\(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \arccos (c x))^2}{x^3} \, dx\) [234]

Optimal result

Integrand size = 29, antiderivative size = 740 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{x^3} \, dx=\frac {40}{9} b^2 c^2 d^2 \sqrt {d-c^2 d x^2}+\frac {5 a b c^3 d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}+\frac {2}{27} b^2 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}+\frac {5 b^2 c^3 d^2 x \sqrt {d-c^2 d x^2} \arccos (c x)}{\sqrt {1-c^2 x^2}}-\frac {b c d^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{x \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{3 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{9 \sqrt {1-c^2 x^2}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2-\frac {5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{2 x^2}+\frac {5 c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2 \text {arctanh}\left (e^{i \arccos (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {b^2 c^2 d^2 \sqrt {d-c^2 d x^2} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{\sqrt {1-c^2 x^2}}-\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {5 b^2 c^2 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,-e^{i \arccos (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {5 b^2 c^2 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,e^{i \arccos (c x)}\right )}{\sqrt {1-c^2 x^2}} \] Output:

40/9*b^2*c^2*d^2*(-c^2*d*x^2+d)^(1/2)+5*a*b*c^3*d^2*x*(-c^2*d*x^2+d)^(1/2) 
/(-c^2*x^2+1)^(1/2)+2/27*b^2*c^2*d^2*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)+5*b 
^2*c^3*d^2*x*(-c^2*d*x^2+d)^(1/2)*arccos(c*x)/(-c^2*x^2+1)^(1/2)-b*c*d^2*( 
-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))/x/(-c^2*x^2+1)^(1/2)-1/3*b*c^3*d^2*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^5*d^2*x 
^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))/(-c^2*x^2+1)^(1/2)-5/2*c^2*d^2*( 
-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2-5/6*c^2*d*(-c^2*d*x^2+d)^(3/2)*(a+ 
b*arccos(c*x))^2-1/2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))^2/x^2+5*c^2*d^ 
2*(-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)-b^2*c^2*d^2*(-c^2*d*x^2+d)^(1/2)*arctanh((-c^2*x^2+1 
)^(1/2))/(-c^2*x^2+1)^(1/2)-5*I*b*c^2*d^2*(-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)+5*I*b*c^2*d 
^2*(-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)+5*b^2*c^2*d^2*(-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)-5*b^2*c^2*d^2*(-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 = 2.44 (sec) , antiderivative size = 988, normalized size of antiderivative = 1.34 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{x^3} \, dx=\frac {-27 a^2 d^3-99 a^2 c^2 d^3 x^2+244 b^2 c^2 d^3 x^2+144 a^2 c^4 d^3 x^4-248 b^2 c^4 d^3 x^4-18 a^2 c^6 d^3 x^6+4 b^2 c^6 d^3 x^6+54 a b c d^3 x \sqrt {1-c^2 x^2}-252 a b c^3 d^3 x^3 \sqrt {1-c^2 x^2}+12 a b c^5 d^3 x^5 \sqrt {1-c^2 x^2}-54 a b d^3 \arccos (c x)-198 a b c^2 d^3 x^2 \arccos (c x)+288 a b c^4 d^3 x^4 \arccos (c x)-36 a b c^6 d^3 x^6 \arccos (c x)+54 b^2 c d^3 x \sqrt {1-c^2 x^2} \arccos (c x)-252 b^2 c^3 d^3 x^3 \sqrt {1-c^2 x^2} \arccos (c x)+12 b^2 c^5 d^3 x^5 \sqrt {1-c^2 x^2} \arccos (c x)-27 b^2 d^3 \arccos (c x)^2-99 b^2 c^2 d^3 x^2 \arccos (c x)^2+144 b^2 c^4 d^3 x^4 \arccos (c x)^2-18 b^2 c^6 d^3 x^6 \arccos (c x)^2-54 b^2 c^2 d^3 x^2 \sqrt {1-c^2 x^2} \coth ^{-1}\left (\sqrt {1-c^2 x^2}\right )-54 i b^2 c^2 d^3 x^2 \sqrt {1-c^2 x^2} \arccos (c x)^2 \arctan \left (e^{i \arccos (c x)}\right )+270 a b c^2 d^3 x^2 \sqrt {1-c^2 x^2} \arccos (c x) \log \left (1-i e^{i \arccos (c x)}\right )+108 b^2 c^2 d^3 x^2 \sqrt {1-c^2 x^2} \arccos (c x)^2 \log \left (1-i e^{i \arccos (c x)}\right )-270 a b c^2 d^3 x^2 \sqrt {1-c^2 x^2} \arccos (c x) \log \left (1+i e^{i \arccos (c x)}\right )-108 b^2 c^2 d^3 x^2 \sqrt {1-c^2 x^2} \arccos (c x)^2 \log \left (1+i e^{i \arccos (c x)}\right )-135 a^2 c^2 d^{5/2} x^2 \sqrt {d-c^2 d x^2} \log (x)+135 a^2 c^2 d^{5/2} x^2 \sqrt {d-c^2 d x^2} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+270 i b c^2 d^3 x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )-270 i b c^2 d^3 x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )-270 b^2 c^2 d^3 x^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arccos (c x)}\right )+270 b^2 c^2 d^3 x^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arccos (c x)}\right )}{54 x^2 \sqrt {d-c^2 d x^2}} \] Input:

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x])^2)/x^3,x]
 

Output:

(-27*a^2*d^3 - 99*a^2*c^2*d^3*x^2 + 244*b^2*c^2*d^3*x^2 + 144*a^2*c^4*d^3* 
x^4 - 248*b^2*c^4*d^3*x^4 - 18*a^2*c^6*d^3*x^6 + 4*b^2*c^6*d^3*x^6 + 54*a* 
b*c*d^3*x*Sqrt[1 - c^2*x^2] - 252*a*b*c^3*d^3*x^3*Sqrt[1 - c^2*x^2] + 12*a 
*b*c^5*d^3*x^5*Sqrt[1 - c^2*x^2] - 54*a*b*d^3*ArcCos[c*x] - 198*a*b*c^2*d^ 
3*x^2*ArcCos[c*x] + 288*a*b*c^4*d^3*x^4*ArcCos[c*x] - 36*a*b*c^6*d^3*x^6*A 
rcCos[c*x] + 54*b^2*c*d^3*x*Sqrt[1 - c^2*x^2]*ArcCos[c*x] - 252*b^2*c^3*d^ 
3*x^3*Sqrt[1 - c^2*x^2]*ArcCos[c*x] + 12*b^2*c^5*d^3*x^5*Sqrt[1 - c^2*x^2] 
*ArcCos[c*x] - 27*b^2*d^3*ArcCos[c*x]^2 - 99*b^2*c^2*d^3*x^2*ArcCos[c*x]^2 
 + 144*b^2*c^4*d^3*x^4*ArcCos[c*x]^2 - 18*b^2*c^6*d^3*x^6*ArcCos[c*x]^2 - 
54*b^2*c^2*d^3*x^2*Sqrt[1 - c^2*x^2]*ArcCoth[Sqrt[1 - c^2*x^2]] - (54*I)*b 
^2*c^2*d^3*x^2*Sqrt[1 - c^2*x^2]*ArcCos[c*x]^2*ArcTan[E^(I*ArcCos[c*x])] + 
 270*a*b*c^2*d^3*x^2*Sqrt[1 - c^2*x^2]*ArcCos[c*x]*Log[1 - I*E^(I*ArcCos[c 
*x])] + 108*b^2*c^2*d^3*x^2*Sqrt[1 - c^2*x^2]*ArcCos[c*x]^2*Log[1 - I*E^(I 
*ArcCos[c*x])] - 270*a*b*c^2*d^3*x^2*Sqrt[1 - c^2*x^2]*ArcCos[c*x]*Log[1 + 
 I*E^(I*ArcCos[c*x])] - 108*b^2*c^2*d^3*x^2*Sqrt[1 - c^2*x^2]*ArcCos[c*x]^ 
2*Log[1 + I*E^(I*ArcCos[c*x])] - 135*a^2*c^2*d^(5/2)*x^2*Sqrt[d - c^2*d*x^ 
2]*Log[x] + 135*a^2*c^2*d^(5/2)*x^2*Sqrt[d - c^2*d*x^2]*Log[d + Sqrt[d]*Sq 
rt[d - c^2*d*x^2]] + (270*I)*b*c^2*d^3*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCos 
[c*x])*PolyLog[2, (-I)*E^(I*ArcCos[c*x])] - (270*I)*b*c^2*d^3*x^2*Sqrt[1 - 
 c^2*x^2]*(a + b*ArcCos[c*x])*PolyLog[2, I*E^(I*ArcCos[c*x])] - 270*b^2...
 

Rubi [A] (warning: unable to verify)

Time = 3.42 (sec) , antiderivative size = 575, normalized size of antiderivative = 0.78, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.759, Rules used = {5201, 5193, 27, 1578, 1192, 25, 1467, 2009, 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.

\(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{x^3} \, dx\)

\(\Big \downarrow \) 5201

\(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arccos (c x))}{x^2}dx}{\sqrt {1-c^2 x^2}}-\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{2 x^2}\)

\(\Big \downarrow \) 5193

\(\displaystyle -\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (b c \int -\frac {-c^4 x^4+6 c^2 x^2+3}{3 x \sqrt {1-c^2 x^2}}dx+\frac {1}{3} c^4 x^3 (a+b \arccos (c x))-2 c^2 x (a+b \arccos (c x))-\frac {a+b \arccos (c x)}{x}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{2 x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} b c \int \frac {-c^4 x^4+6 c^2 x^2+3}{x \sqrt {1-c^2 x^2}}dx+\frac {1}{3} c^4 x^3 (a+b \arccos (c x))-2 c^2 x (a+b \arccos (c x))-\frac {a+b \arccos (c x)}{x}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{2 x^2}\)

\(\Big \downarrow \) 1578

\(\displaystyle -\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} b c \int \frac {-c^4 x^4+6 c^2 x^2+3}{x^2 \sqrt {1-c^2 x^2}}dx^2+\frac {1}{3} c^4 x^3 (a+b \arccos (c x))-2 c^2 x (a+b \arccos (c x))-\frac {a+b \arccos (c x)}{x}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{2 x^2}\)

\(\Big \downarrow \) 1192

\(\displaystyle -\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b \int -\frac {-c^4 x^8-4 c^4 x^4+8 c^4}{1-x^4}d\sqrt {1-c^2 x^2}}{3 c^3}+\frac {1}{3} c^4 x^3 (a+b \arccos (c x))-2 c^2 x (a+b \arccos (c x))-\frac {a+b \arccos (c x)}{x}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{2 x^2}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {b \int \frac {-c^4 x^8-4 c^4 x^4+8 c^4}{1-x^4}d\sqrt {1-c^2 x^2}}{3 c^3}+\frac {1}{3} c^4 x^3 (a+b \arccos (c x))-2 c^2 x (a+b \arccos (c x))-\frac {a+b \arccos (c x)}{x}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{2 x^2}\)

\(\Big \downarrow \) 1467

\(\displaystyle -\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {b \int \left (x^4 c^4+\frac {3 c^4}{1-x^4}+5 c^4\right )d\sqrt {1-c^2 x^2}}{3 c^3}+\frac {1}{3} c^4 x^3 (a+b \arccos (c x))-2 c^2 x (a+b \arccos (c x))-\frac {a+b \arccos (c x)}{x}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{2 x^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arccos (c x))-2 c^2 x (a+b \arccos (c x))-\frac {a+b \arccos (c x)}{x}-\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{2 x^2}\)

\(\Big \downarrow \) 5203

\(\displaystyle -\frac {5}{2} c^2 d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \int \left (1-c^2 x^2\right ) (a+b \arccos (c x))dx}{3 \sqrt {1-c^2 x^2}}+d \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{x}dx+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arccos (c x))-2 c^2 x (a+b \arccos (c x))-\frac {a+b \arccos (c x)}{x}-\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{2 x^2}\)

\(\Big \downarrow \) 5155

\(\displaystyle -\frac {5}{2} c^2 d \left (d \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{x}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (b c \int \frac {x \left (3-c^2 x^2\right )}{3 \sqrt {1-c^2 x^2}}dx-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arccos (c x))-2 c^2 x (a+b \arccos (c x))-\frac {a+b \arccos (c x)}{x}-\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{2 x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {5}{2} c^2 d \left (d \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{x}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (\frac {1}{3} b c \int \frac {x \left (3-c^2 x^2\right )}{\sqrt {1-c^2 x^2}}dx-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arccos (c x))-2 c^2 x (a+b \arccos (c x))-\frac {a+b \arccos (c x)}{x}-\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{2 x^2}\)

\(\Big \downarrow \) 353

\(\displaystyle -\frac {5}{2} c^2 d \left (d \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{x}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (\frac {1}{6} b c \int \frac {3-c^2 x^2}{\sqrt {1-c^2 x^2}}dx^2-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arccos (c x))-2 c^2 x (a+b \arccos (c x))-\frac {a+b \arccos (c x)}{x}-\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{2 x^2}\)

\(\Big \downarrow \) 53

\(\displaystyle -\frac {5}{2} c^2 d \left (d \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{x}dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (\frac {1}{6} b c \int \left (\sqrt {1-c^2 x^2}+\frac {2}{\sqrt {1-c^2 x^2}}\right )dx^2-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arccos (c x))-2 c^2 x (a+b \arccos (c x))-\frac {a+b \arccos (c x)}{x}-\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{2 x^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {5}{2} c^2 d \left (d \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{x}dx+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))+\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arccos (c x))-2 c^2 x (a+b \arccos (c x))-\frac {a+b \arccos (c x)}{x}-\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{2 x^2}\)

\(\Big \downarrow \) 5199

\(\displaystyle -\frac {5}{2} c^2 d \left (d \left (\frac {2 b c \sqrt {d-c^2 d x^2} \int (a+b \arccos (c x))dx}{\sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arccos (c x))^2}{x \sqrt {1-c^2 x^2}}dx}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))+\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arccos (c x))-2 c^2 x (a+b \arccos (c x))-\frac {a+b \arccos (c x)}{x}-\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{2 x^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {5}{2} c^2 d \left (d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arccos (c x))^2}{x \sqrt {1-c^2 x^2}}dx}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2+\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))+\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arccos (c x))-2 c^2 x (a+b \arccos (c x))-\frac {a+b \arccos (c x)}{x}-\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{2 x^2}\)

\(\Big \downarrow \) 5219

\(\displaystyle -\frac {5}{2} c^2 d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arccos (c x))^2}{c x}d\arccos (c x)}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2+\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))+\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arccos (c x))-2 c^2 x (a+b \arccos (c x))-\frac {a+b \arccos (c x)}{x}-\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{2 x^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {5}{2} c^2 d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \int (a+b \arccos (c x))^2 \csc \left (\arccos (c x)+\frac {\pi }{2}\right )d\arccos (c x)}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2+\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))+\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arccos (c x))-2 c^2 x (a+b \arccos (c x))-\frac {a+b \arccos (c x)}{x}-\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{2 x^2}\)

\(\Big \downarrow \) 4669

\(\displaystyle -\frac {5}{2} c^2 d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \left (-2 b \int (a+b \arccos (c x)) \log \left (1-i e^{i \arccos (c x)}\right )d\arccos (c x)+2 b \int (a+b \arccos (c x)) \log \left (1+i e^{i \arccos (c x)}\right )d\arccos (c x)-2 i \arctan \left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2\right )}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2+\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))+\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arccos (c x))-2 c^2 x (a+b \arccos (c x))-\frac {a+b \arccos (c x)}{x}-\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{2 x^2}\)

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {5}{2} c^2 d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \left (2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-i b \int \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )d\arccos (c x)\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-i b \int \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )d\arccos (c x)\right )-2 i \arctan \left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2\right )}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2+\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))+\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arccos (c x))-2 c^2 x (a+b \arccos (c x))-\frac {a+b \arccos (c x)}{x}-\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{2 x^2}\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {5}{2} c^2 d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \left (2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 i \arctan \left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2\right )}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2+\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))+\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arccos (c x))-2 c^2 x (a+b \arccos (c x))-\frac {a+b \arccos (c x)}{x}-\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{2 x^2}\)

\(\Big \downarrow \) 7143

\(\displaystyle -\frac {5}{2} c^2 d \left (d \left (-\frac {\sqrt {d-c^2 d x^2} \left (-2 i \arctan \left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2+2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \operatorname {PolyLog}\left (3,-i e^{i \arccos (c x)}\right )\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \operatorname {PolyLog}\left (3,i e^{i \arccos (c x)}\right )\right )\right )}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2+\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))+\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arccos (c x))-2 c^2 x (a+b \arccos (c x))-\frac {a+b \arccos (c x)}{x}-\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{2 x^2}\)

Input:

Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x])^2)/x^3,x]
 

Output:

-1/2*((d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x])^2)/x^2 - (b*c*d^2*Sqrt[d - 
 c^2*d*x^2]*(-((a + b*ArcCos[c*x])/x) - 2*c^2*x*(a + b*ArcCos[c*x]) + (c^4 
*x^3*(a + b*ArcCos[c*x]))/3 - (b*(-1/3*(c^4*x^6) - 5*c^4*Sqrt[1 - c^2*x^2] 
 - 3*c^4*ArcTanh[Sqrt[1 - c^2*x^2]]))/(3*c^3)))/Sqrt[1 - c^2*x^2] - (5*c^2 
*d*(((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*Sqr 
t[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*ArcCos[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] 
)*PolyLog[2, I*E^(I*ArcCos[c*x])] - b*PolyLog[3, I*E^(I*ArcCos[c*x])])))/S 
qrt[1 - c^2*x^2])))/2
 

Defintions of rubi rules used

rule 25

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

rule 27

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

rule 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 1192

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2/e^(n + 2*p + 1)   Subst[Int[x^( 
2*m + 1)*(e*f - d*g + g*x^2)^n*(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + 
 c*x^4)^p, x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && 
IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m + 1/2]
 

rule 1467

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), 
 x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], 
x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e 
 + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
 

rule 1578

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ 
)^4)^(p_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a 
+ b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int 
egerQ[(m - 1)/2]
 

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 5193

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_) 
^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(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]] /; FreeQ[{a, b, c, d, e, f, m}, 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 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]
 

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 = 0.85 (sec) , antiderivative size = 1365, normalized size of antiderivative = 1.84

method	result	size
default	\(\text {Expression too large to display}\)	\(1365\)
parts	\(\text {Expression too large to display}\)	\(1365\)

Input:

int((-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))^2/x^3,x,method=_RETURNVERBOSE)
 

Output:

a^2*(-1/2/d/x^2*(-c^2*d*x^2+d)^(7/2)-5/2*c^2*(1/5*(-c^2*d*x^2+d)^(5/2)+d*( 
1/3*(-c^2*d*x^2+d)^(3/2)+d*((-c^2*d*x^2+d)^(1/2)-d^(1/2)*ln((2*d+2*d^(1/2) 
*(-c^2*d*x^2+d)^(1/2))/x)))))+b^2*(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*arccos(c*x)+9*arccos(c*x)^2-2)*d^2*c^2/(c^2*x^2-1)-9/8*(-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)^2-2+2*I*arccos(c* 
x))*d^2*c^2/(c^2*x^2-1)-9/8*(-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)^2-2-2*I*arccos(c*x))*d^2*c^2/(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*arccos(c*x)+9*arccos(c*x)^2-2)*d^2* 
c^2/(c^2*x^2-1)-1/2*d^2*(c^2*x^2*arccos(c*x)+2*c*x*(-c^2*x^2+1)^(1/2)-arcc 
os(c*x))*arccos(c*x)*(-d*(c^2*x^2-1))^(1/2)/(c^2*x^2-1)/x^2+I*(-d*(c^2*x^2 
-1))^(1/2)*(-c^2*x^2+1)^(1/2)*(5*I*arccos(c*x)^2*ln(1-I*(c*x+I*(-c^2*x^2+1 
)^(1/2)))-5*I*arccos(c*x)^2*ln(1+I*(c*x+I*(-c^2*x^2+1)^(1/2)))+10*arccos(c 
*x)*polylog(2,I*(c*x+I*(-c^2*x^2+1)^(1/2)))-10*arccos(c*x)*polylog(2,-I*(c 
*x+I*(-c^2*x^2+1)^(1/2)))+10*I*polylog(3,I*(c*x+I*(-c^2*x^2+1)^(1/2)))-10* 
I*polylog(3,-I*(c*x+I*(-c^2*x^2+1)^(1/2)))-4*arctan(c*x+I*(-c^2*x^2+1)^(1/ 
2)))*d^2*c^2/(2*c^2*x^2-2))+2*a*b*(1/72*(-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)*(I+ 
3*arccos(c*x))*d^2*c^2/(c^2*x^2-1)-9/8*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*...
 

Fricas [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{x^3} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arccos \left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))^2/x^3,x, algorithm="frica 
s")
 

Output:

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

Sympy [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{x^3} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}{x^{3}}\, dx \] Input:

integrate((-c**2*d*x**2+d)**(5/2)*(a+b*acos(c*x))**2/x**3,x)
 

Output:

Integral((-d*(c*x - 1)*(c*x + 1))**(5/2)*(a + b*acos(c*x))**2/x**3, x)
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{x^3} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arccos \left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))^2/x^3,x, algorithm="maxim 
a")
 

Output:

1/6*(15*c^2*d^(5/2)*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x) 
) - 3*(-c^2*d*x^2 + d)^(5/2)*c^2 - 5*(-c^2*d*x^2 + d)^(3/2)*c^2*d - 15*sqr 
t(-c^2*d*x^2 + d)*c^2*d^2 - 3*(-c^2*d*x^2 + d)^(7/2)/(d*x^2))*a^2 + sqrt(d 
)*integrate(((b^2*c^4*d^2*x^4 - 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arctan2(sqrt( 
c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 2*(a*b*c^4*d^2*x^4 - 2*a*b*c^2*d^2*x^2 + 
 a*b*d^2)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x))*sqrt(c*x + 1)*sqrt(- 
c*x + 1)/x^3, x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{x^3} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))^2/x^3,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. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^3} \,d x \] Input:

int(((a + b*acos(c*x))^2*(d - c^2*d*x^2)^(5/2))/x^3,x)
 

Output:

int(((a + b*acos(c*x))^2*(d - c^2*d*x^2)^(5/2))/x^3, x)
 

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{x^3} \, dx=\frac {\sqrt {d}\, d^{2} \left (8 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{4} x^{4}-56 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-12 \sqrt {-c^{2} x^{2}+1}\, a^{2}+48 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )}{x^{3}}d x \right ) a b \,x^{2}-96 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )}{x}d x \right ) a b \,c^{2} x^{2}+24 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )^{2}}{x^{3}}d x \right ) b^{2} x^{2}-48 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )^{2}}{x}d x \right ) b^{2} c^{2} x^{2}+48 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x d x \right ) a b \,c^{4} x^{2}+24 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )^{2} x d x \right ) b^{2} c^{4} x^{2}-60 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right )\right ) a^{2} c^{2} x^{2}+65 a^{2} c^{2} x^{2}\right )}{24 x^{2}} \] Input:

int((-c^2*d*x^2+d)^(5/2)*(a+b*acos(c*x))^2/x^3,x)
 

Output:

(sqrt(d)*d**2*(8*sqrt( - c**2*x**2 + 1)*a**2*c**4*x**4 - 56*sqrt( - c**2*x 
**2 + 1)*a**2*c**2*x**2 - 12*sqrt( - c**2*x**2 + 1)*a**2 + 48*int((sqrt( - 
 c**2*x**2 + 1)*acos(c*x))/x**3,x)*a*b*x**2 - 96*int((sqrt( - c**2*x**2 + 
1)*acos(c*x))/x,x)*a*b*c**2*x**2 + 24*int((sqrt( - c**2*x**2 + 1)*acos(c*x 
)**2)/x**3,x)*b**2*x**2 - 48*int((sqrt( - c**2*x**2 + 1)*acos(c*x)**2)/x,x 
)*b**2*c**2*x**2 + 48*int(sqrt( - c**2*x**2 + 1)*acos(c*x)*x,x)*a*b*c**4*x 
**2 + 24*int(sqrt( - c**2*x**2 + 1)*acos(c*x)**2*x,x)*b**2*c**4*x**2 - 60* 
log(tan(asin(c*x)/2))*a**2*c**2*x**2 + 65*a**2*c**2*x**2))/(24*x**2)