Integrand size = 27, antiderivative size = 210 \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )} \, dx=-\frac {b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{d x}-\frac {(a+b \arcsin (c x))^2}{2 d x^2}-\frac {2 c^2 (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{2 i \arcsin (c x)}\right )}{d}+\frac {b^2 c^2 \log (x)}{d}+\frac {i b c^2 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{d}-\frac {i b c^2 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{d}-\frac {b^2 c^2 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )}{2 d}+\frac {b^2 c^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right )}{2 d} \]
-1/2*(a+b*arcsin(c*x))^2/d/x^2-2*c^2*(a+b*arcsin(c*x))^2*arctanh((I*c*x+(- c^2*x^2+1)^(1/2))^2)/d+b^2*c^2*ln(x)/d+I*b*c^2*(a+b*arcsin(c*x))*polylog(2 ,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d-I*b*c^2*(a+b*arcsin(c*x))*polylog(2,(I*c *x+(-c^2*x^2+1)^(1/2))^2)/d-1/2*b^2*c^2*polylog(3,-(I*c*x+(-c^2*x^2+1)^(1/ 2))^2)/d+1/2*b^2*c^2*polylog(3,(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d-b*c*(a+b*ar csin(c*x))*(-c^2*x^2+1)^(1/2)/d/x
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(647\) vs. \(2(210)=420\).
Time = 1.06 (sec) , antiderivative size = 647, normalized size of antiderivative = 3.08 \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )} \, dx=-\frac {\frac {1}{12} i b^2 c^2 \pi ^3+\frac {a^2}{x^2}+\frac {2 a b c \sqrt {1-c^2 x^2}}{x}+4 i a b c^2 \pi \arcsin (c x)+\frac {2 a b \arcsin (c x)}{x^2}+\frac {2 b^2 c \sqrt {1-c^2 x^2} \arcsin (c x)}{x}+\frac {b^2 \arcsin (c x)^2}{x^2}-\frac {4}{3} i b^2 c^2 \arcsin (c x)^3+8 a b c^2 \pi \log \left (1+e^{-i \arcsin (c x)}\right )+2 a b c^2 \pi \log \left (1-i e^{i \arcsin (c x)}\right )+4 a b c^2 \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )-2 a b c^2 \pi \log \left (1+i e^{i \arcsin (c x)}\right )+4 a b c^2 \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )-2 b^2 c^2 \arcsin (c x)^2 \log \left (1-e^{-2 i \arcsin (c x)}\right )-4 a b c^2 \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+2 b^2 c^2 \arcsin (c x)^2 \log \left (1+e^{2 i \arcsin (c x)}\right )-2 a^2 c^2 \log (x)-2 b^2 c^2 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+a^2 c^2 \log \left (1-c^2 x^2\right )-b^2 c^2 \log \left (1-c^2 x^2\right )-8 a b c^2 \pi \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )+2 a b c^2 \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-2 a b c^2 \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-4 i a b c^2 \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-4 i a b c^2 \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )-2 i b^2 c^2 \arcsin (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arcsin (c x)}\right )-2 i b^2 c^2 \arcsin (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )+2 i a b c^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )-b^2 c^2 \operatorname {PolyLog}\left (3,e^{-2 i \arcsin (c x)}\right )+b^2 c^2 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )}{2 d} \]
-1/2*((I/12)*b^2*c^2*Pi^3 + a^2/x^2 + (2*a*b*c*Sqrt[1 - c^2*x^2])/x + (4*I )*a*b*c^2*Pi*ArcSin[c*x] + (2*a*b*ArcSin[c*x])/x^2 + (2*b^2*c*Sqrt[1 - c^2 *x^2]*ArcSin[c*x])/x + (b^2*ArcSin[c*x]^2)/x^2 - ((4*I)/3)*b^2*c^2*ArcSin[ c*x]^3 + 8*a*b*c^2*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 2*a*b*c^2*Pi*Log[1 - I*E^(I*ArcSin[c*x])] + 4*a*b*c^2*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 2*a*b*c^2*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 4*a*b*c^2*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] - 2*b^2*c^2*ArcSin[c*x]^2*Log[1 - E^((-2*I)*ArcSin [c*x])] - 4*a*b*c^2*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 2*b^2*c^2 *ArcSin[c*x]^2*Log[1 + E^((2*I)*ArcSin[c*x])] - 2*a^2*c^2*Log[x] - 2*b^2*c ^2*Log[(c*x)/Sqrt[1 - c^2*x^2]] + a^2*c^2*Log[1 - c^2*x^2] - b^2*c^2*Log[1 - c^2*x^2] - 8*a*b*c^2*Pi*Log[Cos[ArcSin[c*x]/2]] + 2*a*b*c^2*Pi*Log[-Cos [(Pi + 2*ArcSin[c*x])/4]] - 2*a*b*c^2*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (4*I)*a*b*c^2*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (4*I)*a*b*c^2*PolyLog [2, I*E^(I*ArcSin[c*x])] - (2*I)*b^2*c^2*ArcSin[c*x]*PolyLog[2, E^((-2*I)* ArcSin[c*x])] - (2*I)*b^2*c^2*ArcSin[c*x]*PolyLog[2, -E^((2*I)*ArcSin[c*x] )] + (2*I)*a*b*c^2*PolyLog[2, E^((2*I)*ArcSin[c*x])] - b^2*c^2*PolyLog[3, E^((-2*I)*ArcSin[c*x])] + b^2*c^2*PolyLog[3, -E^((2*I)*ArcSin[c*x])])/d
Time = 1.32 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.90, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {5204, 27, 5184, 4919, 3042, 4671, 3011, 2720, 5186, 14, 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.
| \(\displaystyle \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 5204 |
| \(\displaystyle c^2 \int \frac {(a+b \arcsin (c x))^2}{d x \left (1-c^2 x^2\right )}dx+\frac {b c \int \frac {a+b \arcsin (c x)}{x^2 \sqrt {1-c^2 x^2}}dx}{d}-\frac {(a+b \arcsin (c x))^2}{2 d x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}dx}{d}+\frac {b c \int \frac {a+b \arcsin (c x)}{x^2 \sqrt {1-c^2 x^2}}dx}{d}-\frac {(a+b \arcsin (c x))^2}{2 d x^2}\) |
| \(\Big \downarrow \) 5184 |
| \(\displaystyle \frac {c^2 \int \frac {(a+b \arcsin (c x))^2}{c x \sqrt {1-c^2 x^2}}d\arcsin (c x)}{d}+\frac {b c \int \frac {a+b \arcsin (c x)}{x^2 \sqrt {1-c^2 x^2}}dx}{d}-\frac {(a+b \arcsin (c x))^2}{2 d x^2}\) |
| \(\Big \downarrow \) 4919 |
| \(\displaystyle \frac {b c \int \frac {a+b \arcsin (c x)}{x^2 \sqrt {1-c^2 x^2}}dx}{d}+\frac {2 c^2 \int (a+b \arcsin (c x))^2 \csc (2 \arcsin (c x))d\arcsin (c x)}{d}-\frac {(a+b \arcsin (c x))^2}{2 d x^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b c \int \frac {a+b \arcsin (c x)}{x^2 \sqrt {1-c^2 x^2}}dx}{d}+\frac {2 c^2 \int (a+b \arcsin (c x))^2 \csc (2 \arcsin (c x))d\arcsin (c x)}{d}-\frac {(a+b \arcsin (c x))^2}{2 d x^2}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {2 c^2 \left (-b \int (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)+b \int (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )d\arcsin (c x)-\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{d}+\frac {b c \int \frac {a+b \arcsin (c x)}{x^2 \sqrt {1-c^2 x^2}}dx}{d}-\frac {(a+b \arcsin (c x))^2}{2 d x^2}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {2 c^2 \left (b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \int \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )-b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \int \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )-\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{d}+\frac {b c \int \frac {a+b \arcsin (c x)}{x^2 \sqrt {1-c^2 x^2}}dx}{d}-\frac {(a+b \arcsin (c x))^2}{2 d x^2}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {2 c^2 \left (b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{d}+\frac {b c \int \frac {a+b \arcsin (c x)}{x^2 \sqrt {1-c^2 x^2}}dx}{d}-\frac {(a+b \arcsin (c x))^2}{2 d x^2}\) |
| \(\Big \downarrow \) 5186 |
| \(\displaystyle \frac {2 c^2 \left (b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{d}+\frac {b c \left (b c \int \frac {1}{x}dx-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}\right )}{d}-\frac {(a+b \arcsin (c x))^2}{2 d x^2}\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle \frac {2 c^2 \left (b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{d}+\frac {b c \left (b c \log (x)-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}\right )}{d}-\frac {(a+b \arcsin (c x))^2}{2 d x^2}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {2 c^2 \left (-\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2+b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )\right )-b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right )\right )\right )}{d}+\frac {b c \left (b c \log (x)-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}\right )}{d}-\frac {(a+b \arcsin (c x))^2}{2 d x^2}\) |
-1/2*(a + b*ArcSin[c*x])^2/(d*x^2) + (b*c*(-((Sqrt[1 - c^2*x^2]*(a + b*Arc Sin[c*x]))/x) + b*c*Log[x]))/d + (2*c^2*(-((a + b*ArcSin[c*x])^2*ArcTanh[E ^((2*I)*ArcSin[c*x])]) + b*((I/2)*(a + b*ArcSin[c*x])*PolyLog[2, -E^((2*I) *ArcSin[c*x])] - (b*PolyLog[3, -E^((2*I)*ArcSin[c*x])])/4) - b*((I/2)*(a + b*ArcSin[c*x])*PolyLog[2, E^((2*I)*ArcSin[c*x])] - (b*PolyLog[3, E^((2*I) *ArcSin[c*x])])/4)))/d
3.2.90.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[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[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_.)/((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[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*A rcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^ 2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[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/(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[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 609 vs. \(2 (250 ) = 500\).
Time = 0.35 (sec) , antiderivative size = 610, normalized size of antiderivative = 2.90
| method | result | size |
| derivativedivides | \(c^{2} \left (-\frac {a^{2} \left (\frac {1}{2 c^{2} x^{2}}-\ln \left (c x \right )+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b^{2} \left (\frac {\arcsin \left (c x \right ) \left (-2 i c^{2} x^{2}+2 c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )\right )}{2 c^{2} x^{2}}-\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+2 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 a b \left (\frac {-i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )}{2 c^{2} x^{2}}-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\right )\) | \(610\) |
| default | \(c^{2} \left (-\frac {a^{2} \left (\frac {1}{2 c^{2} x^{2}}-\ln \left (c x \right )+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b^{2} \left (\frac {\arcsin \left (c x \right ) \left (-2 i c^{2} x^{2}+2 c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )\right )}{2 c^{2} x^{2}}-\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+2 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 a b \left (\frac {-i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )}{2 c^{2} x^{2}}-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\right )\) | \(610\) |
| parts | \(-\frac {a^{2} \left (\frac {1}{2 x^{2}}-c^{2} \ln \left (x \right )+\frac {c^{2} \ln \left (c x -1\right )}{2}+\frac {c^{2} \ln \left (c x +1\right )}{2}\right )}{d}-\frac {b^{2} c^{2} \left (\frac {\arcsin \left (c x \right ) \left (-2 i c^{2} x^{2}+2 c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )\right )}{2 c^{2} x^{2}}-\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+2 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 a b \,c^{2} \left (\frac {-i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )}{2 c^{2} x^{2}}-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\) | \(616\) |
c^2*(-a^2/d*(1/2/c^2/x^2-ln(c*x)+1/2*ln(c*x-1)+1/2*ln(c*x+1))-b^2/d*(1/2*a rcsin(c*x)*(-2*I*c^2*x^2+2*c*x*(-c^2*x^2+1)^(1/2)+arcsin(c*x))/c^2/x^2-ln( I*c*x+(-c^2*x^2+1)^(1/2)-1)+2*ln(I*c*x+(-c^2*x^2+1)^(1/2))-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*polylog(3,-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)*polylo g(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)+1/2*polylog(3,-(I*c*x+(-c^2*x^2+1)^(1/2 ))^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*polylog(3,I*c*x+(-c^2*x^2+1)^(1/2)))-2*a*b/ d*(1/2*(-I*c^2*x^2+c*x*(-c^2*x^2+1)^(1/2)+arcsin(c*x))/c^2/x^2-arcsin(c*x) *ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+I*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))+arc sin(c*x)*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)-1/2*I*polylog(2,-(I*c*x+(-c^2* x^2+1)^(1/2))^2)-arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+I*polylog(2,I* c*x+(-c^2*x^2+1)^(1/2))))
\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )} x^{3}} \,d x } \]
\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )} \, dx=- \frac {\int \frac {a^{2}}{c^{2} x^{5} - x^{3}}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{2} x^{5} - x^{3}}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{c^{2} x^{5} - x^{3}}\, dx}{d} \]
-(Integral(a**2/(c**2*x**5 - x**3), x) + Integral(b**2*asin(c*x)**2/(c**2* x**5 - x**3), x) + Integral(2*a*b*asin(c*x)/(c**2*x**5 - x**3), x))/d
\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )} x^{3}} \,d x } \]
-1/2*(c^2*log(c*x + 1)/d + c^2*log(c*x - 1)/d - 2*c^2*log(x)/d + 1/(d*x^2) )*a^2 - integrate((b^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a* b*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))/(c^2*d*x^5 - d*x^3), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^3\,\left (d-c^2\,d\,x^2\right )} \,d x \]