Integrand size = 29, antiderivative size = 398 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x^3} \, dx=-\frac {b c \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 x^2}+\frac {c^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {b^2 c^2 \sqrt {d-c^2 d x^2} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{\sqrt {1-c^2 x^2}}-\frac {i b c^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {i b c^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {b^2 c^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {b^2 c^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}} \]
-1/2*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2/x^2-b*c*(a+b*arcsin(c*x))*(- c^2*d*x^2+d)^(1/2)/x/(-c^2*x^2+1)^(1/2)+c^2*(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)-b^2*c^2*ar ctanh((-c^2*x^2+1)^(1/2))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-I*b*c^2* (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)+I*b*c^2*(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)+b^2*c^2*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)-b^2*c^2*polylo g(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 = 4.00 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x^3} \, dx=\frac {1}{8} \left (-\frac {4 a^2 \sqrt {d-c^2 d x^2}}{x^2}-4 a^2 c^2 \sqrt {d} \log (x)+4 a^2 c^2 \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {2 a b c^2 d \sqrt {1-c^2 x^2} \left (-2 \cot \left (\frac {1}{2} \arcsin (c x)\right )-\arcsin (c x) \csc ^2\left (\frac {1}{2} \arcsin (c x)\right )-4 \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )+4 \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )-4 i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )+4 i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )+\arcsin (c x) \sec ^2\left (\frac {1}{2} \arcsin (c x)\right )-2 \tan \left (\frac {1}{2} \arcsin (c x)\right )\right )}{\sqrt {d-c^2 d x^2}}+\frac {b^2 c^2 d \sqrt {1-c^2 x^2} \left (-4 \arcsin (c x) \cot \left (\frac {1}{2} \arcsin (c x)\right )-\arcsin (c x)^2 \csc ^2\left (\frac {1}{2} \arcsin (c x)\right )-4 \arcsin (c x)^2 \log \left (1-e^{i \arcsin (c x)}\right )+4 \arcsin (c x)^2 \log \left (1+e^{i \arcsin (c x)}\right )+8 \log \left (\tan \left (\frac {1}{2} \arcsin (c x)\right )\right )-8 i \arcsin (c x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )+8 i \arcsin (c x) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )+8 \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )-8 \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )+\arcsin (c x)^2 \sec ^2\left (\frac {1}{2} \arcsin (c x)\right )-4 \arcsin (c x) \tan \left (\frac {1}{2} \arcsin (c x)\right )\right )}{\sqrt {d-c^2 d x^2}}\right ) \]
((-4*a^2*Sqrt[d - c^2*d*x^2])/x^2 - 4*a^2*c^2*Sqrt[d]*Log[x] + 4*a^2*c^2*S qrt[d]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (2*a*b*c^2*d*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*ArcSin[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]^2 - 2*Tan[ArcSin[c*x]/2]))/Sqrt[d - c^2 *d*x^2] + (b^2*c^2*d*Sqrt[1 - c^2*x^2]*(-4*ArcSin[c*x]*Cot[ArcSin[c*x]/2] - ArcSin[c*x]^2*Csc[ArcSin[c*x]/2]^2 - 4*ArcSin[c*x]^2*Log[1 - E^(I*ArcSin [c*x])] + 4*ArcSin[c*x]^2*Log[1 + E^(I*ArcSin[c*x])] + 8*Log[Tan[ArcSin[c* x]/2]] - (8*I)*ArcSin[c*x]*PolyLog[2, -E^(I*ArcSin[c*x])] + (8*I)*ArcSin[c *x]*PolyLog[2, E^(I*ArcSin[c*x])] + 8*PolyLog[3, -E^(I*ArcSin[c*x])] - 8*P olyLog[3, E^(I*ArcSin[c*x])] + ArcSin[c*x]^2*Sec[ArcSin[c*x]/2]^2 - 4*ArcS in[c*x]*Tan[ArcSin[c*x]/2]))/Sqrt[d - c^2*d*x^2])/8
Time = 1.20 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.62, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {5196, 5138, 243, 73, 221, 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.
| \(\displaystyle \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x^3} \, dx\) |
| \(\Big \downarrow \) 5196 |
| \(\displaystyle -\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \int \frac {a+b \arcsin (c x)}{x^2}dx}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle -\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \left (b c \int \frac {1}{x \sqrt {1-c^2 x^2}}dx-\frac {a+b \arcsin (c x)}{x}\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} b c \int \frac {1}{x^2 \sqrt {1-c^2 x^2}}dx^2-\frac {a+b \arcsin (c x)}{x}\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \left (-\frac {b \int \frac {1}{\frac {1}{c^2}-\frac {x^4}{c^2}}d\sqrt {1-c^2 x^2}}{c}-\frac {a+b \arcsin (c x)}{x}\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \left (-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 5218 |
| \(\displaystyle -\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{c x}d\arcsin (c x)}{2 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \left (-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {c^2 \sqrt {d-c^2 d x^2} \int (a+b \arcsin (c x))^2 \csc (\arcsin (c x))d\arcsin (c x)}{2 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \left (-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle -\frac {c^2 \sqrt {d-c^2 d x^2} \left (-2 b \int (a+b \arcsin (c x)) \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+2 b \int (a+b \arcsin (c x)) \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{2 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \left (-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {c^2 \sqrt {d-c^2 d x^2} \left (2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{2 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \left (-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {c^2 \sqrt {d-c^2 d x^2} \left (2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{2 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \left (-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {c^2 \sqrt {d-c^2 d x^2} \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2+2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )\right )\right )}{2 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \left (-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 x^2}\) |
-1/2*(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/x^2 + (b*c*Sqrt[d - c^2*d *x^2]*(-((a + b*ArcSin[c*x])/x) - b*c*ArcTanh[Sqrt[1 - c^2*x^2]]))/Sqrt[1 - c^2*x^2] - (c^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])])))/(2*Sqrt[1 - c ^2*x^2])
3.3.16.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(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_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2 *x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
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 + 1))), x] + (-Simp[b*c*(n/(f*(m + 1)))*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] + Simp[(c^2/(f^2*(m + 1)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int [(f*x)^(m + 2)*((a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2]), x], x]) /; FreeQ[ {a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 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 = 0.28 (sec) , antiderivative size = 589, normalized size of antiderivative = 1.48
| method | result | size |
| default | \(a^{2} \left (-\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{2 d \,x^{2}}-\frac {c^{2} \left (\sqrt {-c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )\right )}{2}\right )+b^{2} \left (-\frac {\left (c^{2} x^{2} \arcsin \left (c x \right )-2 c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right ) \arcsin \left (c x \right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{2 x^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-4 \,\operatorname {arctanh}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\left (c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{2 x^{2} \left (c^{2} x^{2}-1\right )}+\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right ) c^{2}}{2 c^{2} x^{2}-2}\right )\) | \(589\) |
| parts | \(a^{2} \left (-\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{2 d \,x^{2}}-\frac {c^{2} \left (\sqrt {-c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )\right )}{2}\right )+b^{2} \left (-\frac {\left (c^{2} x^{2} \arcsin \left (c x \right )-2 c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right ) \arcsin \left (c x \right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{2 x^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-4 \,\operatorname {arctanh}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\left (c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{2 x^{2} \left (c^{2} x^{2}-1\right )}+\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right ) c^{2}}{2 c^{2} x^{2}-2}\right )\) | \(589\) |
a^2*(-1/2/d/x^2*(-c^2*d*x^2+d)^(3/2)-1/2*c^2*((-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*(-1/2*(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 )/x^2/(c^2*x^2-1)-1/2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*(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*arcsin(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))-4*arctanh(I*c*x+(-c^2*x^2+1)^(1/2 )))*c^2/(c^2*x^2-1))+2*a*b*(-1/2*(c^2*x^2*arcsin(c*x)-c*x*(-c^2*x^2+1)^(1/ 2)-arcsin(c*x))*(-d*(c^2*x^2-1))^(1/2)/x^2/(c^2*x^2-1)+I*(-d*(c^2*x^2-1))^ (1/2)*(-c^2*x^2+1)^(1/2)*(I*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-I*a rcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-polylog(2,I*c*x+(-c^2*x^2+1)^(1/ 2))+polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2)))*c^2/(2*c^2*x^2-2))
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x^3} \, dx=\int { \frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x^3} \, dx=\int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{3}}\, dx \]
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x^3} \, dx=\int { \frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
1/2*(c^2*sqrt(d)*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x)) - sqrt(-c^2*d*x^2 + d)*c^2 - (-c^2*d*x^2 + d)^(3/2)/(d*x^2))*a^2 + sqrt(d)* integrate((b^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b*arctan 2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1)/x^3, x)
Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x^3} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2}}{x^3} \,d x \]