Integrand size = 29, antiderivative size = 333 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {(a+b \arcsin (c x))^2}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x (a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 i c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \text {arctanh}\left (e^{2 i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {4 b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {i b^2 c \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {i b^2 c \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}} \]
-(a+b*arcsin(c*x))^2/d/x/(-c^2*d*x^2+d)^(1/2)+2*c^2*x*(a+b*arcsin(c*x))^2/ d/(-c^2*d*x^2+d)^(1/2)-2*I*c*(a+b*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)/d/(-c^ 2*d*x^2+d)^(1/2)-4*b*c*(a+b*arcsin(c*x))*arctanh((I*c*x+(-c^2*x^2+1)^(1/2) )^2)*(-c^2*x^2+1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2)+4*b*c*(a+b*arcsin(c*x))*ln( 1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)*(-c^2*x^2+1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2)- I*b^2*c*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)*(-c^2*x^2+1)^(1/2)/d/(-c^ 2*d*x^2+d)^(1/2)-I*b^2*c*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)*(-c^2*x^2 +1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2)
Time = 1.18 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-a^2+2 a^2 c^2 x^2-2 a b \arcsin (c x)+4 a b c^2 x^2 \arcsin (c x)-b^2 \arcsin (c x)^2+2 b^2 c^2 x^2 \arcsin (c x)^2-2 i b^2 c x \sqrt {1-c^2 x^2} \arcsin (c x)^2+2 b^2 c x \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+2 b^2 c x \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1+e^{2 i \arcsin (c x)}\right )+2 a b c x \sqrt {1-c^2 x^2} \log (c x)+a b c x \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )-i b^2 c x \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )-i b^2 c x \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{d x \sqrt {d-c^2 d x^2}} \]
(-a^2 + 2*a^2*c^2*x^2 - 2*a*b*ArcSin[c*x] + 4*a*b*c^2*x^2*ArcSin[c*x] - b^ 2*ArcSin[c*x]^2 + 2*b^2*c^2*x^2*ArcSin[c*x]^2 - (2*I)*b^2*c*x*Sqrt[1 - c^2 *x^2]*ArcSin[c*x]^2 + 2*b^2*c*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - E^(( 2*I)*ArcSin[c*x])] + 2*b^2*c*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + E^((2 *I)*ArcSin[c*x])] + 2*a*b*c*x*Sqrt[1 - c^2*x^2]*Log[c*x] + a*b*c*x*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2] - I*b^2*c*x*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(( 2*I)*ArcSin[c*x])] - I*b^2*c*x*Sqrt[1 - c^2*x^2]*PolyLog[2, E^((2*I)*ArcSi n[c*x])])/(d*x*Sqrt[d - c^2*d*x^2])
Time = 1.83 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.83, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {5204, 5160, 5180, 3042, 4202, 2620, 2715, 2838, 5184, 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.
| \(\displaystyle \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 5204 |
| \(\displaystyle 2 c^2 \int \frac {(a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}}dx+\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}dx}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{d x \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5160 |
| \(\displaystyle 2 c^2 \left (\frac {x (a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2}dx}{d \sqrt {d-c^2 d x^2}}\right )+\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}dx}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{d x \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5180 |
| \(\displaystyle 2 c^2 \left (\frac {x (a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \int \frac {c x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{c d \sqrt {d-c^2 d x^2}}\right )+\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}dx}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{d x \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}dx}{d \sqrt {d-c^2 d x^2}}+2 c^2 \left (\frac {x (a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \int (a+b \arcsin (c x)) \tan (\arcsin (c x))d\arcsin (c x)}{c d \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \arcsin (c x))^2}{d x \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 4202 |
| \(\displaystyle 2 c^2 \left (\frac {x (a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1+e^{2 i \arcsin (c x)}}d\arcsin (c x)\right )}{c d \sqrt {d-c^2 d x^2}}\right )+\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}dx}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{d x \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}dx}{d \sqrt {d-c^2 d x^2}}+2 c^2 \left (\frac {x (a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (\frac {1}{2} i b \int \log \left (1+e^{2 i \arcsin (c x)}\right )d\arcsin (c x)-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )\right )}{c d \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \arcsin (c x))^2}{d x \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}dx}{d \sqrt {d-c^2 d x^2}}+2 c^2 \left (\frac {x (a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \log \left (1+e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )\right )}{c d \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \arcsin (c x))^2}{d x \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}dx}{d \sqrt {d-c^2 d x^2}}+2 c^2 \left (\frac {x (a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )\right )\right )}{c d \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \arcsin (c x))^2}{d x \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5184 |
| \(\displaystyle \frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{c x \sqrt {1-c^2 x^2}}d\arcsin (c x)}{d \sqrt {d-c^2 d x^2}}+2 c^2 \left (\frac {x (a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )\right )\right )}{c d \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \arcsin (c x))^2}{d x \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 4919 |
| \(\displaystyle \frac {4 b c \sqrt {1-c^2 x^2} \int (a+b \arcsin (c x)) \csc (2 \arcsin (c x))d\arcsin (c x)}{d \sqrt {d-c^2 d x^2}}+2 c^2 \left (\frac {x (a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )\right )\right )}{c d \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \arcsin (c x))^2}{d x \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 b c \sqrt {1-c^2 x^2} \int (a+b \arcsin (c x)) \csc (2 \arcsin (c x))d\arcsin (c x)}{d \sqrt {d-c^2 d x^2}}+2 c^2 \left (\frac {x (a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )\right )\right )}{c d \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \arcsin (c x))^2}{d x \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {4 b c \sqrt {1-c^2 x^2} \left (-\frac {1}{2} b \int \log \left (1-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)+\frac {1}{2} b \int \log \left (1+e^{2 i \arcsin (c x)}\right )d\arcsin (c x)-\left (\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )\right )}{d \sqrt {d-c^2 d x^2}}+2 c^2 \left (\frac {x (a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )\right )\right )}{c d \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \arcsin (c x))^2}{d x \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {4 b c \sqrt {1-c^2 x^2} \left (\frac {1}{4} i b \int e^{-2 i \arcsin (c x)} \log \left (1-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}-\frac {1}{4} i b \int e^{-2 i \arcsin (c x)} \log \left (1+e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}-\left (\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )\right )}{d \sqrt {d-c^2 d x^2}}+2 c^2 \left (\frac {x (a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )\right )\right )}{c d \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \arcsin (c x))^2}{d x \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {4 b c \sqrt {1-c^2 x^2} \left (-\left (\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )+\frac {1}{4} i b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )-\frac {1}{4} i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+2 c^2 \left (\frac {x (a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )\right )\right )}{c d \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \arcsin (c x))^2}{d x \sqrt {d-c^2 d x^2}}\) |
-((a + b*ArcSin[c*x])^2/(d*x*Sqrt[d - c^2*d*x^2])) + 2*c^2*((x*(a + b*ArcS in[c*x])^2)/(d*Sqrt[d - c^2*d*x^2]) - (2*b*Sqrt[1 - c^2*x^2]*(((I/2)*(a + b*ArcSin[c*x])^2)/b - (2*I)*((-1/2*I)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I) *ArcSin[c*x])] - (b*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/4)))/(c*d*Sqrt[d - c^2*d*x^2])) + (4*b*c*Sqrt[1 - c^2*x^2]*(-((a + b*ArcSin[c*x])*ArcTanh[E^ ((2*I)*ArcSin[c*x])]) + (I/4)*b*PolyLog[2, -E^((2*I)*ArcSin[c*x])] - (I/4) *b*PolyLog[2, E^((2*I)*ArcSin[c*x])]))/(d*Sqrt[d - c^2*d*x^2])
3.3.51.3.1 Defintions of rubi rules used
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_.) + (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*( e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt Q[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_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x _Symbol] :> Simp[x*((a + b*ArcSin[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*ArcSin[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_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[-e^(-1) Subst[Int[(a + b*x)^n*Tan[x], x], x, ArcSin[c*x ]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[1/d Subst[Int[(a + b*x)^n/(Cos[x]*Sin[x]), x], x, ArcSi n[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 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]
Time = 0.24 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.54
| method | result | size |
| default | \(a^{2} \left (-\frac {1}{d x \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 c^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}\right )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i c x \sqrt {-c^{2} x^{2}+1}+2 c^{2} x^{2}-1\right ) \arcsin \left (c x \right )^{2}}{\left (c^{2} x^{2}-1\right ) d^{2} x}+\frac {i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+4 \arcsin \left (c x \right )^{2}+2 \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) c}{\left (c^{2} x^{2}-1\right ) d^{2}}\right )+2 a b \left (\frac {4 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i c x \sqrt {-c^{2} x^{2}+1}+2 c^{2} x^{2}-1\right ) \arcsin \left (c x \right )}{\left (c^{2} x^{2}-1\right ) d^{2} x}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{4}-1\right ) c}{d^{2} \left (c^{2} x^{2}-1\right )}\right )\) | \(513\) |
| parts | \(a^{2} \left (-\frac {1}{d x \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 c^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}\right )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i c x \sqrt {-c^{2} x^{2}+1}+2 c^{2} x^{2}-1\right ) \arcsin \left (c x \right )^{2}}{\left (c^{2} x^{2}-1\right ) d^{2} x}+\frac {i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+4 \arcsin \left (c x \right )^{2}+2 \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) c}{\left (c^{2} x^{2}-1\right ) d^{2}}\right )+2 a b \left (\frac {4 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i c x \sqrt {-c^{2} x^{2}+1}+2 c^{2} x^{2}-1\right ) \arcsin \left (c x \right )}{\left (c^{2} x^{2}-1\right ) d^{2} x}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{4}-1\right ) c}{d^{2} \left (c^{2} x^{2}-1\right )}\right )\) | \(513\) |
a^2*(-1/d/x/(-c^2*d*x^2+d)^(1/2)+2*c^2/d*x/(-c^2*d*x^2+d)^(1/2))+b^2*(-(-d *(c^2*x^2-1))^(1/2)*(2*I*c*x*(-c^2*x^2+1)^(1/2)+2*c^2*x^2-1)*arcsin(c*x)^2 /(c^2*x^2-1)/d^2/x+I*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/(c^2*x^2-1) /d^2*(2*I*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+2*I*arcsin(c*x)*ln(1+ (I*c*x+(-c^2*x^2+1)^(1/2))^2)+2*I*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2 ))+4*arcsin(c*x)^2+2*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))+polylog(2,-(I*c* x+(-c^2*x^2+1)^(1/2))^2)+2*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2)))*c)+2*a*b*( 4*I*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/(c^2*x^2-1)/d^2*arcsin(c*x)* c-(-d*(c^2*x^2-1))^(1/2)*(2*I*c*x*(-c^2*x^2+1)^(1/2)+2*c^2*x^2-1)*arcsin(c *x)/(c^2*x^2-1)/d^2/x-(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1 )/d^2*ln((I*c*x+(-c^2*x^2+1)^(1/2))^4-1)*c)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
integral(sqrt(-c^2*d*x^2 + d)*(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2 )/(c^4*d^2*x^6 - 2*c^2*d^2*x^4 + d^2*x^2), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{2} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
a*b*c*(log(c*x + 1)/d^(3/2) + log(c*x - 1)/d^(3/2) + 2*log(x)/d^(3/2)) + 2 *(2*c^2*x/(sqrt(-c^2*d*x^2 + d)*d) - 1/(sqrt(-c^2*d*x^2 + d)*d*x))*a*b*arc sin(c*x) + (2*c^2*x/(sqrt(-c^2*d*x^2 + d)*d) - 1/(sqrt(-c^2*d*x^2 + d)*d*x ))*a^2 - b^2*integrate(arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2/((c^2* d*x^4 - d*x^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)), x)/sqrt(d)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]