Integrand size = 29, antiderivative size = 728 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^3} \, dx=\frac {40}{9} b^2 c^2 d^2 \sqrt {d-c^2 d x^2}+\frac {5 a b c^3 d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}+\frac {2}{27} b^2 c^2 d \left (d-c^2 d x^2\right )^{3/2}+\frac {5 b^2 c^3 d^2 x \sqrt {d-c^2 d x^2} \arcsin (c x)}{\sqrt {1-c^2 x^2}}-\frac {b c d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{9 \sqrt {1-c^2 x^2}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{2 x^2}+\frac {5 c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {b^2 c^2 d^2 \sqrt {d-c^2 d x^2} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{\sqrt {1-c^2 x^2}}-\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {5 b^2 c^2 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {5 b^2 c^2 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}} \] Output:
40/9*b^2*c^2*d^2*(-c^2*d*x^2+d)^(1/2)+5*a*b*c^3*d^2*x*(-c^2*d*x^2+d)^(1/2) /(-c^2*x^2+1)^(1/2)+2/27*b^2*c^2*d*(-c^2*d*x^2+d)^(3/2)+5*b^2*c^3*d^2*x*(- c^2*d*x^2+d)^(1/2)*arcsin(c*x)/(-c^2*x^2+1)^(1/2)-b*c*d^2*(-c^2*d*x^2+d)^( 1/2)*(a+b*arcsin(c*x))/x/(-c^2*x^2+1)^(1/2)-1/3*b*c^3*d^2*x*(-c^2*d*x^2+d) ^(1/2)*(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2)-2/9*b*c^5*d^2*x^3*(-c^2*d*x^2+ d)^(1/2)*(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2)-5/2*c^2*d^2*(-c^2*d*x^2+d)^( 1/2)*(a+b*arcsin(c*x))^2-5/6*c^2*d*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^ 2-1/2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2/x^2+5*c^2*d^2*(-c^2*d*x^2+d )^(1/2)*(a+b*arcsin(c*x))^2*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))/(-c^2*x^2+1) ^(1/2)-b^2*c^2*d^2*(-c^2*d*x^2+d)^(1/2)*arctanh((-c^2*x^2+1)^(1/2))/(-c^2* x^2+1)^(1/2)-5*I*b*c^2*d^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))*polylog( 2,-I*c*x-(-c^2*x^2+1)^(1/2))/(-c^2*x^2+1)^(1/2)+5*I*b*c^2*d^2*(-c^2*d*x^2+ d)^(1/2)*(a+b*arcsin(c*x))*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))/(-c^2*x^2+1 )^(1/2)+5*b^2*c^2*d^2*(-c^2*d*x^2+d)^(1/2)*polylog(3,-I*c*x-(-c^2*x^2+1)^( 1/2))/(-c^2*x^2+1)^(1/2)-5*b^2*c^2*d^2*(-c^2*d*x^2+d)^(1/2)*polylog(3,I*c* x+(-c^2*x^2+1)^(1/2))/(-c^2*x^2+1)^(1/2)
Time = 7.16 (sec) , antiderivative size = 1073, normalized size of antiderivative = 1.47 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^3} \, dx =\text {Too large to display} \] Input:
Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/x^3,x]
Output:
Sqrt[-(d*(-1 + c^2*x^2))]*((-7*a^2*c^2*d^2)/3 - (a^2*d^2)/(2*x^2) + (a^2*c ^4*d^2*x^2)/3) - (5*a^2*c^2*d^(5/2)*Log[x])/2 + (5*a^2*c^2*d^(5/2)*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/2 - 4*a*b*c^2*d^2*Sqrt[d*(1 - c^2*x^2 )]*(-((c*x)/Sqrt[1 - c^2*x^2]) + ArcSin[c*x] + (ArcSin[c*x]*(Log[1 - E^(I* ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] + (I*(PolyL og[2, -E^(I*ArcSin[c*x])] - PolyLog[2, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x ^2]) - 2*b^2*c^2*d^2*Sqrt[d*(1 - c^2*x^2)]*(-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])]))/Sqrt[1 - c^2*x^2] + ((2*I)*ArcSin[c*x]*(Pol yLog[2, -E^(I*ArcSin[c*x])] - PolyLog[2, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2 *x^2] + (2*(-PolyLog[3, -E^(I*ArcSin[c*x])] + PolyLog[3, E^(I*ArcSin[c*x]) ]))/Sqrt[1 - c^2*x^2]) - (a*b*c^2*d^2*Sqrt[d*(1 - c^2*x^2)]*(-9*c*x + 9*Sq rt[1 - c^2*x^2]*ArcSin[c*x] + 3*ArcSin[c*x]*Cos[3*ArcSin[c*x]] - Sin[3*Arc Sin[c*x]]))/(18*Sqrt[1 - c^2*x^2]) - (b^2*c^2*d^2*Sqrt[d*(1 - c^2*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]) + (a*b*c^2*d^3*Sqrt[1 - c^2*x^2]*(-2*Cot[ArcSin[c*x]/2] - ArcSin[c *x]*Csc[ArcSin[c*x]/2]^2 - 4*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 4*Ar cSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - (4*I)*PolyLog[2, -E^(I*ArcSin[c*x]) ] + (4*I)*PolyLog[2, E^(I*ArcSin[c*x])] + ArcSin[c*x]*Sec[ArcSin[c*x]/2...
Time = 3.59 (sec) , antiderivative size = 558, normalized size of antiderivative = 0.77, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.759, Rules used = {5200, 5192, 27, 1578, 1192, 25, 1467, 2009, 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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^3} \, dx\) |
| \(\Big \downarrow \) 5200 |
| \(\displaystyle \frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{x^2}dx}{\sqrt {1-c^2 x^2}}-\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 5192 |
| \(\displaystyle -\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-b c \int -\frac {-c^4 x^4+6 c^2 x^2+3}{3 x \sqrt {1-c^2 x^2}}dx+\frac {1}{3} c^4 x^3 (a+b \arcsin (c x))-2 c^2 x (a+b \arcsin (c x))-\frac {a+b \arcsin (c x)}{x}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} b c \int \frac {-c^4 x^4+6 c^2 x^2+3}{x \sqrt {1-c^2 x^2}}dx+\frac {1}{3} c^4 x^3 (a+b \arcsin (c x))-2 c^2 x (a+b \arcsin (c x))-\frac {a+b \arcsin (c x)}{x}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle -\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{6} b c \int \frac {-c^4 x^4+6 c^2 x^2+3}{x^2 \sqrt {1-c^2 x^2}}dx^2+\frac {1}{3} c^4 x^3 (a+b \arcsin (c x))-2 c^2 x (a+b \arcsin (c x))-\frac {a+b \arcsin (c x)}{x}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 1192 |
| \(\displaystyle -\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {b \int -\frac {-c^4 x^8-4 c^4 x^4+8 c^4}{1-x^4}d\sqrt {1-c^2 x^2}}{3 c^3}+\frac {1}{3} c^4 x^3 (a+b \arcsin (c x))-2 c^2 x (a+b \arcsin (c x))-\frac {a+b \arcsin (c x)}{x}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b \int \frac {-c^4 x^8-4 c^4 x^4+8 c^4}{1-x^4}d\sqrt {1-c^2 x^2}}{3 c^3}+\frac {1}{3} c^4 x^3 (a+b \arcsin (c x))-2 c^2 x (a+b \arcsin (c x))-\frac {a+b \arcsin (c x)}{x}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 1467 |
| \(\displaystyle -\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b \int \left (x^4 c^4+\frac {3 c^4}{1-x^4}+5 c^4\right )d\sqrt {1-c^2 x^2}}{3 c^3}+\frac {1}{3} c^4 x^3 (a+b \arcsin (c x))-2 c^2 x (a+b \arcsin (c x))-\frac {a+b \arcsin (c x)}{x}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arcsin (c x))-2 c^2 x (a+b \arcsin (c x))-\frac {a+b \arcsin (c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (-\frac {2 b c d \sqrt {d-c^2 d x^2} \int \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx}{3 \sqrt {1-c^2 x^2}}+d \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x}dx+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arcsin (c x))-2 c^2 x (a+b \arcsin (c x))-\frac {a+b \arcsin (c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 5154 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x}dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-b c \int \frac {x \left (3-c^2 x^2\right )}{3 \sqrt {1-c^2 x^2}}dx-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arcsin (c x))-2 c^2 x (a+b \arcsin (c x))-\frac {a+b \arcsin (c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x}dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} b c \int \frac {x \left (3-c^2 x^2\right )}{\sqrt {1-c^2 x^2}}dx-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arcsin (c x))-2 c^2 x (a+b \arcsin (c x))-\frac {a+b \arcsin (c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 353 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x}dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} b c \int \frac {3-c^2 x^2}{\sqrt {1-c^2 x^2}}dx^2-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arcsin (c x))-2 c^2 x (a+b \arcsin (c x))-\frac {a+b \arcsin (c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x}dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} b c \int \left (\sqrt {1-c^2 x^2}+\frac {2}{\sqrt {1-c^2 x^2}}\right )dx^2-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arcsin (c x))-2 c^2 x (a+b \arcsin (c x))-\frac {a+b \arcsin (c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x}dx+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))-\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arcsin (c x))-2 c^2 x (a+b \arcsin (c x))-\frac {a+b \arcsin (c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 5198 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \left (-\frac {2 b c \sqrt {d-c^2 d x^2} \int (a+b \arcsin (c x))dx}{\sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}}dx}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))-\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arcsin (c x))-2 c^2 x (a+b \arcsin (c x))-\frac {a+b \arcsin (c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}}dx}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))-\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arcsin (c x))-2 c^2 x (a+b \arcsin (c x))-\frac {a+b \arcsin (c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 5218 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{c x}d\arcsin (c x)}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))-\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arcsin (c x))-2 c^2 x (a+b \arcsin (c x))-\frac {a+b \arcsin (c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \left (\frac {\sqrt {d-c^2 d x^2} \int (a+b \arcsin (c x))^2 \csc (\arcsin (c x))d\arcsin (c x)}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))-\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arcsin (c x))-2 c^2 x (a+b \arcsin (c x))-\frac {a+b \arcsin (c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \left (\frac {\sqrt {d-c^2 d x^2} \left (-2 b \int (a+b \arcsin (c x)) \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+2 b \int (a+b \arcsin (c x)) \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))-\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arcsin (c x))-2 c^2 x (a+b \arcsin (c x))-\frac {a+b \arcsin (c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \left (\frac {\sqrt {d-c^2 d x^2} \left (2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))-\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arcsin (c x))-2 c^2 x (a+b \arcsin (c x))-\frac {a+b \arcsin (c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \left (\frac {\sqrt {d-c^2 d x^2} \left (2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))-\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arcsin (c x))-2 c^2 x (a+b \arcsin (c x))-\frac {a+b \arcsin (c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \left (\frac {\sqrt {d-c^2 d x^2} \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2+2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )\right )\right )}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))-\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{3} c^4 x^3 (a+b \arcsin (c x))-2 c^2 x (a+b \arcsin (c x))-\frac {a+b \arcsin (c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {1-c^2 x^2}\right )}{3 c^3}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{2 x^2}\) |
Input:
Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/x^3,x]
Output:
-1/2*((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/x^2 + (b*c*d^2*Sqrt[d - c^2*d*x^2]*(-((a + b*ArcSin[c*x])/x) - 2*c^2*x*(a + b*ArcSin[c*x]) + (c^4 *x^3*(a + b*ArcSin[c*x]))/3 + (b*(-1/3*(c^4*x^6) - 5*c^4*Sqrt[1 - c^2*x^2] - 3*c^4*ArcTanh[Sqrt[1 - c^2*x^2]]))/(3*c^3)))/Sqrt[1 - c^2*x^2] - (5*c^2 *d*(((d - c^2*d*x^2)^(3/2)*(a + b*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*S qrt[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]))/Sqr t[1 - c^2*x^2] + (Sqrt[d - c^2*d*x^2]*(-2*(a + b*ArcSin[c*x])^2*ArcTanh[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])]) - 2*b*(I*(a + b*ArcSin[c*x])*PolyLo g[2, E^(I*ArcSin[c*x])] - b*PolyLog[3, E^(I*ArcSin[c*x])])))/Sqrt[1 - c^2* x^2])))/2
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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2/e^(n + 2*p + 1) Subst[Int[x^( 2*m + 1)*(e*f - d*g + g*x^2)^n*(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4)^p, x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m + 1/2]
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/2]
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_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Simp[ (a + b*ArcSin[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/Sqrt[1 - c ^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0 ] && IGtQ[p, 0]
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 + 1))), x] + (-Simp[2*e*(p/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(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} , x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[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*((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]
Time = 0.91 (sec) , antiderivative size = 1326, normalized size of antiderivative = 1.82
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1326\) |
| parts | \(\text {Expression too large to display}\) | \(1326\) |
Input:
int((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2/x^3,x,method=_RETURNVERBOSE)
Output:
a^2*(-1/2/d/x^2*(-c^2*d*x^2+d)^(7/2)-5/2*c^2*(1/5*(-c^2*d*x^2+d)^(5/2)+d*( 1/3*(-c^2*d*x^2+d)^(3/2)+d*((-c^2*d*x^2+d)^(1/2)-d^(1/2)*ln((2*d+2*d^(1/2) *(-c^2*d*x^2+d)^(1/2))/x)))))+b^2*(1/216*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4 -5*c^2*x^2-4*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+3*I*(-c^2*x^2+1)^(1/2)*x*c+1)*(6 *I*arcsin(c*x)+9*arcsin(c*x)^2-2)*c^2*d^2/(c^2*x^2-1)-9/8*(-d*(c^2*x^2-1)) ^(1/2)*(c^2*x^2-I*c*x*(-c^2*x^2+1)^(1/2)-1)*(arcsin(c*x)^2-2+2*I*arcsin(c* x))*c^2*d^2/(c^2*x^2-1)-9/8*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*c *x+c^2*x^2-1)*(arcsin(c*x)^2-2-2*I*arcsin(c*x))*c^2*d^2/(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)*c^2*d^ 2/(c^2*x^2-1)-1/2*d^2*(c^2*x^2*arcsin(c*x)-2*c*x*(-c^2*x^2+1)^(1/2)-arcsin (c*x))*arcsin(c*x)*(-d*(c^2*x^2-1))^(1/2)/(c^2*x^2-1)/x^2-1/2*(-d*(c^2*x^2 -1))^(1/2)*(-c^2*x^2+1)^(1/2)*(5*arcsin(c*x)^2*ln(1+I*c*x+(-c^2*x^2+1)^(1/ 2))-5*arcsin(c*x)^2*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-10*I*arcsin(c*x)*polylo g(2,-I*c*x-(-c^2*x^2+1)^(1/2))+10*I*arcsin(c*x)*polylog(2,I*c*x+(-c^2*x^2+ 1)^(1/2))-4*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))+10*polylog(3,-I*c*x-(-c^2*x^ 2+1)^(1/2))-10*polylog(3,I*c*x+(-c^2*x^2+1)^(1/2)))*c^2*d^2/(c^2*x^2-1))+2 *a*b*(1/72*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2-4*I*(-c^2*x^2+1)^(1 /2)*x^3*c^3+3*I*(-c^2*x^2+1)^(1/2)*x*c+1)*(I+3*arcsin(c*x))*c^2*d^2/(c^2*x ^2-1)-9/8*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*c*x*(-c^2*x^2+1)^(1/2)-1)*(...
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^3} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \] Input:
integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2/x^3,x, algorithm="frica s")
Output:
integral((a^2*c^4*d^2*x^4 - 2*a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^4*d^2*x^4 - 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arcsin(c*x)^2 + 2*(a*b*c^4*d^2*x^4 - 2*a*b *c^2*d^2*x^2 + a*b*d^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d)/x^3, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^3} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{3}}\, dx \] Input:
integrate((-c**2*d*x**2+d)**(5/2)*(a+b*asin(c*x))**2/x**3,x)
Output:
Integral((-d*(c*x - 1)*(c*x + 1))**(5/2)*(a + b*asin(c*x))**2/x**3, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^3} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \] Input:
integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2/x^3,x, algorithm="maxim a")
Output:
1/6*(15*c^2*d^(5/2)*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x) ) - 3*(-c^2*d*x^2 + d)^(5/2)*c^2 - 5*(-c^2*d*x^2 + d)^(3/2)*c^2*d - 15*sqr t(-c^2*d*x^2 + d)*c^2*d^2 - 3*(-c^2*d*x^2 + d)^(7/2)/(d*x^2))*a^2 + sqrt(d )*integrate(((b^2*c^4*d^2*x^4 - 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*c^4*d^2*x^4 - 2*a*b*c^2*d^2*x^2 + a*b*d^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(- c*x + 1)/x^3, x)
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2/x^3,x, algorithm="giac" )
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^3} \,d x \] Input:
int(((a + b*asin(c*x))^2*(d - c^2*d*x^2)^(5/2))/x^3,x)
Output:
int(((a + b*asin(c*x))^2*(d - c^2*d*x^2)^(5/2))/x^3, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{x^3} \, dx=\frac {\sqrt {d}\, d^{2} \left (8 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{4} x^{4}-56 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-12 \sqrt {-c^{2} x^{2}+1}\, a^{2}+48 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )}{x^{3}}d x \right ) a b \,x^{2}-96 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )}{x}d x \right ) a b \,c^{2} x^{2}+24 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2}}{x^{3}}d x \right ) b^{2} x^{2}-48 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2}}{x}d x \right ) b^{2} c^{2} x^{2}+48 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x d x \right ) a b \,c^{4} x^{2}+24 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} x d x \right ) b^{2} c^{4} x^{2}-60 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right )\right ) a^{2} c^{2} x^{2}+65 a^{2} c^{2} x^{2}\right )}{24 x^{2}} \] Input:
int((-c^2*d*x^2+d)^(5/2)*(a+b*asin(c*x))^2/x^3,x)
Output:
(sqrt(d)*d**2*(8*sqrt( - c**2*x**2 + 1)*a**2*c**4*x**4 - 56*sqrt( - c**2*x **2 + 1)*a**2*c**2*x**2 - 12*sqrt( - c**2*x**2 + 1)*a**2 + 48*int((sqrt( - c**2*x**2 + 1)*asin(c*x))/x**3,x)*a*b*x**2 - 96*int((sqrt( - c**2*x**2 + 1)*asin(c*x))/x,x)*a*b*c**2*x**2 + 24*int((sqrt( - c**2*x**2 + 1)*asin(c*x )**2)/x**3,x)*b**2*x**2 - 48*int((sqrt( - c**2*x**2 + 1)*asin(c*x)**2)/x,x )*b**2*c**2*x**2 + 48*int(sqrt( - c**2*x**2 + 1)*asin(c*x)*x,x)*a*b*c**4*x **2 + 24*int(sqrt( - c**2*x**2 + 1)*asin(c*x)**2*x,x)*b**2*c**4*x**2 - 60* log(tan(asin(c*x)/2))*a**2*c**2*x**2 + 65*a**2*c**2*x**2))/(24*x**2)