\(\int \frac {(d-c^2 d x^2)^{3/2} (a+b \arccos (c x))^2}{x} \, dx\) [224]

Optimal result

Integrand size = 29, antiderivative size = 545 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x} \, dx=-\frac {22}{9} b^2 d \sqrt {d-c^2 d x^2}-\frac {2 a b c d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {2}{27} b^2 d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}-\frac {2 b^2 c d x \sqrt {d-c^2 d x^2} \arccos (c x)}{\sqrt {1-c^2 x^2}}-\frac {2 b c d x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{9 \sqrt {1-c^2 x^2}}+d \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2-\frac {2 d \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2 \text {arctanh}\left (e^{i \arccos (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 i b d \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 i b d \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,-e^{i \arccos (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 b^2 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,e^{i \arccos (c x)}\right )}{\sqrt {1-c^2 x^2}} \] Output:

-22/9*b^2*d*(-c^2*d*x^2+d)^(1/2)-2*a*b*c*d*x*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^ 
2+1)^(1/2)-2/27*b^2*d*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)-2*b^2*c*d*x*(-c^2* 
d*x^2+d)^(1/2)*arccos(c*x)/(-c^2*x^2+1)^(1/2)-2/3*b*c*d*x*(-c^2*d*x^2+d)^( 
1/2)*(a+b*arccos(c*x))/(-c^2*x^2+1)^(1/2)+2/9*b*c^3*d*x^3*(-c^2*d*x^2+d)^( 
1/2)*(a+b*arccos(c*x))/(-c^2*x^2+1)^(1/2)+d*(-c^2*d*x^2+d)^(1/2)*(a+b*arcc 
os(c*x))^2+1/3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))^2-2*d*(-c^2*d*x^2+d) 
^(1/2)*(a+b*arccos(c*x))^2*arctanh(c*x+I*(-c^2*x^2+1)^(1/2))/(-c^2*x^2+1)^ 
(1/2)+2*I*b*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))*polylog(2,-c*x-I*(-c^ 
2*x^2+1)^(1/2))/(-c^2*x^2+1)^(1/2)-2*I*b*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arcco 
s(c*x))*polylog(2,c*x+I*(-c^2*x^2+1)^(1/2))/(-c^2*x^2+1)^(1/2)-2*b^2*d*(-c 
^2*d*x^2+d)^(1/2)*polylog(3,-c*x-I*(-c^2*x^2+1)^(1/2))/(-c^2*x^2+1)^(1/2)+ 
2*b^2*d*(-c^2*d*x^2+d)^(1/2)*polylog(3,c*x+I*(-c^2*x^2+1)^(1/2))/(-c^2*x^2 
+1)^(1/2)
 

Mathematica [A] (verified)

Time = 2.42 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.07 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x} \, dx=-\frac {1}{3} a^2 d \left (-4+c^2 x^2\right ) \sqrt {d-c^2 d x^2}-\frac {a b d \sqrt {d-c^2 d x^2} \left (-9 c x-12 \left (1-c^2 x^2\right )^{3/2} \arccos (c x)+\cos (3 \arccos (c x))\right )}{18 \sqrt {1-c^2 x^2}}-\frac {1}{54} b^2 d \sqrt {d-c^2 d x^2} \left (28-4 c^2 x^2+9 \arccos (c x)^2 (-1+\cos (2 \arccos (c x)))+\frac {3 \arccos (c x) (-9 c x+\cos (3 \arccos (c x)))}{\sqrt {1-c^2 x^2}}\right )+a^2 d^{3/2} \log (c x)-a^2 d^{3/2} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {2 a b d \sqrt {d-c^2 d x^2} \left (c x+\sqrt {1-c^2 x^2} \arccos (c x)-\arccos (c x) \log \left (1-i e^{i \arccos (c x)}\right )+\arccos (c x) \log \left (1+i e^{i \arccos (c x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )+i \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )\right )}{\sqrt {1-c^2 x^2}}+\frac {b^2 d \sqrt {d-c^2 d x^2} \left (-2 \sqrt {1-c^2 x^2}+2 c x \arccos (c x)+\sqrt {1-c^2 x^2} \arccos (c x)^2-\arccos (c x)^2 \log \left (1-i e^{i \arccos (c x)}\right )+\arccos (c x)^2 \log \left (1+i e^{i \arccos (c x)}\right )-2 i \arccos (c x) \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )+2 i \arccos (c x) \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )+2 \operatorname {PolyLog}\left (3,-i e^{i \arccos (c x)}\right )-2 \operatorname {PolyLog}\left (3,i e^{i \arccos (c x)}\right )\right )}{\sqrt {1-c^2 x^2}} \] Input:

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

Output:

-1/3*(a^2*d*(-4 + c^2*x^2)*Sqrt[d - c^2*d*x^2]) - (a*b*d*Sqrt[d - c^2*d*x^ 
2]*(-9*c*x - 12*(1 - c^2*x^2)^(3/2)*ArcCos[c*x] + Cos[3*ArcCos[c*x]]))/(18 
*Sqrt[1 - c^2*x^2]) - (b^2*d*Sqrt[d - c^2*d*x^2]*(28 - 4*c^2*x^2 + 9*ArcCo 
s[c*x]^2*(-1 + Cos[2*ArcCos[c*x]]) + (3*ArcCos[c*x]*(-9*c*x + Cos[3*ArcCos 
[c*x]]))/Sqrt[1 - c^2*x^2]))/54 + a^2*d^(3/2)*Log[c*x] - a^2*d^(3/2)*Log[d 
 + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (2*a*b*d*Sqrt[d - c^2*d*x^2]*(c*x + Sqrt 
[1 - c^2*x^2]*ArcCos[c*x] - ArcCos[c*x]*Log[1 - I*E^(I*ArcCos[c*x])] + Arc 
Cos[c*x]*Log[1 + I*E^(I*ArcCos[c*x])] - I*PolyLog[2, (-I)*E^(I*ArcCos[c*x] 
)] + I*PolyLog[2, I*E^(I*ArcCos[c*x])]))/Sqrt[1 - c^2*x^2] + (b^2*d*Sqrt[d 
 - c^2*d*x^2]*(-2*Sqrt[1 - c^2*x^2] + 2*c*x*ArcCos[c*x] + Sqrt[1 - c^2*x^2 
]*ArcCos[c*x]^2 - ArcCos[c*x]^2*Log[1 - I*E^(I*ArcCos[c*x])] + ArcCos[c*x] 
^2*Log[1 + I*E^(I*ArcCos[c*x])] - (2*I)*ArcCos[c*x]*PolyLog[2, (-I)*E^(I*A 
rcCos[c*x])] + (2*I)*ArcCos[c*x]*PolyLog[2, I*E^(I*ArcCos[c*x])] + 2*PolyL 
og[3, (-I)*E^(I*ArcCos[c*x])] - 2*PolyLog[3, I*E^(I*ArcCos[c*x])]))/Sqrt[1 
 - c^2*x^2]
 

Rubi [A] (verified)

Time = 1.93 (sec) , antiderivative size = 394, normalized size of antiderivative = 0.72, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {5203, 5155, 27, 353, 53, 2009, 5199, 2009, 5219, 3042, 4669, 3011, 2720, 7143}

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.

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

\(\Big \downarrow \) 5203

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

\(\Big \downarrow \) 5155

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 353

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

\(\Big \downarrow \) 53

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5199

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5219

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

\(\Big \downarrow \) 3042

\(\displaystyle d \left (-\frac {\sqrt {d-c^2 d x^2} \int (a+b \arccos (c x))^2 \csc \left (\arccos (c x)+\frac {\pi }{2}\right )d\arccos (c x)}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2+\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))+\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

((d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x])^2)/3 + (2*b*c*d*Sqrt[d - c^2*d* 
x^2]*((b*c*((-4*Sqrt[1 - c^2*x^2])/c^2 - (2*(1 - c^2*x^2)^(3/2))/(3*c^2))) 
/6 + x*(a + b*ArcCos[c*x]) - (c^2*x^3*(a + b*ArcCos[c*x]))/3))/(3*Sqrt[1 - 
 c^2*x^2]) + d*(Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2 + (2*b*c*Sqrt[d 
- c^2*d*x^2]*(a*x - (b*Sqrt[1 - c^2*x^2])/c + b*x*ArcCos[c*x]))/Sqrt[1 - c 
^2*x^2] - (Sqrt[d - c^2*d*x^2]*((-2*I)*(a + b*ArcCos[c*x])^2*ArcTan[E^(I*A 
rcCos[c*x])] + 2*b*(I*(a + b*ArcCos[c*x])*PolyLog[2, (-I)*E^(I*ArcCos[c*x] 
)] - b*PolyLog[3, (-I)*E^(I*ArcCos[c*x])]) - 2*b*(I*(a + b*ArcCos[c*x])*Po 
lyLog[2, I*E^(I*ArcCos[c*x])] - b*PolyLog[3, I*E^(I*ArcCos[c*x])])))/Sqrt[ 
1 - c^2*x^2])
 

Defintions of rubi rules used

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 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

rule 353

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

rule 2009

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

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

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

rule 4669

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol 
] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si 
mp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], 
 x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x 
))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 5155

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbo 
l] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcCos[c*x])   u, x 
] + Simp[b*c   Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; Fr 
eeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
 

rule 5199

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 + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt[1 
 - c^2*x^2]]   Int[(f*x)^m*((a + b*ArcCos[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*ArcCos[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 5203

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 + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1))   Int[(f*x) 
^m*(d + e*x^2)^(p - 1)*(a + b*ArcCos[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*ArcCos[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 5219

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

rule 7143

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

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 843, normalized size of antiderivative = 1.55

method	result	size
default	\(-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}-3 i \sqrt {-c^{2} x^{2}+1}\, x c +1\right ) \left (6 i \arccos \left (c x \right ) b^{2}+9 \arccos \left (c x \right )^{2} b^{2}+6 i a b +18 \arccos \left (c x \right ) a b +9 a^{2}-2 b^{2}\right ) d}{216 \left (c^{2} x^{2}-1\right )}+\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )^{2} b^{2}+2 \arccos \left (c x \right ) a b +a^{2}-2 b^{2}+2 i \arccos \left (c x \right ) b^{2}+2 i a b \right ) d}{8 \left (c^{2} x^{2}-1\right )}+\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )^{2} b^{2}+2 \arccos \left (c x \right ) a b +a^{2}-2 b^{2}-2 i \arccos \left (c x \right ) b^{2}-2 i a b \right ) d}{8 \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+4 c^{4} x^{4}+3 i \sqrt {-c^{2} x^{2}+1}\, x c -5 c^{2} x^{2}+1\right ) \left (-6 i \arccos \left (c x \right ) b^{2}+9 \arccos \left (c x \right )^{2} b^{2}-6 i a b +18 \arccos \left (c x \right ) a b +9 a^{2}-2 b^{2}\right ) d}{216 \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arccos \left (c x \right )^{2} \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right ) b^{2}-i \arccos \left (c x \right )^{2} \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right ) b^{2}+2 i \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right ) a b -2 i \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right ) a b +2 \arccos \left (c x \right ) \operatorname {polylog}\left (2, i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right ) b^{2}-2 \arccos \left (c x \right ) \operatorname {polylog}\left (2, -i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right ) b^{2}+2 i \operatorname {polylog}\left (3, i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right ) b^{2}-2 i \operatorname {polylog}\left (3, -i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right ) b^{2}+2 \operatorname {polylog}\left (2, i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right ) a b -2 \operatorname {polylog}\left (2, -i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right ) a b +2 a^{2} \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right ) d}{c^{2} x^{2}-1}\)	\(843\)

Input:

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

Output:

-1/216*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2+4*I*(-c^2*x^2+1)^(1/2)* 
x^3*c^3-3*I*(-c^2*x^2+1)^(1/2)*c*x+1)*(6*I*b^2*arccos(c*x)+9*arccos(c*x)^2 
*b^2+6*I*a*b+18*arccos(c*x)*a*b+9*a^2-2*b^2)*d/(c^2*x^2-1)+5/8*(-d*(c^2*x^ 
2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*c*x+c^2*x^2-1)*(arccos(c*x)^2*b^2+2*arcc 
os(c*x)*a*b+a^2-2*b^2+2*I*arccos(c*x)*b^2+2*I*a*b)*d/(c^2*x^2-1)+5/8*(-d*( 
c^2*x^2-1))^(1/2)*(-I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(arccos(c*x)^2*b^2 
+2*arccos(c*x)*a*b+a^2-2*b^2-2*I*b^2*arccos(c*x)-2*I*a*b)*d/(c^2*x^2-1)-1/ 
216*(-d*(c^2*x^2-1))^(1/2)*(-4*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+4*c^4*x^4+3*I* 
(-c^2*x^2+1)^(1/2)*c*x-5*c^2*x^2+1)*(-6*I*b^2*arccos(c*x)+9*arccos(c*x)^2* 
b^2-6*I*a*b+18*arccos(c*x)*a*b+9*a^2-2*b^2)*d/(c^2*x^2-1)-I*(-d*(c^2*x^2-1 
))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*(I*arccos(c*x)^2*ln(1-I*(c*x+I*(-c 
^2*x^2+1)^(1/2)))*b^2-I*arccos(c*x)^2*ln(1+I*(c*x+I*(-c^2*x^2+1)^(1/2)))*b 
^2+2*I*arccos(c*x)*ln(1-I*(c*x+I*(-c^2*x^2+1)^(1/2)))*a*b-2*I*arccos(c*x)* 
ln(1+I*(c*x+I*(-c^2*x^2+1)^(1/2)))*a*b+2*arccos(c*x)*polylog(2,I*(c*x+I*(- 
c^2*x^2+1)^(1/2)))*b^2-2*arccos(c*x)*polylog(2,-I*(c*x+I*(-c^2*x^2+1)^(1/2 
)))*b^2+2*I*polylog(3,I*(c*x+I*(-c^2*x^2+1)^(1/2)))*b^2-2*I*polylog(3,-I*( 
c*x+I*(-c^2*x^2+1)^(1/2)))*b^2+2*polylog(2,I*(c*x+I*(-c^2*x^2+1)^(1/2)))*a 
*b-2*polylog(2,-I*(c*x+I*(-c^2*x^2+1)^(1/2)))*a*b+2*a^2*arctan(c*x+I*(-c^2 
*x^2+1)^(1/2)))*d
 

Fricas [F]

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

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

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, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/3*(3*d^(3/2)*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x)) - 
(-c^2*d*x^2 + d)^(3/2) - 3*sqrt(-c^2*d*x^2 + d)*d)*a^2 - sqrt(d)*integrate 
(((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, x)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(d)*d*( - sqrt( - c**2*x**2 + 1)*a**2*c**2*x**2 + 4*sqrt( - c**2*x**2 
 + 1)*a**2 + 6*int((sqrt( - c**2*x**2 + 1)*acos(c*x))/x,x)*a*b + 3*int((sq 
rt( - c**2*x**2 + 1)*acos(c*x)**2)/x,x)*b**2 - 6*int(sqrt( - c**2*x**2 + 1 
)*acos(c*x)*x,x)*a*b*c**2 - 3*int(sqrt( - c**2*x**2 + 1)*acos(c*x)**2*x,x) 
*b**2*c**2 + 3*log(tan(asin(c*x)/2))*a**2 - 4*a**2))/3