Integrand size = 27, antiderivative size = 433 \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {b c}{4 d^2 x \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {5 b c^3 x}{12 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {3 b c \sqrt {1-c^2 x^2}}{4 d^2 x \sqrt {d-c^2 d x^2}}+\frac {5 c^2 (a+b \arcsin (c x))}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \arcsin (c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 (a+b \arcsin (c x))}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {13 b c^2 \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 i b c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}} \]
5/6*c^2*(a+b*arcsin(c*x))/d/(-c^2*d*x^2+d)^(3/2)+1/2*(-a-b*arcsin(c*x))/d/ x^2/(-c^2*d*x^2+d)^(3/2)+5/2*c^2*(a+b*arcsin(c*x))/d^2/(-c^2*d*x^2+d)^(1/2 )+1/4*b*c/d^2/x/(-c^2*x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)-5/12*b*c^3*x/d^2/( -c^2*x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)-3/4*b*c*(-c^2*x^2+1)^(1/2)/d^2/x/(- c^2*d*x^2+d)^(1/2)-5*c^2*(a+b*arcsin(c*x))*arctanh(I*c*x+(-c^2*x^2+1)^(1/2 ))*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)-13/6*b*c^2*arctanh(c*x)*(-c ^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)+5/2*I*b*c^2*polylog(2,-I*c*x-(-c^ 2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)-5/2*I*b*c^2*po lylog(2,I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1 /2)
Time = 7.35 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.24 \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\sqrt {-d \left (-1+c^2 x^2\right )} \left (-\frac {a}{2 d^3 x^2}+\frac {a c^2}{3 d^3 \left (-1+c^2 x^2\right )^2}-\frac {2 a c^2}{d^3 \left (-1+c^2 x^2\right )}\right )+\frac {5 a c^2 \log (x)}{2 d^{5/2}}-\frac {5 a c^2 \log \left (d+\sqrt {d} \sqrt {-d \left (-1+c^2 x^2\right )}\right )}{2 d^{5/2}}+\frac {b c^2 \sqrt {1-c^2 x^2} \left (-\frac {2 (-1+\arcsin (c x))}{-1+c x}+52 \arcsin (c x)-6 \cot \left (\frac {1}{2} \arcsin (c x)\right )-3 \arcsin (c x) \csc ^2\left (\frac {1}{2} \arcsin (c x)\right )+60 \arcsin (c x) \left (\log \left (1-e^{i \arcsin (c x)}\right )-\log \left (1+e^{i \arcsin (c x)}\right )\right )+52 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-52 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+60 i \left (\operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-\operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )+3 \arcsin (c x) \sec ^2\left (\frac {1}{2} \arcsin (c x)\right )+\frac {4 \arcsin (c x) \sin \left (\frac {1}{2} \arcsin (c x)\right )}{\left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )^3}+\frac {52 \arcsin (c x) \sin \left (\frac {1}{2} \arcsin (c x)\right )}{\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )}-\frac {4 \arcsin (c x) \sin \left (\frac {1}{2} \arcsin (c x)\right )}{\left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )^3}+\frac {2 (1+\arcsin (c x))}{\left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )^2}-\frac {52 \arcsin (c x) \sin \left (\frac {1}{2} \arcsin (c x)\right )}{\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )}-6 \tan \left (\frac {1}{2} \arcsin (c x)\right )\right )}{24 d^2 \sqrt {d \left (1-c^2 x^2\right )}} \]
Sqrt[-(d*(-1 + c^2*x^2))]*(-1/2*a/(d^3*x^2) + (a*c^2)/(3*d^3*(-1 + c^2*x^2 )^2) - (2*a*c^2)/(d^3*(-1 + c^2*x^2))) + (5*a*c^2*Log[x])/(2*d^(5/2)) - (5 *a*c^2*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/(2*d^(5/2)) + (b*c^2*Sq rt[1 - c^2*x^2]*((-2*(-1 + ArcSin[c*x]))/(-1 + c*x) + 52*ArcSin[c*x] - 6*C ot[ArcSin[c*x]/2] - 3*ArcSin[c*x]*Csc[ArcSin[c*x]/2]^2 + 60*ArcSin[c*x]*(L og[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]) + 52*Log[Cos[ArcSi n[c*x]/2] - Sin[ArcSin[c*x]/2]] - 52*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c *x]/2]] + (60*I)*(PolyLog[2, -E^(I*ArcSin[c*x])] - PolyLog[2, E^(I*ArcSin[ c*x])]) + 3*ArcSin[c*x]*Sec[ArcSin[c*x]/2]^2 + (4*ArcSin[c*x]*Sin[ArcSin[c *x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3 + (52*ArcSin[c*x]*Sin[ ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) - (4*ArcSin[c*x] *Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^3 + (2*(1 + ArcSin[c*x]))/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2 - (52*ArcSin[c* x]*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) - 6*Tan[A rcSin[c*x]/2]))/(24*d^2*Sqrt[d*(1 - c^2*x^2)])
Time = 1.43 (sec) , antiderivative size = 375, normalized size of antiderivative = 0.87, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {5204, 253, 264, 219, 5208, 215, 219, 5208, 219, 5218, 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.
| \(\displaystyle \int \frac {a+b \arcsin (c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5204 |
| \(\displaystyle \frac {5}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^{5/2}}dx+\frac {b c \sqrt {1-c^2 x^2} \int \frac {1}{x^2 \left (1-c^2 x^2\right )^2}dx}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {a+b \arcsin (c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {5}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^{5/2}}dx+\frac {b c \sqrt {1-c^2 x^2} \left (\frac {3}{2} \int \frac {1}{x^2 \left (1-c^2 x^2\right )}dx+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {a+b \arcsin (c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {5}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^{5/2}}dx+\frac {b c \sqrt {1-c^2 x^2} \left (\frac {3}{2} \left (c^2 \int \frac {1}{1-c^2 x^2}dx-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {a+b \arcsin (c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^{5/2}}dx-\frac {a+b \arcsin (c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {b c \sqrt {1-c^2 x^2} \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5208 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}-\frac {b c \sqrt {1-c^2 x^2} \int \frac {1}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {a+b \arcsin (c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {b c \sqrt {1-c^2 x^2} \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}-\frac {b c \sqrt {1-c^2 x^2} \left (\frac {1}{2} \int \frac {1}{1-c^2 x^2}dx+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )-\frac {a+b \arcsin (c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {b c \sqrt {1-c^2 x^2} \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {a+b \arcsin (c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {1-c^2 x^2} \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\right )-\frac {a+b \arcsin (c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {b c \sqrt {1-c^2 x^2} \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5208 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\frac {\int \frac {a+b \arcsin (c x)}{x \sqrt {d-c^2 d x^2}}dx}{d}-\frac {b c \sqrt {1-c^2 x^2} \int \frac {1}{1-c^2 x^2}dx}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {a+b \arcsin (c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {1-c^2 x^2} \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\right )-\frac {a+b \arcsin (c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {b c \sqrt {1-c^2 x^2} \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\frac {\int \frac {a+b \arcsin (c x)}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {a+b \arcsin (c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {1-c^2 x^2} \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\right )-\frac {a+b \arcsin (c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {b c \sqrt {1-c^2 x^2} \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5218 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\frac {\sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{c x}d\arcsin (c x)}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {a+b \arcsin (c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {1-c^2 x^2} \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\right )-\frac {a+b \arcsin (c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {b c \sqrt {1-c^2 x^2} \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\frac {\sqrt {1-c^2 x^2} \int (a+b \arcsin (c x)) \csc (\arcsin (c x))d\arcsin (c x)}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {a+b \arcsin (c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {1-c^2 x^2} \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\right )-\frac {a+b \arcsin (c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {b c \sqrt {1-c^2 x^2} \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\frac {\sqrt {1-c^2 x^2} \left (-b \int \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {a+b \arcsin (c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {1-c^2 x^2} \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\right )-\frac {a+b \arcsin (c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {b c \sqrt {1-c^2 x^2} \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\frac {\sqrt {1-c^2 x^2} \left (i b \int e^{-i \arcsin (c x)} \log \left (1-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {a+b \arcsin (c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {1-c^2 x^2} \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\right )-\frac {a+b \arcsin (c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {b c \sqrt {1-c^2 x^2} \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {5}{2} c^2 \left (\frac {\frac {\sqrt {1-c^2 x^2} \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {a+b \arcsin (c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {1-c^2 x^2} \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\right )-\frac {a+b \arcsin (c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {b c \sqrt {1-c^2 x^2} \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\) |
-1/2*(a + b*ArcSin[c*x])/(d*x^2*(d - c^2*d*x^2)^(3/2)) + (b*c*Sqrt[1 - c^2 *x^2]*(1/(2*x*(1 - c^2*x^2)) + (3*(-x^(-1) + c*ArcTanh[c*x]))/2))/(2*d^2*S qrt[d - c^2*d*x^2]) + (5*c^2*((a + b*ArcSin[c*x])/(3*d*(d - c^2*d*x^2)^(3/ 2)) - (b*c*Sqrt[1 - c^2*x^2]*(x/(2*(1 - c^2*x^2)) + ArcTanh[c*x]/(2*c)))/( 3*d^2*Sqrt[d - c^2*d*x^2]) + ((a + b*ArcSin[c*x])/(d*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[1 - c^2*x^2]*ArcTanh[c*x])/(d*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c ^2*x^2]*(-2*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])] + I*b*PolyLog[2 , -E^(I*ArcSin[c*x])] - I*b*PolyLog[2, E^(I*ArcSin[c*x])]))/(d*Sqrt[d - c^ 2*d*x^2]))/d))/2
3.2.38.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
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_.) + 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 + 1)*((a + b *ArcSin[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*ArcSin[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*ArcSin[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_.) + 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 + 1)*((a + b*ArcSin[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*ArcSin[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*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] && LtQ[p, -1] && !G tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 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]
Time = 0.27 (sec) , antiderivative size = 605, normalized size of antiderivative = 1.40
| method | result | size |
| default | \(-\frac {a}{2 d \,x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a \,c^{2}}{6 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a \,c^{2}}{2 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {5 a \,c^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{2 d^{\frac {5}{2}}}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (-5 i x^{3} c^{3}+15 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{6} x^{6}+15 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{6} x^{6}+26 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{6} x^{6}+3 i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )+15 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{2} c^{2}-20 i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{2} x^{2}-30 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}-30 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}-52 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+2 i x^{5} c^{5}+15 i \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+3 i c x +15 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}+15 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}+26 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-30 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{4} c^{4}+15 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{6} c^{6}\right )}{6 d^{3} \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) x^{2}}\) | \(605\) |
| parts | \(-\frac {a}{2 d \,x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a \,c^{2}}{6 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a \,c^{2}}{2 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {5 a \,c^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{2 d^{\frac {5}{2}}}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (-5 i x^{3} c^{3}+15 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{6} x^{6}+15 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{6} x^{6}+26 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{6} x^{6}+3 i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )+15 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{2} c^{2}-20 i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{2} x^{2}-30 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}-30 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}-52 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+2 i x^{5} c^{5}+15 i \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+3 i c x +15 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}+15 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}+26 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-30 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{4} c^{4}+15 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{6} c^{6}\right )}{6 d^{3} \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) x^{2}}\) | \(605\) |
-1/2*a/d/x^2/(-c^2*d*x^2+d)^(3/2)+5/6*a*c^2/d/(-c^2*d*x^2+d)^(3/2)+5/2*a*c ^2/d^2/(-c^2*d*x^2+d)^(1/2)-5/2*a*c^2/d^(5/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^ 2+d)^(1/2))/x)-1/6*I*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*(-5*I*x^3 *c^3+15*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))*c^6*x^6+15*dilog(I*c*x+(-c^2*x^2 +1)^(1/2))*c^6*x^6+26*arctan(I*c*x+(-c^2*x^2+1)^(1/2))*c^6*x^6+3*I*(-c^2*x ^2+1)^(1/2)*arcsin(c*x)+15*I*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))*x^ 2*c^2-20*I*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^2*c^2-30*dilog(1+I*c*x+(-c^2*x ^2+1)^(1/2))*c^4*x^4-30*dilog(I*c*x+(-c^2*x^2+1)^(1/2))*c^4*x^4-52*arctan( I*c*x+(-c^2*x^2+1)^(1/2))*c^4*x^4+2*I*x^5*c^5+15*I*arcsin(c*x)*(-c^2*x^2+1 )^(1/2)*x^4*c^4+3*I*c*x+15*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))*c^2*x^2+15*di log(I*c*x+(-c^2*x^2+1)^(1/2))*c^2*x^2+26*arctan(I*c*x+(-c^2*x^2+1)^(1/2))* c^2*x^2-30*I*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))*x^4*c^4+15*I*arcsi n(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))*x^6*c^6)/d^3/(c^6*x^6-3*c^4*x^4+3*c^ 2*x^2-1)/x^2
\[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
integral(-sqrt(-c^2*d*x^2 + d)*(b*arcsin(c*x) + a)/(c^6*d^3*x^9 - 3*c^4*d^ 3*x^7 + 3*c^2*d^3*x^5 - d^3*x^3), x)
\[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {a + b \operatorname {asin}{\left (c x \right )}}{x^{3} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
-1/6*a*(15*c^2*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x))/d^( 5/2) - 15*c^2/(sqrt(-c^2*d*x^2 + d)*d^2) - 5*c^2/((-c^2*d*x^2 + d)^(3/2)*d ) + 3/((-c^2*d*x^2 + d)^(3/2)*d*x^2)) + b*integrate(arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/((c^4*d^2*x^7 - 2*c^2*d^2*x^5 + d^2*x^3)*sqrt(c*x + 1 )*sqrt(-c*x + 1)), x)/sqrt(d)
\[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
Timed out. \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{x^3\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]