Integrand size = 29, antiderivative size = 546 \[ \int \frac {x^5 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {b^2}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {16 a b x \sqrt {1-c^2 x^2}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \left (1-c^2 x^2\right )}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {16 b^2 x \sqrt {1-c^2 x^2} \arccos (c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {b x^3 (a+b \arccos (c x))}{3 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {11 b x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arccos (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \arccos (c x))^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{3 c^6 d^3}-\frac {22 i b \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \arctan \left (e^{i \arccos (c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {11 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}} \] Output:
1/3*b^2/c^6/d^2/(-c^2*d*x^2+d)^(1/2)+16/3*a*b*x*(-c^2*x^2+1)^(1/2)/c^5/d^2 /(-c^2*d*x^2+d)^(1/2)+2*b^2*(-c^2*x^2+1)/c^6/d^2/(-c^2*d*x^2+d)^(1/2)+16/3 *b^2*x*(-c^2*x^2+1)^(1/2)*arccos(c*x)/c^5/d^2/(-c^2*d*x^2+d)^(1/2)-1/3*b*x ^3*(a+b*arccos(c*x))/c^3/d^2/(-c^2*x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)-11/3* b*x*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))/c^5/d^2/(-c^2*d*x^2+d)^(1/2)+1/3* x^4*(a+b*arccos(c*x))^2/c^2/d/(-c^2*d*x^2+d)^(3/2)-4/3*x^2*(a+b*arccos(c*x ))^2/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-8/3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x ))^2/c^6/d^3-22/3*I*b*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))*arctan(c*x+I*(- c^2*x^2+1)^(1/2))/c^6/d^2/(-c^2*d*x^2+d)^(1/2)+11/3*I*b^2*(-c^2*x^2+1)^(1/ 2)*polylog(2,-I*(c*x+I*(-c^2*x^2+1)^(1/2)))/c^6/d^2/(-c^2*d*x^2+d)^(1/2)-1 1/3*I*b^2*(-c^2*x^2+1)^(1/2)*polylog(2,I*(c*x+I*(-c^2*x^2+1)^(1/2)))/c^6/d ^2/(-c^2*d*x^2+d)^(1/2)
Time = 1.46 (sec) , antiderivative size = 532, normalized size of antiderivative = 0.97 \[ \int \frac {x^5 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {\sqrt {d-c^2 d x^2} \left (64 a^2-22 b^2-96 a^2 c^2 x^2+24 a^2 c^4 x^4+50 a b \arccos (c x)+25 b^2 \arccos (c x)^2+28 b^2 \cos (2 \arccos (c x))-72 a b \arccos (c x) \cos (2 \arccos (c x))-36 b^2 \arccos (c x)^2 \cos (2 \arccos (c x))-6 b^2 \cos (4 \arccos (c x))+6 a b \arccos (c x) \cos (4 \arccos (c x))+3 b^2 \arccos (c x)^2 \cos (4 \arccos (c x))-66 b^2 \sqrt {1-c^2 x^2} \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )+66 b^2 \sqrt {1-c^2 x^2} \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )+66 a b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arccos (c x)\right )\right )-66 a b \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {1}{2} \arccos (c x)\right )\right )-88 i b^2 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )+88 i b^2 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )+8 a b \sin (2 \arccos (c x))+8 b^2 \arccos (c x) \sin (2 \arccos (c x))+22 b^2 \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right ) \sin (3 \arccos (c x))-22 b^2 \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right ) \sin (3 \arccos (c x))-22 a b \log \left (\cos \left (\frac {1}{2} \arccos (c x)\right )\right ) \sin (3 \arccos (c x))+22 a b \log \left (\sin \left (\frac {1}{2} \arccos (c x)\right )\right ) \sin (3 \arccos (c x))-6 a b \sin (4 \arccos (c x))-6 b^2 \arccos (c x) \sin (4 \arccos (c x))\right )}{24 c^6 d^3 \left (-1+c^2 x^2\right )^2} \] Input:
Integrate[(x^5*(a + b*ArcCos[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]
Output:
-1/24*(Sqrt[d - c^2*d*x^2]*(64*a^2 - 22*b^2 - 96*a^2*c^2*x^2 + 24*a^2*c^4* x^4 + 50*a*b*ArcCos[c*x] + 25*b^2*ArcCos[c*x]^2 + 28*b^2*Cos[2*ArcCos[c*x] ] - 72*a*b*ArcCos[c*x]*Cos[2*ArcCos[c*x]] - 36*b^2*ArcCos[c*x]^2*Cos[2*Arc Cos[c*x]] - 6*b^2*Cos[4*ArcCos[c*x]] + 6*a*b*ArcCos[c*x]*Cos[4*ArcCos[c*x] ] + 3*b^2*ArcCos[c*x]^2*Cos[4*ArcCos[c*x]] - 66*b^2*Sqrt[1 - c^2*x^2]*ArcC os[c*x]*Log[1 - E^(I*ArcCos[c*x])] + 66*b^2*Sqrt[1 - c^2*x^2]*ArcCos[c*x]* Log[1 + E^(I*ArcCos[c*x])] + 66*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcCos[c*x]/ 2]] - 66*a*b*Sqrt[1 - c^2*x^2]*Log[Sin[ArcCos[c*x]/2]] - (88*I)*b^2*(1 - c ^2*x^2)^(3/2)*PolyLog[2, -E^(I*ArcCos[c*x])] + (88*I)*b^2*(1 - c^2*x^2)^(3 /2)*PolyLog[2, E^(I*ArcCos[c*x])] + 8*a*b*Sin[2*ArcCos[c*x]] + 8*b^2*ArcCo s[c*x]*Sin[2*ArcCos[c*x]] + 22*b^2*ArcCos[c*x]*Log[1 - E^(I*ArcCos[c*x])]* Sin[3*ArcCos[c*x]] - 22*b^2*ArcCos[c*x]*Log[1 + E^(I*ArcCos[c*x])]*Sin[3*A rcCos[c*x]] - 22*a*b*Log[Cos[ArcCos[c*x]/2]]*Sin[3*ArcCos[c*x]] + 22*a*b*L og[Sin[ArcCos[c*x]/2]]*Sin[3*ArcCos[c*x]] - 6*a*b*Sin[4*ArcCos[c*x]] - 6*b ^2*ArcCos[c*x]*Sin[4*ArcCos[c*x]]))/(c^6*d^3*(-1 + c^2*x^2)^2)
Time = 2.45 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.01, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {5207, 5207, 243, 53, 2009, 5183, 2009, 5211, 241, 5165, 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.
| \(\displaystyle \int \frac {x^5 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5207 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \int \frac {x^4 (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {4 \int \frac {x^3 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}}dx}{3 c^2 d}+\frac {x^4 (a+b \arccos (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5207 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 \int \frac {x^2 (a+b \arccos (c x))}{1-c^2 x^2}dx}{2 c^2}+\frac {b \int \frac {x^3}{\left (1-c^2 x^2\right )^{3/2}}dx}{2 c}+\frac {x^3 (a+b \arccos (c x))}{2 c^2 \left (1-c^2 x^2\right )}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {4 \left (\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x^2 (a+b \arccos (c x))}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}-\frac {2 \int \frac {x (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x^2 (a+b \arccos (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^4 (a+b \arccos (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 \int \frac {x^2 (a+b \arccos (c x))}{1-c^2 x^2}dx}{2 c^2}+\frac {b \int \frac {x^2}{\left (1-c^2 x^2\right )^{3/2}}dx^2}{4 c}+\frac {x^3 (a+b \arccos (c x))}{2 c^2 \left (1-c^2 x^2\right )}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {4 \left (\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x^2 (a+b \arccos (c x))}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}-\frac {2 \int \frac {x (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x^2 (a+b \arccos (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^4 (a+b \arccos (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 \int \frac {x^2 (a+b \arccos (c x))}{1-c^2 x^2}dx}{2 c^2}+\frac {b \int \left (\frac {1}{c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{c^2 \sqrt {1-c^2 x^2}}\right )dx^2}{4 c}+\frac {x^3 (a+b \arccos (c x))}{2 c^2 \left (1-c^2 x^2\right )}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {4 \left (\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x^2 (a+b \arccos (c x))}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}-\frac {2 \int \frac {x (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x^2 (a+b \arccos (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^4 (a+b \arccos (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \left (\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x^2 (a+b \arccos (c x))}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}-\frac {2 \int \frac {x (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x^2 (a+b \arccos (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 \int \frac {x^2 (a+b \arccos (c x))}{1-c^2 x^2}dx}{2 c^2}+\frac {x^3 (a+b \arccos (c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {2}{c^4 \sqrt {1-c^2 x^2}}\right )}{4 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arccos (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5183 |
| \(\displaystyle -\frac {4 \left (-\frac {2 \left (-\frac {2 b \sqrt {1-c^2 x^2} \int (a+b \arccos (c x))dx}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{c^2 d}\right )}{c^2 d}+\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x^2 (a+b \arccos (c x))}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arccos (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 \int \frac {x^2 (a+b \arccos (c x))}{1-c^2 x^2}dx}{2 c^2}+\frac {x^3 (a+b \arccos (c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {2}{c^4 \sqrt {1-c^2 x^2}}\right )}{4 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arccos (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \left (\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x^2 (a+b \arccos (c x))}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arccos (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\right )}{3 c^2 d}+\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 \int \frac {x^2 (a+b \arccos (c x))}{1-c^2 x^2}dx}{2 c^2}+\frac {x^3 (a+b \arccos (c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {2}{c^4 \sqrt {1-c^2 x^2}}\right )}{4 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arccos (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5211 |
| \(\displaystyle -\frac {4 \left (\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int \frac {a+b \arccos (c x)}{1-c^2 x^2}dx}{c^2}-\frac {b \int \frac {x}{\sqrt {1-c^2 x^2}}dx}{c}-\frac {x (a+b \arccos (c x))}{c^2}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arccos (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\right )}{3 c^2 d}+\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 \left (\frac {\int \frac {a+b \arccos (c x)}{1-c^2 x^2}dx}{c^2}-\frac {b \int \frac {x}{\sqrt {1-c^2 x^2}}dx}{c}-\frac {x (a+b \arccos (c x))}{c^2}\right )}{2 c^2}+\frac {x^3 (a+b \arccos (c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {2}{c^4 \sqrt {1-c^2 x^2}}\right )}{4 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arccos (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle -\frac {4 \left (\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int \frac {a+b \arccos (c x)}{1-c^2 x^2}dx}{c^2}-\frac {x (a+b \arccos (c x))}{c^2}+\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arccos (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\right )}{3 c^2 d}+\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 \left (\frac {\int \frac {a+b \arccos (c x)}{1-c^2 x^2}dx}{c^2}-\frac {x (a+b \arccos (c x))}{c^2}+\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{2 c^2}+\frac {x^3 (a+b \arccos (c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {2}{c^4 \sqrt {1-c^2 x^2}}\right )}{4 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arccos (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5165 |
| \(\displaystyle -\frac {4 \left (\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {\int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}d\arccos (c x)}{c^3}-\frac {x (a+b \arccos (c x))}{c^2}+\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arccos (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\right )}{3 c^2 d}+\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 \left (-\frac {\int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}d\arccos (c x)}{c^3}-\frac {x (a+b \arccos (c x))}{c^2}+\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{2 c^2}+\frac {x^3 (a+b \arccos (c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {2}{c^4 \sqrt {1-c^2 x^2}}\right )}{4 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arccos (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {4 \left (\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {\int (a+b \arccos (c x)) \csc (\arccos (c x))d\arccos (c x)}{c^3}-\frac {x (a+b \arccos (c x))}{c^2}+\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arccos (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\right )}{3 c^2 d}+\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 \left (-\frac {\int (a+b \arccos (c x)) \csc (\arccos (c x))d\arccos (c x)}{c^3}-\frac {x (a+b \arccos (c x))}{c^2}+\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{2 c^2}+\frac {x^3 (a+b \arccos (c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {2}{c^4 \sqrt {1-c^2 x^2}}\right )}{4 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arccos (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle -\frac {4 \left (\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {-b \int \log \left (1-e^{i \arccos (c x)}\right )d\arccos (c x)+b \int \log \left (1+e^{i \arccos (c x)}\right )d\arccos (c x)-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{c^3}-\frac {x (a+b \arccos (c x))}{c^2}+\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arccos (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\right )}{3 c^2 d}+\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 \left (-\frac {-b \int \log \left (1-e^{i \arccos (c x)}\right )d\arccos (c x)+b \int \log \left (1+e^{i \arccos (c x)}\right )d\arccos (c x)-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{c^3}-\frac {x (a+b \arccos (c x))}{c^2}+\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{2 c^2}+\frac {x^3 (a+b \arccos (c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {2}{c^4 \sqrt {1-c^2 x^2}}\right )}{4 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arccos (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {4 \left (\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {i b \int e^{-i \arccos (c x)} \log \left (1-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}-i b \int e^{-i \arccos (c x)} \log \left (1+e^{i \arccos (c x)}\right )de^{i \arccos (c x)}-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{c^3}-\frac {x (a+b \arccos (c x))}{c^2}+\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arccos (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\right )}{3 c^2 d}+\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 \left (-\frac {i b \int e^{-i \arccos (c x)} \log \left (1-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}-i b \int e^{-i \arccos (c x)} \log \left (1+e^{i \arccos (c x)}\right )de^{i \arccos (c x)}-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{c^3}-\frac {x (a+b \arccos (c x))}{c^2}+\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{2 c^2}+\frac {x^3 (a+b \arccos (c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {2}{c^4 \sqrt {1-c^2 x^2}}\right )}{4 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arccos (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {4 \left (\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{c^3}-\frac {x (a+b \arccos (c x))}{c^2}+\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arccos (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\right )}{3 c^2 d}+\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 \left (-\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{c^3}-\frac {x (a+b \arccos (c x))}{c^2}+\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{2 c^2}+\frac {x^3 (a+b \arccos (c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {2}{c^4 \sqrt {1-c^2 x^2}}\right )}{4 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arccos (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
Input:
Int[(x^5*(a + b*ArcCos[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]
Output:
(x^4*(a + b*ArcCos[c*x])^2)/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) + (2*b*Sqrt[1 - c^2*x^2]*((b*(2/(c^4*Sqrt[1 - c^2*x^2]) + (2*Sqrt[1 - c^2*x^2])/c^4))/(4 *c) + (x^3*(a + b*ArcCos[c*x]))/(2*c^2*(1 - c^2*x^2)) - (3*((b*Sqrt[1 - c^ 2*x^2])/c^3 - (x*(a + b*ArcCos[c*x]))/c^2 - (-2*(a + b*ArcCos[c*x])*ArcTan h[E^(I*ArcCos[c*x])] + I*b*PolyLog[2, -E^(I*ArcCos[c*x])] - I*b*PolyLog[2, E^(I*ArcCos[c*x])])/c^3))/(2*c^2)))/(3*c*d^2*Sqrt[d - c^2*d*x^2]) - (4*(( x^2*(a + b*ArcCos[c*x])^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) - (2*(-((Sqrt[d - c ^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(c^2*d)) - (2*b*Sqrt[1 - c^2*x^2]*(a*x - (b*Sqrt[1 - c^2*x^2])/c + b*x*ArcCos[c*x]))/(c*Sqrt[d - c^2*d*x^2])))/(c^2 *d) + (2*b*Sqrt[1 - c^2*x^2]*((b*Sqrt[1 - c^2*x^2])/c^3 - (x*(a + b*ArcCos [c*x]))/c^2 - (-2*(a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])] + I*b*Pol yLog[2, -E^(I*ArcCos[c*x])] - I*b*PolyLog[2, E^(I*ArcCos[c*x])])/c^3))/(c* d*Sqrt[d - c^2*d*x^2])))/(3*c^2*d)
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])
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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]
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]
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]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbo l] :> Simp[-(c*d)^(-1) Subst[Int[(a + b*x)^n*Csc[x], x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n, x], x] - Simp [b*f*(n/(2*c*(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}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IG tQ[m, 1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] - S imp[b*f*(n/(c*(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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m , 1] && NeQ[m + 2*p + 1, 0]
Time = 0.79 (sec) , antiderivative size = 803, normalized size of antiderivative = 1.47
| method | result | size |
| default | \(a^{2} \left (-\frac {x^{4}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {\frac {4 x^{2}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {8}{3 d \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}}{c^{2}}\right )+b^{2} \left (-\frac {\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}-2+2 i \arccos \left (c x \right )\right )}{2 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {\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}-2-2 i \arccos \left (c x \right )\right )}{2 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 \arccos \left (c x \right )^{2} x^{2} c^{2}+\sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x c -c^{2} x^{2}-5 \arccos \left (c x \right )^{2}+1\right )}{3 \left (c^{2} x^{2}-1\right )^{2} d^{3} c^{6}}+\frac {11 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\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 )+i\right )}{2 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {\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 )-i\right )}{2 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (12 c^{2} x^{2} \arccos \left (c x \right )+c x \sqrt {-c^{2} x^{2}+1}-10 \arccos \left (c x \right )\right )}{6 \left (c^{2} x^{2}-1\right )^{2} d^{3} c^{6}}-\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )}{6 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{6 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}\right )\) | \(803\) |
| parts | \(a^{2} \left (-\frac {x^{4}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {\frac {4 x^{2}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {8}{3 d \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}}{c^{2}}\right )+b^{2} \left (-\frac {\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}-2+2 i \arccos \left (c x \right )\right )}{2 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {\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}-2-2 i \arccos \left (c x \right )\right )}{2 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 \arccos \left (c x \right )^{2} x^{2} c^{2}+\sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x c -c^{2} x^{2}-5 \arccos \left (c x \right )^{2}+1\right )}{3 \left (c^{2} x^{2}-1\right )^{2} d^{3} c^{6}}+\frac {11 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\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 )+i\right )}{2 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {\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 )-i\right )}{2 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (12 c^{2} x^{2} \arccos \left (c x \right )+c x \sqrt {-c^{2} x^{2}+1}-10 \arccos \left (c x \right )\right )}{6 \left (c^{2} x^{2}-1\right )^{2} d^{3} c^{6}}-\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )}{6 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{6 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}\right )\) | \(803\) |
Input:
int(x^5*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
Output:
a^2*(-x^4/c^2/d/(-c^2*d*x^2+d)^(3/2)+4/c^2*(x^2/c^2/d/(-c^2*d*x^2+d)^(3/2) -2/3/d/c^4/(-c^2*d*x^2+d)^(3/2)))+b^2*(-1/2*(-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^3/c^6/(c ^2*x^2-1)-1/2*(-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^3/c^6/(c^2*x^2-1)+1/3*(-d*(c^2*x^2-1) )^(1/2)*(6*arccos(c*x)^2*x^2*c^2+(-c^2*x^2+1)^(1/2)*arccos(c*x)*x*c-c^2*x^ 2-5*arccos(c*x)^2+1)/(c^2*x^2-1)^2/d^3/c^6+11/3*I*(-c^2*x^2+1)^(1/2)*(-d*( c^2*x^2-1))^(1/2)*(I*arccos(c*x)*ln(1-c*x-I*(-c^2*x^2+1)^(1/2))-I*arccos(c *x)*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))-polylog(2,-c*x-I*(-c^2*x^2+1)^(1/2))+po lylog(2,c*x+I*(-c^2*x^2+1)^(1/2)))/d^3/c^6/(c^2*x^2-1))+2*a*b*(-1/2*(-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)+I)/d^3/ c^6/(c^2*x^2-1)-1/2*(-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)-I)/d^3/c^6/(c^2*x^2-1)+1/6*(-d*(c^2*x^2-1))^(1/2)*(12* c^2*x^2*arccos(c*x)+c*x*(-c^2*x^2+1)^(1/2)-10*arccos(c*x))/(c^2*x^2-1)^2/d ^3/c^6-11/6*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^3/c^6/(c^2*x^2-1)* ln(I*(-c^2*x^2+1)^(1/2)+c*x-1)+11/6*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1 /2)/d^3/c^6/(c^2*x^2-1)*ln(1+c*x+I*(-c^2*x^2+1)^(1/2)))
\[ \int \frac {x^5 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2} x^{5}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^5*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="frica s")
Output:
integral(-(b^2*x^5*arccos(c*x)^2 + 2*a*b*x^5*arccos(c*x) + a^2*x^5)*sqrt(- c^2*d*x^2 + d)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
\[ \int \frac {x^5 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^{5} \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(x**5*(a+b*acos(c*x))**2/(-c**2*d*x**2+d)**(5/2),x)
Output:
Integral(x**5*(a + b*acos(c*x))**2/(-d*(c*x - 1)*(c*x + 1))**(5/2), x)
\[ \int \frac {x^5 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2} x^{5}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^5*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxim a")
Output:
-1/3*a^2*(3*x^4/((-c^2*d*x^2 + d)^(3/2)*c^2*d) - 12*x^2/((-c^2*d*x^2 + d)^ (3/2)*c^4*d) + 8/((-c^2*d*x^2 + d)^(3/2)*c^6*d)) - 1/3*((3*b^2*c^4*x^4 - 1 2*b^2*c^2*x^2 + 8*b^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*sqrt(d)*arctan2(sqrt(c *x + 1)*sqrt(-c*x + 1), c*x)^2 + 3*(c^10*d^3*x^4 - 2*c^8*d^3*x^2 + c^6*d^3 )*integrate(2/3*(3*sqrt(c*x + 1)*sqrt(-c*x + 1)*a*b*c^5*sqrt(d)*x^5*arctan 2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + (3*b^2*c^6*x^6 - 15*b^2*c^4*x^4 + 2 0*b^2*c^2*x^2 - 8*b^2)*sqrt(d)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)) /(c^11*d^3*x^6 - 3*c^9*d^3*x^4 + 3*c^7*d^3*x^2 - c^5*d^3), x))/(c^10*d^3*x ^4 - 2*c^8*d^3*x^2 + c^6*d^3)
Exception generated. \[ \int \frac {x^5 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^5*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac" )
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x^5 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^5\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \] Input:
int((x^5*(a + b*acos(c*x))^2)/(d - c^2*d*x^2)^(5/2),x)
Output:
int((x^5*(a + b*acos(c*x))^2)/(d - c^2*d*x^2)^(5/2), x)
\[ \int \frac {x^5 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {6 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right ) x^{5}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) a b \,c^{8} x^{2}-6 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right ) x^{5}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) a b \,c^{6}+3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right )^{2} x^{5}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b^{2} c^{8} x^{2}-3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right )^{2} x^{5}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b^{2} c^{6}+3 a^{2} c^{4} x^{4}-12 a^{2} c^{2} x^{2}+8 a^{2}}{3 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, c^{6} d^{2} \left (c^{2} x^{2}-1\right )} \] Input:
int(x^5*(a+b*acos(c*x))^2/(-c^2*d*x^2+d)^(5/2),x)
Output:
(6*sqrt( - c**2*x**2 + 1)*int((acos(c*x)*x**5)/(sqrt( - c**2*x**2 + 1)*c** 4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x**2 + 1)),x)*a *b*c**8*x**2 - 6*sqrt( - c**2*x**2 + 1)*int((acos(c*x)*x**5)/(sqrt( - c**2 *x**2 + 1)*c**4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x **2 + 1)),x)*a*b*c**6 + 3*sqrt( - c**2*x**2 + 1)*int((acos(c*x)**2*x**5)/( sqrt( - c**2*x**2 + 1)*c**4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sq rt( - c**2*x**2 + 1)),x)*b**2*c**8*x**2 - 3*sqrt( - c**2*x**2 + 1)*int((ac os(c*x)**2*x**5)/(sqrt( - c**2*x**2 + 1)*c**4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x**2 + 1)),x)*b**2*c**6 + 3*a**2*c**4*x**4 - 12*a**2*c**2*x**2 + 8*a**2)/(3*sqrt(d)*sqrt( - c**2*x**2 + 1)*c**6*d**2*(c **2*x**2 - 1))