Integrand size = 29, antiderivative size = 545 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x} \, dx=-\frac {22}{9} b^2 d \sqrt {d-c^2 d x^2}-\frac {2 a b c d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {2}{27} b^2 d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}-\frac {2 b^2 c d x \sqrt {d-c^2 d x^2} \arcsin (c x)}{\sqrt {1-c^2 x^2}}-\frac {2 b c d x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{9 \sqrt {1-c^2 x^2}}+d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 i b d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 i b d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 b^2 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}} \]
1/3*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2-22/9*b^2*d*(-c^2*d*x^2+d)^(1/ 2)-2/27*b^2*d*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)+d*(a+b*arcsin(c*x))^2*(-c^ 2*d*x^2+d)^(1/2)-2*a*b*c*d*x*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-2*b^2 *c*d*x*arcsin(c*x)*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-2/3*b*c*d*x*(a+ b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+2/9*b*c^3*d*x^3*(a+ b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-2*d*(a+b*arcsin(c*x ))^2*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^( 1/2)+2*I*b*d*(a+b*arcsin(c*x))*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))*(-c^2* d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-2*I*b*d*(a+b*arcsin(c*x))*polylog(2,I*c* x+(-c^2*x^2+1)^(1/2))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-2*b^2*d*poly log(3,-I*c*x-(-c^2*x^2+1)^(1/2))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+2 *b^2*d*polylog(3,I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+ 1)^(1/2)
Time = 2.14 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.06 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x} \, dx=-\frac {1}{3} a^2 d \left (-4+c^2 x^2\right ) \sqrt {d-c^2 d x^2}+a^2 d^{3/2} \log (c x)-a^2 d^{3/2} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {2 a b d \sqrt {d-c^2 d x^2} \left (-c x+\sqrt {1-c^2 x^2} \arcsin (c x)+\arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )-\arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )+i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {b^2 d \sqrt {d-c^2 d x^2} \left (2 \sqrt {1-c^2 x^2}+2 c x \arcsin (c x)-\sqrt {1-c^2 x^2} \arcsin (c x)^2-\arcsin (c x)^2 \left (\log \left (1-e^{i \arcsin (c x)}\right )-\log \left (1+e^{i \arcsin (c x)}\right )\right )-2 i \arcsin (c x) \left (\operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-\operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )+2 \left (\operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )-\operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {a b d \sqrt {d-c^2 d x^2} \left (9 c x-3 \arcsin (c x) \left (3 \sqrt {1-c^2 x^2}+\cos (3 \arcsin (c x))\right )+\sin (3 \arcsin (c x))\right )}{18 \sqrt {1-c^2 x^2}}+\frac {b^2 d \sqrt {d-c^2 d x^2} \left (27 \sqrt {1-c^2 x^2} \left (-2+\arcsin (c x)^2\right )+\left (-2+9 \arcsin (c x)^2\right ) \cos (3 \arcsin (c x))-6 \arcsin (c x) (9 c x+\sin (3 \arcsin (c x)))\right )}{108 \sqrt {1-c^2 x^2}} \]
-1/3*(a^2*d*(-4 + c^2*x^2)*Sqrt[d - c^2*d*x^2]) + a^2*d^(3/2)*Log[c*x] - a ^2*d^(3/2)*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (2*a*b*d*Sqrt[d - c^2*d* x^2]*(-(c*x) + Sqrt[1 - c^2*x^2]*ArcSin[c*x] + ArcSin[c*x]*Log[1 - E^(I*Ar cSin[c*x])] - ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + I*PolyLog[2, -E^(I* ArcSin[c*x])] - I*PolyLog[2, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] - (b^2 *d*Sqrt[d - c^2*d*x^2]*(2*Sqrt[1 - c^2*x^2] + 2*c*x*ArcSin[c*x] - Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2 - ArcSin[c*x]^2*(Log[1 - E^(I*ArcSin[c*x])] - Log[ 1 + E^(I*ArcSin[c*x])]) - (2*I)*ArcSin[c*x]*(PolyLog[2, -E^(I*ArcSin[c*x]) ] - PolyLog[2, E^(I*ArcSin[c*x])]) + 2*(PolyLog[3, -E^(I*ArcSin[c*x])] - P olyLog[3, E^(I*ArcSin[c*x])])))/Sqrt[1 - c^2*x^2] - (a*b*d*Sqrt[d - c^2*d* x^2]*(9*c*x - 3*ArcSin[c*x]*(3*Sqrt[1 - c^2*x^2] + Cos[3*ArcSin[c*x]]) + S in[3*ArcSin[c*x]]))/(18*Sqrt[1 - c^2*x^2]) + (b^2*d*Sqrt[d - c^2*d*x^2]*(2 7*Sqrt[1 - c^2*x^2]*(-2 + ArcSin[c*x]^2) + (-2 + 9*ArcSin[c*x]^2)*Cos[3*Ar cSin[c*x]] - 6*ArcSin[c*x]*(9*c*x + Sin[3*ArcSin[c*x]])))/(108*Sqrt[1 - c^ 2*x^2])
Time = 1.77 (sec) , antiderivative size = 378, normalized size of antiderivative = 0.69, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {5202, 5154, 27, 353, 53, 2009, 5198, 2009, 5218, 3042, 4671, 3011, 2720, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x} \, dx\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle -\frac {2 b c d \sqrt {d-c^2 d x^2} \int \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx}{3 \sqrt {1-c^2 x^2}}+d \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x}dx+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5154 |
| \(\displaystyle d \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x}dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-b c \int \frac {x \left (3-c^2 x^2\right )}{3 \sqrt {1-c^2 x^2}}dx-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x}dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} b c \int \frac {x \left (3-c^2 x^2\right )}{\sqrt {1-c^2 x^2}}dx-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 353 |
| \(\displaystyle d \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x}dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} b c \int \frac {3-c^2 x^2}{\sqrt {1-c^2 x^2}}dx^2-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle d \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x}dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} b c \int \left (\sqrt {1-c^2 x^2}+\frac {2}{\sqrt {1-c^2 x^2}}\right )dx^2-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle d \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x}dx+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))-\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5198 |
| \(\displaystyle d \left (-\frac {2 b c \sqrt {d-c^2 d x^2} \int (a+b \arcsin (c x))dx}{\sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}}dx}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))-\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}}dx}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))-\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5218 |
| \(\displaystyle d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{c x}d\arcsin (c x)}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))-\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle d \left (\frac {\sqrt {d-c^2 d x^2} \int (a+b \arcsin (c x))^2 \csc (\arcsin (c x))d\arcsin (c x)}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))-\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle d \left (\frac {\sqrt {d-c^2 d x^2} \left (-2 b \int (a+b \arcsin (c x)) \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+2 b \int (a+b \arcsin (c x)) \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))-\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle d \left (\frac {\sqrt {d-c^2 d x^2} \left (2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))-\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle d \left (\frac {\sqrt {d-c^2 d x^2} \left (2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))-\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle d \left (\frac {\sqrt {d-c^2 d x^2} \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2+2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )\right )\right )}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))-\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\) |
((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/3 - (2*b*c*d*Sqrt[d - c^2*d* x^2]*(-1/6*(b*c*((-4*Sqrt[1 - c^2*x^2])/c^2 - (2*(1 - c^2*x^2)^(3/2))/(3*c ^2))) + x*(a + b*ArcSin[c*x]) - (c^2*x^3*(a + b*ArcSin[c*x]))/3))/(3*Sqrt[ 1 - c^2*x^2]) + d*(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 - (2*b*c*Sqrt [d - c^2*d*x^2]*(a*x + (b*Sqrt[1 - c^2*x^2])/c + b*x*ArcSin[c*x]))/Sqrt[1 - c^2*x^2] + (Sqrt[d - c^2*d*x^2]*(-2*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*A rcSin[c*x])] + 2*b*(I*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])] - b*PolyLog[3, -E^(I*ArcSin[c*x])]) - 2*b*(I*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*ArcSin[c*x])] - b*PolyLog[3, E^(I*ArcSin[c*x])])))/Sqrt[1 - c^2*x^2] )
3.3.22.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_.)*((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]
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]]]
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]
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_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbo l] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcSin[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]
Int[((a_.) + ArcSin[(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*ArcS in[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*ArcSin[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*ArcSin[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])
Int[((a_.) + ArcSin[(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*ArcS in[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*ArcSin[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*ArcSin[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]
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)* (x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e* x^2]] Subst[Int[(a + b*x)^n*Sin[x]^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a , b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1071 vs. \(2 (529 ) = 1058\).
Time = 0.28 (sec) , antiderivative size = 1072, normalized size of antiderivative = 1.97
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1072\) |
| parts | \(\text {Expression too large to display}\) | \(1072\) |
1/3*(-c^2*d*x^2+d)^(3/2)*a^2-a^2*d^(3/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^ (1/2))/x)+a^2*d*(-c^2*d*x^2+d)^(1/2)+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^3*x^3*(-c^2*x^2+1)^(1/2)+3*I*(-c^2*x^2+1)^(1/2)*x *c+1)*(6*I*arcsin(c*x)+9*arcsin(c*x)^2-2)*d/(c^2*x^2-1)+5/8*(-d*(c^2*x^2-1 ))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*(arcsin(c*x)^2-2+2*I*arcsin( c*x))*d/(c^2*x^2-1)+5/8*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c ^2*x^2-1)*(arcsin(c*x)^2-2-2*I*arcsin(c*x))*d/(c^2*x^2-1)-1/216*(-d*(c^2*x ^2-1))^(1/2)*(4*I*c^3*x^3*(-c^2*x^2+1)^(1/2)+4*c^4*x^4-3*I*(-c^2*x^2+1)^(1 /2)*x*c-5*c^2*x^2+1)*(-6*I*arcsin(c*x)+9*arcsin(c*x)^2-2)*d/(c^2*x^2-1)+(- d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*(arcsin(c*x)^2*ln(1+I* c*x+(-c^2*x^2+1)^(1/2))-arcsin(c*x)^2*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-2*I*a rcsin(c*x)*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))+2*I*arcsin(c*x)*polylog(2, I*c*x+(-c^2*x^2+1)^(1/2))+2*polylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))-2*polylog (3,I*c*x+(-c^2*x^2+1)^(1/2)))*d)+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^3*x^3*(-c^2*x^2+1)^(1/2)+3*I*(-c^2*x^2+1)^(1/2)*x*c+ 1)*(I+3*arcsin(c*x))*d/(c^2*x^2-1)+5/8*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*( -c^2*x^2+1)^(1/2)*x*c-1)*(arcsin(c*x)+I)*d/(c^2*x^2-1)+5/8*(-d*(c^2*x^2-1) )^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(arcsin(c*x)-I)*d/(c^2*x^2-1) -1/72*(-d*(c^2*x^2-1))^(1/2)*(4*I*c^3*x^3*(-c^2*x^2+1)^(1/2)+4*c^4*x^4-3*I *(-c^2*x^2+1)^(1/2)*x*c-5*c^2*x^2+1)*(-I+3*arcsin(c*x))*d/(c^2*x^2-1)-I...
\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
integral(-(a^2*c^2*d*x^2 - a^2*d + (b^2*c^2*d*x^2 - b^2*d)*arcsin(c*x)^2 + 2*(a*b*c^2*d*x^2 - a*b*d)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d)/x, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x}\, dx \]
\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
-1/3*(3*d^(3/2)*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x)) - (-c^2*d*x^2 + d)^(3/2) - 3*sqrt(-c^2*d*x^2 + d)*d)*a^2 - sqrt(d)*integrate (((b^2*c^2*d*x^2 - b^2*d)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2 *(a*b*c^2*d*x^2 - a*b*d)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt( c*x + 1)*sqrt(-c*x + 1)/x, x)
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x} \,d x \]