\(\int \frac {(a+b \arccos (c x))^2}{x^2 (d-c^2 d x^2)^{3/2}} \, dx\) [253]

Optimal result

Integrand size = 29, antiderivative size = 333 \[ \int \frac {(a+b \arccos (c x))^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {(a+b \arccos (c x))^2}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x (a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 i c \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \text {arctanh}\left (e^{2 i \arccos (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {4 b c \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \log \left (1+e^{2 i \arccos (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {i b^2 c \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {i b^2 c \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{d \sqrt {d-c^2 d x^2}} \] Output:

-(a+b*arccos(c*x))^2/d/x/(-c^2*d*x^2+d)^(1/2)+2*c^2*x*(a+b*arccos(c*x))^2/ 
d/(-c^2*d*x^2+d)^(1/2)-2*I*c*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))^2/d/(-c^ 
2*d*x^2+d)^(1/2)-4*b*c*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))*arctanh((c*x+I 
*(-c^2*x^2+1)^(1/2))^2)/d/(-c^2*d*x^2+d)^(1/2)+4*b*c*(-c^2*x^2+1)^(1/2)*(a 
+b*arccos(c*x))*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))^2)/d/(-c^2*d*x^2+d)^(1/2)- 
I*b^2*c*(-c^2*x^2+1)^(1/2)*polylog(2,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)/d/(-c^ 
2*d*x^2+d)^(1/2)-I*b^2*c*(-c^2*x^2+1)^(1/2)*polylog(2,(c*x+I*(-c^2*x^2+1)^ 
(1/2))^2)/d/(-c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 1.21 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b \arccos (c x))^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-a^2+2 a^2 c^2 x^2-2 a b \arccos (c x)+4 a b c^2 x^2 \arccos (c x)-b^2 \arccos (c x)^2+2 b^2 c^2 x^2 \arccos (c x)^2+2 i b^2 c x \sqrt {1-c^2 x^2} \arccos (c x)^2-2 b^2 c x \sqrt {1-c^2 x^2} \arccos (c x) \log \left (1-e^{2 i \arccos (c x)}\right )-2 b^2 c x \sqrt {1-c^2 x^2} \arccos (c x) \log \left (1+e^{2 i \arccos (c x)}\right )-2 a b c x \sqrt {1-c^2 x^2} \log (c x)-a b c x \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )+i b^2 c x \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )+i b^2 c x \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{d x \sqrt {d-c^2 d x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.89 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.83, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.552, Rules used = {5205, 5161, 5181, 3042, 25, 4200, 25, 2620, 2715, 2838, 5185, 4919, 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.

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

\(\Big \downarrow \) 5205

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

\(\Big \downarrow \) 5161

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

\(\Big \downarrow \) 5181

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4200

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 5185

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

\(\Big \downarrow \) 4919

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 4200

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))), x] 
, x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
 

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 4919

Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n   Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n 
, x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]
 

rule 5161

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x 
_Symbol] :> Simp[x*((a + b*ArcCos[c*x])^n/(d*Sqrt[d + e*x^2])), x] + Simp[b 
*c*(n/d)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]   Int[x*((a + b*ArcCos[c*x 
])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d 
 + e, 0] && GtQ[n, 0]
 

rule 5181

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

rule 5185

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

rule 5205

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 + 1)*((a + b 
*ArcCos[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)) 
)   Int[(f*x)^(m + 2)*(d + e*x^2)^p*(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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
 

Maple [A] (verified)

Time = 0.65 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.55

method	result	size
default	\(a^{2} \left (-\frac {1}{d x \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 c^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}\right )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x c +2 c^{2} x^{2}-1\right ) \arccos \left (c x \right )^{2}}{d^{2} x \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 i \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )+2 i \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+4 \arccos \left (c x \right )^{2}+\operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )\right ) c}{\left (c^{2} x^{2}-1\right ) d^{2}}\right )+2 a b \left (-\frac {4 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) c}{\left (c^{2} x^{2}-1\right ) d^{2}}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x c +2 c^{2} x^{2}-1\right ) \arccos \left (c x \right )}{d^{2} x \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{4}-1\right ) c}{\left (c^{2} x^{2}-1\right ) d^{2}}\right )\)	\(517\)
parts	\(a^{2} \left (-\frac {1}{d x \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 c^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}\right )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x c +2 c^{2} x^{2}-1\right ) \arccos \left (c x \right )^{2}}{d^{2} x \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 i \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )+2 i \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+4 \arccos \left (c x \right )^{2}+\operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )\right ) c}{\left (c^{2} x^{2}-1\right ) d^{2}}\right )+2 a b \left (-\frac {4 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) c}{\left (c^{2} x^{2}-1\right ) d^{2}}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x c +2 c^{2} x^{2}-1\right ) \arccos \left (c x \right )}{d^{2} x \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{4}-1\right ) c}{\left (c^{2} x^{2}-1\right ) d^{2}}\right )\)	\(517\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(-c^2*d*x^2 + d)*(b^2*arccos(c*x)^2 + 2*a*b*arccos(c*x) + a^2 
)/(c^4*d^2*x^6 - 2*c^2*d^2*x^4 + d^2*x^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 2*sqrt( - c**2*x**2 + 1)*int(acos(c*x)/(sqrt( - c**2*x**2 + 1)*c**2*x* 
*4 - sqrt( - c**2*x**2 + 1)*x**2),x)*a*b*x - sqrt( - c**2*x**2 + 1)*int(ac 
os(c*x)**2/(sqrt( - c**2*x**2 + 1)*c**2*x**4 - sqrt( - c**2*x**2 + 1)*x**2 
),x)*b**2*x + 2*a**2*c**2*x**2 - a**2)/(sqrt(d)*sqrt( - c**2*x**2 + 1)*d*x 
)