (-\mathit {acos} \left (c x \right )^{3} b^{2} c^{3} x^{3}-3 \mathit {acos} \left (c x \right )^{2} a b \,c^{3} x^{3}+3 \mathit {asin} \left (c x \right ) a^{2} c^{3} x^{3}+4 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}\, a^{2}-6 \left (\int \frac {\mathit {acos} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, x^{2}}d x \right ) a b \,c^{2} x^{3}-3 \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, x^{2}}d x \right ) b^{2} c^{2} x^{3}+6 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )}{x^{4}}d x \right ) a b \,x^{3}+3 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )^{2}}{x^{4}}d x \right ) b^{2} x^{3}\right )}{3 x^{3}} \] Input:

int((-c^2*d*x^2+d)^(3/2)*(a+b*acos(c*x))^2/x^4,x)
 

Output:

(sqrt(d)*d*( - acos(c*x)**3*b**2*c**3*x**3 - 3*acos(c*x)**2*a*b*c**3*x**3 
+ 3*asin(c*x)*a**2*c**3*x**3 + 4*sqrt( - c**2*x**2 + 1)*a**2*c**2*x**2 - s 
qrt( - c**2*x**2 + 1)*a**2 - 6*int(acos(c*x)/(sqrt( - c**2*x**2 + 1)*x**2) 
,x)*a*b*c**2*x**3 - 3*int(acos(c*x)**2/(sqrt( - c**2*x**2 + 1)*x**2),x)*b* 
*2*c**2*x**3 + 6*int((sqrt( - c**2*x**2 + 1)*acos(c*x))/x**4,x)*a*b*x**3 + 
 3*int((sqrt( - c**2*x**2 + 1)*acos(c*x)**2)/x**4,x)*b**2*x**3))/(3*x**3)