Integrand size = 29, antiderivative size = 452 \[ \int \frac {(a+b \arccos (c x))^2}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {b^2 c^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c (a+b \arccos (c x))}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {(a+b \arccos (c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {4 c^2 x (a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^2 x (a+b \arccos (c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 i c \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \text {arctanh}\left (e^{2 i \arccos (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {16 b c \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \log \left (1+e^{2 i \arccos (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b^2 c \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {i b^2 c \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}} \] Output:
1/3*b^2*c^2*x/d^2/(-c^2*d*x^2+d)^(1/2)-1/3*b*c*(a+b*arccos(c*x))/d^2/(-c^2 *x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)-(a+b*arccos(c*x))^2/d/x/(-c^2*d*x^2+d)^ (3/2)+4/3*c^2*x*(a+b*arccos(c*x))^2/d/(-c^2*d*x^2+d)^(3/2)+8/3*c^2*x*(a+b* arccos(c*x))^2/d^2/(-c^2*d*x^2+d)^(1/2)-8/3*I*c*(-c^2*x^2+1)^(1/2)*(a+b*ar ccos(c*x))^2/d^2/(-c^2*d*x^2+d)^(1/2)-4*b*c*(-c^2*x^2+1)^(1/2)*(a+b*arccos (c*x))*arctanh((c*x+I*(-c^2*x^2+1)^(1/2))^2)/d^2/(-c^2*d*x^2+d)^(1/2)+16/3 *b*c*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))^ 2)/d^2/(-c^2*d*x^2+d)^(1/2)-5/3*I*b^2*c*(-c^2*x^2+1)^(1/2)*polylog(2,-(c*x +I*(-c^2*x^2+1)^(1/2))^2)/d^2/(-c^2*d*x^2+d)^(1/2)-I*b^2*c*(-c^2*x^2+1)^(1 /2)*polylog(2,(c*x+I*(-c^2*x^2+1)^(1/2))^2)/d^2/(-c^2*d*x^2+d)^(1/2)
Time = 1.95 (sec) , antiderivative size = 434, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b \arccos (c x))^2}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {3 a^2-12 a^2 c^2 x^2-b^2 c^2 x^2+8 a^2 c^4 x^4+b^2 c^4 x^4-a b c x \sqrt {1-c^2 x^2}+6 a b \arccos (c x)-24 a b c^2 x^2 \arccos (c x)+16 a b c^4 x^4 \arccos (c x)-b^2 c x \sqrt {1-c^2 x^2} \arccos (c x)+3 b^2 \arccos (c x)^2-12 b^2 c^2 x^2 \arccos (c x)^2+8 b^2 c^4 x^4 \arccos (c x)^2-8 i b^2 c x \left (1-c^2 x^2\right )^{3/2} \arccos (c x)^2+10 b^2 c x \left (1-c^2 x^2\right )^{3/2} \arccos (c x) \log \left (1-e^{2 i \arccos (c x)}\right )+6 b^2 c x \left (1-c^2 x^2\right )^{3/2} \arccos (c x) \log \left (1+e^{2 i \arccos (c x)}\right )+6 a b c x \left (1-c^2 x^2\right )^{3/2} \log (c x)+5 a b c x \left (1-c^2 x^2\right )^{3/2} \log \left (1-c^2 x^2\right )-3 i b^2 c x \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )-5 i b^2 c x \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{3 d x \left (d-c^2 d x^2\right )^{3/2}} \] Input:
Integrate[(a + b*ArcCos[c*x])^2/(x^2*(d - c^2*d*x^2)^(5/2)),x]
Output:
-1/3*(3*a^2 - 12*a^2*c^2*x^2 - b^2*c^2*x^2 + 8*a^2*c^4*x^4 + b^2*c^4*x^4 - a*b*c*x*Sqrt[1 - c^2*x^2] + 6*a*b*ArcCos[c*x] - 24*a*b*c^2*x^2*ArcCos[c*x ] + 16*a*b*c^4*x^4*ArcCos[c*x] - b^2*c*x*Sqrt[1 - c^2*x^2]*ArcCos[c*x] + 3 *b^2*ArcCos[c*x]^2 - 12*b^2*c^2*x^2*ArcCos[c*x]^2 + 8*b^2*c^4*x^4*ArcCos[c *x]^2 - (8*I)*b^2*c*x*(1 - c^2*x^2)^(3/2)*ArcCos[c*x]^2 + 10*b^2*c*x*(1 - c^2*x^2)^(3/2)*ArcCos[c*x]*Log[1 - E^((2*I)*ArcCos[c*x])] + 6*b^2*c*x*(1 - c^2*x^2)^(3/2)*ArcCos[c*x]*Log[1 + E^((2*I)*ArcCos[c*x])] + 6*a*b*c*x*(1 - c^2*x^2)^(3/2)*Log[c*x] + 5*a*b*c*x*(1 - c^2*x^2)^(3/2)*Log[1 - c^2*x^2] - (3*I)*b^2*c*x*(1 - c^2*x^2)^(3/2)*PolyLog[2, -E^((2*I)*ArcCos[c*x])] - (5*I)*b^2*c*x*(1 - c^2*x^2)^(3/2)*PolyLog[2, E^((2*I)*ArcCos[c*x])])/(d*x* (d - c^2*d*x^2)^(3/2))
Time = 3.28 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.01, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.724, Rules used = {5205, 5163, 5161, 5181, 3042, 25, 4200, 25, 2620, 2715, 2838, 5183, 208, 5209, 208, 5185, 4919, 3042, 4671, 2715, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a+b \arccos (c x))^2}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5205 |
| \(\displaystyle -\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \int \frac {(a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}}dx-\frac {(a+b \arccos (c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5163 |
| \(\displaystyle 4 c^2 \left (\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \int \frac {(a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}}dx}{3 d}+\frac {x (a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arccos (c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5161 |
| \(\displaystyle 4 c^2 \left (\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {x (a+b \arccos (c x))}{1-c^2 x^2}dx}{d \sqrt {d-c^2 d x^2}}+\frac {x (a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arccos (c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5181 |
| \(\displaystyle 4 c^2 \left (\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \int \frac {c x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}d\arccos (c x)}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arccos (c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \int -\left ((a+b \arccos (c x)) \tan \left (\arccos (c x)+\frac {\pi }{2}\right )\right )d\arccos (c x)}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \arccos (c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {2 b \sqrt {1-c^2 x^2} \int (a+b \arccos (c x)) \tan \left (\arccos (c x)+\frac {\pi }{2}\right )d\arccos (c x)}{c d \sqrt {d-c^2 d x^2}}+\frac {x (a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \arccos (c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4200 |
| \(\displaystyle 4 c^2 \left (\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (2 i \int -\frac {e^{2 i \arccos (c x)} (a+b \arccos (c x))}{1-e^{2 i \arccos (c x)}}d\arccos (c x)-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arccos (c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 4 c^2 \left (\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \int \frac {e^{2 i \arccos (c x)} (a+b \arccos (c x))}{1-e^{2 i \arccos (c x)}}d\arccos (c x)-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arccos (c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{2} i b \int \log \left (1-e^{2 i \arccos (c x)}\right )d\arccos (c x)\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \arccos (c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{4} b \int e^{-2 i \arccos (c x)} \log \left (1-e^{2 i \arccos (c x)}\right )de^{2 i \arccos (c x)}\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \arccos (c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle 4 c^2 \left (\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arccos (c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5183 |
| \(\displaystyle 4 c^2 \left (\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {b \int \frac {1}{\left (1-c^2 x^2\right )^{3/2}}dx}{2 c}+\frac {a+b \arccos (c x)}{2 c^2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \arccos (c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle -\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {a+b \arccos (c x)}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b x}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \arccos (c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5209 |
| \(\displaystyle -\frac {2 b c \sqrt {1-c^2 x^2} \left (\int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )}dx+\frac {1}{2} b c \int \frac {1}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {a+b \arccos (c x)}{2 \left (1-c^2 x^2\right )}\right )}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {a+b \arccos (c x)}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b x}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \arccos (c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle -\frac {2 b c \sqrt {1-c^2 x^2} \left (\int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )}dx+\frac {a+b \arccos (c x)}{2 \left (1-c^2 x^2\right )}+\frac {b c x}{2 \sqrt {1-c^2 x^2}}\right )}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {a+b \arccos (c x)}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b x}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \arccos (c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5185 |
| \(\displaystyle -\frac {2 b c \sqrt {1-c^2 x^2} \left (-\int \frac {a+b \arccos (c x)}{c x \sqrt {1-c^2 x^2}}d\arccos (c x)+\frac {a+b \arccos (c x)}{2 \left (1-c^2 x^2\right )}+\frac {b c x}{2 \sqrt {1-c^2 x^2}}\right )}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {a+b \arccos (c x)}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b x}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \arccos (c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4919 |
| \(\displaystyle -\frac {2 b c \sqrt {1-c^2 x^2} \left (-2 \int (a+b \arccos (c x)) \csc (2 \arccos (c x))d\arccos (c x)+\frac {a+b \arccos (c x)}{2 \left (1-c^2 x^2\right )}+\frac {b c x}{2 \sqrt {1-c^2 x^2}}\right )}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {a+b \arccos (c x)}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b x}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \arccos (c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 b c \sqrt {1-c^2 x^2} \left (-2 \int (a+b \arccos (c x)) \csc (2 \arccos (c x))d\arccos (c x)+\frac {a+b \arccos (c x)}{2 \left (1-c^2 x^2\right )}+\frac {b c x}{2 \sqrt {1-c^2 x^2}}\right )}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {a+b \arccos (c x)}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b x}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \arccos (c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle -\frac {2 b c \sqrt {1-c^2 x^2} \left (-2 \left (-\frac {1}{2} b \int \log \left (1-e^{2 i \arccos (c x)}\right )d\arccos (c x)+\frac {1}{2} b \int \log \left (1+e^{2 i \arccos (c x)}\right )d\arccos (c x)-\left (\text {arctanh}\left (e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))\right )\right )+\frac {a+b \arccos (c x)}{2 \left (1-c^2 x^2\right )}+\frac {b c x}{2 \sqrt {1-c^2 x^2}}\right )}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {a+b \arccos (c x)}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b x}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \arccos (c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {2 b c \sqrt {1-c^2 x^2} \left (-2 \left (\frac {1}{4} i b \int e^{-2 i \arccos (c x)} \log \left (1-e^{2 i \arccos (c x)}\right )de^{2 i \arccos (c x)}-\frac {1}{4} i b \int e^{-2 i \arccos (c x)} \log \left (1+e^{2 i \arccos (c x)}\right )de^{2 i \arccos (c x)}-\left (\text {arctanh}\left (e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))\right )\right )+\frac {a+b \arccos (c x)}{2 \left (1-c^2 x^2\right )}+\frac {b c x}{2 \sqrt {1-c^2 x^2}}\right )}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {a+b \arccos (c x)}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b x}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \arccos (c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {2 b c \sqrt {1-c^2 x^2} \left (-2 \left (-\left (\text {arctanh}\left (e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))\right )+\frac {1}{4} i b \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )-\frac {1}{4} i b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )\right )+\frac {a+b \arccos (c x)}{2 \left (1-c^2 x^2\right )}+\frac {b c x}{2 \sqrt {1-c^2 x^2}}\right )}{d^2 \sqrt {d-c^2 d x^2}}+4 c^2 \left (\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {a+b \arccos (c x)}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b x}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {(a+b \arccos (c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
Input:
Int[(a + b*ArcCos[c*x])^2/(x^2*(d - c^2*d*x^2)^(5/2)),x]
Output:
-((a + b*ArcCos[c*x])^2/(d*x*(d - c^2*d*x^2)^(3/2))) - (2*b*c*Sqrt[1 - c^2 *x^2]*((b*c*x)/(2*Sqrt[1 - c^2*x^2]) + (a + b*ArcCos[c*x])/(2*(1 - c^2*x^2 )) - 2*(-((a + b*ArcCos[c*x])*ArcTanh[E^((2*I)*ArcCos[c*x])]) + (I/4)*b*Po lyLog[2, -E^((2*I)*ArcCos[c*x])] - (I/4)*b*PolyLog[2, E^((2*I)*ArcCos[c*x] )])))/(d^2*Sqrt[d - c^2*d*x^2]) + 4*c^2*((x*(a + b*ArcCos[c*x])^2)/(3*d*(d - c^2*d*x^2)^(3/2)) + (2*b*c*Sqrt[1 - c^2*x^2]*((b*x)/(2*c*Sqrt[1 - c^2*x ^2]) + (a + b*ArcCos[c*x])/(2*c^2*(1 - c^2*x^2))))/(3*d^2*Sqrt[d - c^2*d*x ^2]) + (2*((x*(a + b*ArcCos[c*x])^2)/(d*Sqrt[d - c^2*d*x^2]) - (2*b*Sqrt[1 - c^2*x^2]*(((-1/2*I)*(a + b*ArcCos[c*x])^2)/b - (2*I)*((I/2)*(a + b*ArcC os[c*x])*Log[1 - E^((2*I)*ArcCos[c*x])] + (b*PolyLog[2, E^((2*I)*ArcCos[c* x])])/4)))/(c*d*Sqrt[d - c^2*d*x^2])))/(3*d))
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
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[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol ] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I Int[(c + d*x)^ m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))), x] , x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
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[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n , x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x _Symbol] :> Simp[x*((a + b*ArcCos[c*x])^n/(d*Sqrt[d + e*x^2])), x] + Simp[b *c*(n/d)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]] Int[x*((a + b*ArcCos[c*x ])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_ Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(2*d*(p + 1 ))), x] + (Simp[(2*p + 3)/(2*d*(p + 1)) Int[(d + e*x^2)^(p + 1)*(a + b*Ar cCos[c*x])^n, x], x] - Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2 *x^2)^p] Int[x*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x ]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[1/e Subst[Int[(a + b*x)^n*Cot[x], x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
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_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[-d^(-1) Subst[Int[(a + b*x)^n/(Cos[x]*Sin[x]), x], x, A rcCos[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n , 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b *ArcCos[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)) ) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] + Simp[b* c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m + 1)*( 1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1)) Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n, x], x] - Simp[b*c *(n/(2*f*(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] && LtQ[p, -1] && !G tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3778 vs. \(2 (447 ) = 894\).
Time = 0.76 (sec) , antiderivative size = 3779, normalized size of antiderivative = 8.36
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(3779\) |
| parts | \(\text {Expression too large to display}\) | \(3779\) |
Input:
int((a+b*arccos(c*x))^2/x^2/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
Output:
-272/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^ 3*x^2*arccos(c*x)*(-c^2*x^2+1)^(1/2)*c^3+128/3*I*a*b*(-d*(c^2*x^2-1))^(1/2 )/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^4*arccos(c*x)*(-c^2*x^2+1)^(1/ 2)*c^5-8*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^ 3*x*(-c^2*x^2+1)*c^2+32/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4 +26*c^2*x^2-9)/d^3*x^7*(-c^2*x^2+1)*c^8-88/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(8 *c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^5*(-c^2*x^2+1)*c^6-44*b^2*(-d*(c^2 *x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x*arccos(c*x)^2*c^2 -64/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x ^5*arccos(c*x)^2*c^6+56*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+2 6*c^2*x^2-9)/d^3*x^3*arccos(c*x)^2*c^4-3*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6 *x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*arccos(c*x)*(-c^2*x^2+1)^(1/2)*c+3*I*b^2 *(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*(-c^2*x^2+ 1)^(1/2)*c+80/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^ 2-9)/d^3*x^3*(-c^2*x^2+1)*c^4+112*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25 *c^4*x^4+26*c^2*x^2-9)/d^3*x^3*arccos(c*x)*c^4-128/3*a*b*(-d*(c^2*x^2-1))^ (1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^5*arccos(c*x)*c^6-88*a*b*( -d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x*arccos(c*x )*c^2+8/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d ^3*x^2*(-c^2*x^2+1)^(1/2)*c^3+10/3*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2...
\[ \int \frac {(a+b \arccos (c x))^2}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}} \,d x } \] Input:
integrate((a+b*arccos(c*x))^2/x^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="frica s")
Output:
integral(-sqrt(-c^2*d*x^2 + d)*(b^2*arccos(c*x)^2 + 2*a*b*arccos(c*x) + a^ 2)/(c^6*d^3*x^8 - 3*c^4*d^3*x^6 + 3*c^2*d^3*x^4 - d^3*x^2), x)
\[ \int \frac {(a+b \arccos (c x))^2}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}{x^{2} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((a+b*acos(c*x))**2/x**2/(-c**2*d*x**2+d)**(5/2),x)
Output:
Integral((a + b*acos(c*x))**2/(x**2*(-d*(c*x - 1)*(c*x + 1))**(5/2)), x)
\[ \int \frac {(a+b \arccos (c x))^2}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}} \,d x } \] Input:
integrate((a+b*arccos(c*x))^2/x^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxim a")
Output:
1/3*a^2*(8*c^2*x/(sqrt(-c^2*d*x^2 + d)*d^2) + 4*c^2*x/((-c^2*d*x^2 + d)^(3 /2)*d) - 3/((-c^2*d*x^2 + d)^(3/2)*d*x)) - sqrt(d)*integrate((b^2*arctan2( sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 2*a*b*arctan2(sqrt(c*x + 1)*sqrt(-c *x + 1), c*x))*sqrt(c*x + 1)*sqrt(-c*x + 1)/(c^6*d^3*x^8 - 3*c^4*d^3*x^6 + 3*c^2*d^3*x^4 - d^3*x^2), x)
Exception generated. \[ \int \frac {(a+b \arccos (c x))^2}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a+b*arccos(c*x))^2/x^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac" )
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {(a+b \arccos (c x))^2}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2}{x^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \] Input:
int((a + b*acos(c*x))^2/(x^2*(d - c^2*d*x^2)^(5/2)),x)
Output:
int((a + b*acos(c*x))^2/(x^2*(d - c^2*d*x^2)^(5/2)), x)
\[ \int \frac {(a+b \arccos (c x))^2}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {6 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{6}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{4}+\sqrt {-c^{2} x^{2}+1}\, x^{2}}d x \right ) a b \,c^{2} x^{3}-6 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{6}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{4}+\sqrt {-c^{2} x^{2}+1}\, x^{2}}d x \right ) a b x +3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{6}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{4}+\sqrt {-c^{2} x^{2}+1}\, x^{2}}d x \right ) b^{2} c^{2} x^{3}-3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{6}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{4}+\sqrt {-c^{2} x^{2}+1}\, x^{2}}d x \right ) b^{2} x +8 a^{2} c^{4} x^{4}-12 a^{2} c^{2} x^{2}+3 a^{2}}{3 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} x \left (c^{2} x^{2}-1\right )} \] Input:
int((a+b*acos(c*x))^2/x^2/(-c^2*d*x^2+d)^(5/2),x)
Output:
(6*sqrt( - c**2*x**2 + 1)*int(acos(c*x)/(sqrt( - c**2*x**2 + 1)*c**4*x**6 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**4 + sqrt( - c**2*x**2 + 1)*x**2),x)*a*b *c**2*x**3 - 6*sqrt( - c**2*x**2 + 1)*int(acos(c*x)/(sqrt( - c**2*x**2 + 1 )*c**4*x**6 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**4 + sqrt( - c**2*x**2 + 1)* x**2),x)*a*b*x + 3*sqrt( - c**2*x**2 + 1)*int(acos(c*x)**2/(sqrt( - c**2*x **2 + 1)*c**4*x**6 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**4 + sqrt( - c**2*x** 2 + 1)*x**2),x)*b**2*c**2*x**3 - 3*sqrt( - c**2*x**2 + 1)*int(acos(c*x)**2 /(sqrt( - c**2*x**2 + 1)*c**4*x**6 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**4 + sqrt( - c**2*x**2 + 1)*x**2),x)*b**2*x + 8*a**2*c**4*x**4 - 12*a**2*c**2*x **2 + 3*a**2)/(3*sqrt(d)*sqrt( - c**2*x**2 + 1)*d**2*x*(c**2*x**2 - 1))