Integrand size = 23, antiderivative size = 291 \[ \int \frac {(a+b \arcsin (c+d x))^3}{(c e+d e x)^4} \, dx=-\frac {b^2 (a+b \arcsin (c+d x))}{d e^4 (c+d x)}-\frac {b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{2 d e^4 (c+d x)^2}-\frac {(a+b \arcsin (c+d x))^3}{3 d e^4 (c+d x)^3}-\frac {b (a+b \arcsin (c+d x))^2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {b^3 \text {arctanh}\left (\sqrt {1-(c+d x)^2}\right )}{d e^4}+\frac {i b^2 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {i b^2 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {b^3 \operatorname {PolyLog}\left (3,-e^{i \arcsin (c+d x)}\right )}{d e^4}+\frac {b^3 \operatorname {PolyLog}\left (3,e^{i \arcsin (c+d x)}\right )}{d e^4} \]
-b^2*(a+b*arcsin(d*x+c))/d/e^4/(d*x+c)-1/3*(a+b*arcsin(d*x+c))^3/d/e^4/(d* x+c)^3-b*(a+b*arcsin(d*x+c))^2*arctanh(I*(d*x+c)+(1-(d*x+c)^2)^(1/2))/d/e^ 4-b^3*arctanh((1-(d*x+c)^2)^(1/2))/d/e^4+I*b^2*(a+b*arcsin(d*x+c))*polylog (2,-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))/d/e^4-I*b^2*(a+b*arcsin(d*x+c))*polylog (2,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))/d/e^4-b^3*polylog(3,-I*(d*x+c)-(1-(d*x+c )^2)^(1/2))/d/e^4+b^3*polylog(3,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))/d/e^4-1/2*b *(a+b*arcsin(d*x+c))^2*(1-(d*x+c)^2)^(1/2)/d/e^4/(d*x+c)^2
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(732\) vs. \(2(291)=582\).
Time = 8.26 (sec) , antiderivative size = 732, normalized size of antiderivative = 2.52 \[ \int \frac {(a+b \arcsin (c+d x))^3}{(c e+d e x)^4} \, dx=-\frac {a^3}{3 d e^4 (c+d x)^3}-\frac {a^2 b \sqrt {1-c^2-2 c d x-d^2 x^2}}{2 d e^4 (c+d x)^2}-\frac {a^2 b \arcsin (c+d x)}{d e^4 (c+d x)^3}+\frac {a^2 b \log (c+d x)}{2 d e^4}-\frac {a^2 b \log \left (1+\sqrt {1-c^2-2 c d x-d^2 x^2}\right )}{2 d e^4}+\frac {a b^2 \left (8 i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )-\frac {2 \left (2+4 \arcsin (c+d x)^2-2 \cos (2 \arcsin (c+d x))-3 (c+d x) \arcsin (c+d x) \log \left (1-e^{i \arcsin (c+d x)}\right )+3 (c+d x) \arcsin (c+d x) \log \left (1+e^{i \arcsin (c+d x)}\right )+4 i (c+d x)^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )+2 \arcsin (c+d x) \sin (2 \arcsin (c+d x))+\arcsin (c+d x) \log \left (1-e^{i \arcsin (c+d x)}\right ) \sin (3 \arcsin (c+d x))-\arcsin (c+d x) \log \left (1+e^{i \arcsin (c+d x)}\right ) \sin (3 \arcsin (c+d x))\right )}{(c+d x)^3}\right )}{8 d e^4}+\frac {b^3 \left (-24 \arcsin (c+d x) \cot \left (\frac {1}{2} \arcsin (c+d x)\right )-4 \arcsin (c+d x)^3 \cot \left (\frac {1}{2} \arcsin (c+d x)\right )-6 \arcsin (c+d x)^2 \csc ^2\left (\frac {1}{2} \arcsin (c+d x)\right )-(c+d x) \arcsin (c+d x)^3 \csc ^4\left (\frac {1}{2} \arcsin (c+d x)\right )+24 \arcsin (c+d x)^2 \log \left (1-e^{i \arcsin (c+d x)}\right )-24 \arcsin (c+d x)^2 \log \left (1+e^{i \arcsin (c+d x)}\right )+48 \log \left (\tan \left (\frac {1}{2} \arcsin (c+d x)\right )\right )+48 i \arcsin (c+d x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )-48 i \arcsin (c+d x) \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )-48 \operatorname {PolyLog}\left (3,-e^{i \arcsin (c+d x)}\right )+48 \operatorname {PolyLog}\left (3,e^{i \arcsin (c+d x)}\right )+6 \arcsin (c+d x)^2 \sec ^2\left (\frac {1}{2} \arcsin (c+d x)\right )-\frac {16 \arcsin (c+d x)^3 \sin ^4\left (\frac {1}{2} \arcsin (c+d x)\right )}{(c+d x)^3}-24 \arcsin (c+d x) \tan \left (\frac {1}{2} \arcsin (c+d x)\right )-4 \arcsin (c+d x)^3 \tan \left (\frac {1}{2} \arcsin (c+d x)\right )\right )}{48 d e^4} \]
-1/3*a^3/(d*e^4*(c + d*x)^3) - (a^2*b*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/( 2*d*e^4*(c + d*x)^2) - (a^2*b*ArcSin[c + d*x])/(d*e^4*(c + d*x)^3) + (a^2* b*Log[c + d*x])/(2*d*e^4) - (a^2*b*Log[1 + Sqrt[1 - c^2 - 2*c*d*x - d^2*x^ 2]])/(2*d*e^4) + (a*b^2*((8*I)*PolyLog[2, -E^(I*ArcSin[c + d*x])] - (2*(2 + 4*ArcSin[c + d*x]^2 - 2*Cos[2*ArcSin[c + d*x]] - 3*(c + d*x)*ArcSin[c + d*x]*Log[1 - E^(I*ArcSin[c + d*x])] + 3*(c + d*x)*ArcSin[c + d*x]*Log[1 + E^(I*ArcSin[c + d*x])] + (4*I)*(c + d*x)^3*PolyLog[2, E^(I*ArcSin[c + d*x] )] + 2*ArcSin[c + d*x]*Sin[2*ArcSin[c + d*x]] + ArcSin[c + d*x]*Log[1 - E^ (I*ArcSin[c + d*x])]*Sin[3*ArcSin[c + d*x]] - ArcSin[c + d*x]*Log[1 + E^(I *ArcSin[c + d*x])]*Sin[3*ArcSin[c + d*x]]))/(c + d*x)^3))/(8*d*e^4) + (b^3 *(-24*ArcSin[c + d*x]*Cot[ArcSin[c + d*x]/2] - 4*ArcSin[c + d*x]^3*Cot[Arc Sin[c + d*x]/2] - 6*ArcSin[c + d*x]^2*Csc[ArcSin[c + d*x]/2]^2 - (c + d*x) *ArcSin[c + d*x]^3*Csc[ArcSin[c + d*x]/2]^4 + 24*ArcSin[c + d*x]^2*Log[1 - E^(I*ArcSin[c + d*x])] - 24*ArcSin[c + d*x]^2*Log[1 + E^(I*ArcSin[c + d*x ])] + 48*Log[Tan[ArcSin[c + d*x]/2]] + (48*I)*ArcSin[c + d*x]*PolyLog[2, - E^(I*ArcSin[c + d*x])] - (48*I)*ArcSin[c + d*x]*PolyLog[2, E^(I*ArcSin[c + d*x])] - 48*PolyLog[3, -E^(I*ArcSin[c + d*x])] + 48*PolyLog[3, E^(I*ArcSi n[c + d*x])] + 6*ArcSin[c + d*x]^2*Sec[ArcSin[c + d*x]/2]^2 - (16*ArcSin[c + d*x]^3*Sin[ArcSin[c + d*x]/2]^4)/(c + d*x)^3 - 24*ArcSin[c + d*x]*Tan[A rcSin[c + d*x]/2] - 4*ArcSin[c + d*x]^3*Tan[ArcSin[c + d*x]/2]))/(48*d*...
Time = 1.15 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.82, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {5304, 27, 5138, 5204, 5138, 243, 73, 219, 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.
| \(\displaystyle \int \frac {(a+b \arcsin (c+d x))^3}{(c e+d e x)^4} \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int \frac {(a+b \arcsin (c+d x))^3}{e^4 (c+d x)^4}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \arcsin (c+d x))^3}{(c+d x)^4}d(c+d x)}{d e^4}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {b \int \frac {(a+b \arcsin (c+d x))^2}{(c+d x)^3 \sqrt {1-(c+d x)^2}}d(c+d x)-\frac {(a+b \arcsin (c+d x))^3}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 5204 |
| \(\displaystyle \frac {b \left (b \int \frac {a+b \arcsin (c+d x)}{(c+d x)^2}d(c+d x)+\frac {1}{2} \int \frac {(a+b \arcsin (c+d x))^2}{(c+d x) \sqrt {1-(c+d x)^2}}d(c+d x)-\frac {\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{2 (c+d x)^2}\right )-\frac {(a+b \arcsin (c+d x))^3}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {b \left (b \left (b \int \frac {1}{(c+d x) \sqrt {1-(c+d x)^2}}d(c+d x)-\frac {a+b \arcsin (c+d x)}{c+d x}\right )+\frac {1}{2} \int \frac {(a+b \arcsin (c+d x))^2}{(c+d x) \sqrt {1-(c+d x)^2}}d(c+d x)-\frac {\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{2 (c+d x)^2}\right )-\frac {(a+b \arcsin (c+d x))^3}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {b \left (b \left (\frac {1}{2} b \int \frac {1}{\sqrt {-c-d x+1} (c+d x)^2}d(c+d x)^2-\frac {a+b \arcsin (c+d x)}{c+d x}\right )+\frac {1}{2} \int \frac {(a+b \arcsin (c+d x))^2}{(c+d x) \sqrt {1-(c+d x)^2}}d(c+d x)-\frac {\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{2 (c+d x)^2}\right )-\frac {(a+b \arcsin (c+d x))^3}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {b \left (b \left (-b \int \frac {1}{1-(c+d x)^4}d\sqrt {-c-d x+1}-\frac {a+b \arcsin (c+d x)}{c+d x}\right )+\frac {1}{2} \int \frac {(a+b \arcsin (c+d x))^2}{(c+d x) \sqrt {1-(c+d x)^2}}d(c+d x)-\frac {\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{2 (c+d x)^2}\right )-\frac {(a+b \arcsin (c+d x))^3}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {1}{2} \int \frac {(a+b \arcsin (c+d x))^2}{(c+d x) \sqrt {1-(c+d x)^2}}d(c+d x)+b \left (-\frac {a+b \arcsin (c+d x)}{c+d x}-b \text {arctanh}\left (\sqrt {-c-d x+1}\right )\right )-\frac {\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{2 (c+d x)^2}\right )-\frac {(a+b \arcsin (c+d x))^3}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 5218 |
| \(\displaystyle \frac {b \left (\frac {1}{2} \int \frac {(a+b \arcsin (c+d x))^2}{c+d x}d\arcsin (c+d x)+b \left (-\frac {a+b \arcsin (c+d x)}{c+d x}-b \text {arctanh}\left (\sqrt {-c-d x+1}\right )\right )-\frac {\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{2 (c+d x)^2}\right )-\frac {(a+b \arcsin (c+d x))^3}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b \left (\frac {1}{2} \int (a+b \arcsin (c+d x))^2 \csc (\arcsin (c+d x))d\arcsin (c+d x)+b \left (-\frac {a+b \arcsin (c+d x)}{c+d x}-b \text {arctanh}\left (\sqrt {-c-d x+1}\right )\right )-\frac {\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{2 (c+d x)^2}\right )-\frac {(a+b \arcsin (c+d x))^3}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {-\frac {(a+b \arcsin (c+d x))^3}{3 (c+d x)^3}+b \left (\frac {1}{2} \left (-2 b \int (a+b \arcsin (c+d x)) \log \left (1-e^{i \arcsin (c+d x)}\right )d\arcsin (c+d x)+2 b \int (a+b \arcsin (c+d x)) \log \left (1+e^{i \arcsin (c+d x)}\right )d\arcsin (c+d x)-2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2\right )+b \left (-\frac {a+b \arcsin (c+d x)}{c+d x}-b \text {arctanh}\left (\sqrt {-c-d x+1}\right )\right )-\frac {\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{2 (c+d x)^2}\right )}{d e^4}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {-\frac {(a+b \arcsin (c+d x))^3}{3 (c+d x)^3}+b \left (\frac {1}{2} \left (2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))-i b \int \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )d\arcsin (c+d x)\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))-i b \int \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )d\arcsin (c+d x)\right )-2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2\right )+b \left (-\frac {a+b \arcsin (c+d x)}{c+d x}-b \text {arctanh}\left (\sqrt {-c-d x+1}\right )\right )-\frac {\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{2 (c+d x)^2}\right )}{d e^4}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {-\frac {(a+b \arcsin (c+d x))^3}{3 (c+d x)^3}+b \left (\frac {1}{2} \left (-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))-b \int e^{-i \arcsin (c+d x)} \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )de^{i \arcsin (c+d x)}\right )+2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))-b \int e^{-i \arcsin (c+d x)} \operatorname {PolyLog}(2,-c-d x)de^{i \arcsin (c+d x)}\right )-2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2\right )+b \left (-\frac {a+b \arcsin (c+d x)}{c+d x}-b \text {arctanh}\left (\sqrt {-c-d x+1}\right )\right )-\frac {\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{2 (c+d x)^2}\right )}{d e^4}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {-\frac {(a+b \arcsin (c+d x))^3}{3 (c+d x)^3}+b \left (\frac {1}{2} \left (-2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))-b \operatorname {PolyLog}\left (3,e^{i \arcsin (c+d x)}\right )\right )+2 b \left (-b \operatorname {PolyLog}(3,-c-d x)+i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))\right )\right )+b \left (-\frac {a+b \arcsin (c+d x)}{c+d x}-b \text {arctanh}\left (\sqrt {-c-d x+1}\right )\right )-\frac {\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{2 (c+d x)^2}\right )}{d e^4}\) |
(-1/3*(a + b*ArcSin[c + d*x])^3/(c + d*x)^3 + b*(-1/2*(Sqrt[1 - (c + d*x)^ 2]*(a + b*ArcSin[c + d*x])^2)/(c + d*x)^2 + b*(-((a + b*ArcSin[c + d*x])/( c + d*x)) - b*ArcTanh[Sqrt[1 - c - d*x]]) + (-2*(a + b*ArcSin[c + d*x])^2* ArcTanh[E^(I*ArcSin[c + d*x])] - 2*b*(I*(a + b*ArcSin[c + d*x])*PolyLog[2, E^(I*ArcSin[c + d*x])] - b*PolyLog[3, E^(I*ArcSin[c + d*x])]) + 2*b*(I*(a + b*ArcSin[c + d*x])*PolyLog[2, -E^(I*ArcSin[c + d*x])] - b*PolyLog[3, -c - d*x]))/2))/(d*e^4)
3.3.5.3.1 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[((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[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(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_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b *ArcSin[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)) ) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] - Simp[b* c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m + 1)*( 1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(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[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A rcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
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 = 1.25 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.89
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{3}}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b^{3} \left (-\frac {\arcsin \left (d x +c \right ) \left (3 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\, \left (d x +c \right )+2 \arcsin \left (d x +c \right )^{2}+6 \left (d x +c \right )^{2}\right )}{6 \left (d x +c \right )^{3}}+\frac {\arcsin \left (d x +c \right )^{2} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{2}-i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-\frac {\arcsin \left (d x +c \right )^{2} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{2}+i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-\operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-2 \,\operatorname {arctanh}\left (i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e^{4}}+\frac {3 a \,b^{2} \left (-\frac {\arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\, \left (d x +c \right )+\arcsin \left (d x +c \right )^{2}+\left (d x +c \right )^{2}}{3 \left (d x +c \right )^{3}}-\frac {\arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{3}+\frac {i \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{3}+\frac {\arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{3}-\frac {i \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{3}\right )}{e^{4}}+\frac {3 a^{2} b \left (-\frac {\arcsin \left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {1-\left (d x +c \right )^{2}}}{6 \left (d x +c \right )^{2}}-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )}{6}\right )}{e^{4}}}{d}\) | \(550\) |
| default | \(\frac {-\frac {a^{3}}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b^{3} \left (-\frac {\arcsin \left (d x +c \right ) \left (3 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\, \left (d x +c \right )+2 \arcsin \left (d x +c \right )^{2}+6 \left (d x +c \right )^{2}\right )}{6 \left (d x +c \right )^{3}}+\frac {\arcsin \left (d x +c \right )^{2} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{2}-i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-\frac {\arcsin \left (d x +c \right )^{2} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{2}+i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-\operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-2 \,\operatorname {arctanh}\left (i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e^{4}}+\frac {3 a \,b^{2} \left (-\frac {\arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\, \left (d x +c \right )+\arcsin \left (d x +c \right )^{2}+\left (d x +c \right )^{2}}{3 \left (d x +c \right )^{3}}-\frac {\arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{3}+\frac {i \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{3}+\frac {\arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{3}-\frac {i \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{3}\right )}{e^{4}}+\frac {3 a^{2} b \left (-\frac {\arcsin \left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {1-\left (d x +c \right )^{2}}}{6 \left (d x +c \right )^{2}}-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )}{6}\right )}{e^{4}}}{d}\) | \(550\) |
| parts | \(-\frac {a^{3}}{3 e^{4} \left (d x +c \right )^{3} d}+\frac {b^{3} \left (-\frac {\arcsin \left (d x +c \right ) \left (3 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\, \left (d x +c \right )+2 \arcsin \left (d x +c \right )^{2}+6 \left (d x +c \right )^{2}\right )}{6 \left (d x +c \right )^{3}}+\frac {\arcsin \left (d x +c \right )^{2} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{2}-i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-\frac {\arcsin \left (d x +c \right )^{2} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{2}+i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-\operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-2 \,\operatorname {arctanh}\left (i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e^{4} d}+\frac {3 a \,b^{2} \left (-\frac {\arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\, \left (d x +c \right )+\arcsin \left (d x +c \right )^{2}+\left (d x +c \right )^{2}}{3 \left (d x +c \right )^{3}}-\frac {\arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{3}+\frac {i \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{3}+\frac {\arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{3}-\frac {i \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{3}\right )}{e^{4} d}+\frac {3 a^{2} b \left (-\frac {\arcsin \left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {1-\left (d x +c \right )^{2}}}{6 \left (d x +c \right )^{2}}-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )}{6}\right )}{e^{4} d}\) | \(558\) |
1/d*(-1/3*a^3/e^4/(d*x+c)^3+b^3/e^4*(-1/6/(d*x+c)^3*arcsin(d*x+c)*(3*arcsi n(d*x+c)*(1-(d*x+c)^2)^(1/2)*(d*x+c)+2*arcsin(d*x+c)^2+6*(d*x+c)^2)+1/2*ar csin(d*x+c)^2*ln(1-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))-I*arcsin(d*x+c)*polylog( 2,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))+polylog(3,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))- 1/2*arcsin(d*x+c)^2*ln(1+I*(d*x+c)+(1-(d*x+c)^2)^(1/2))+I*arcsin(d*x+c)*po lylog(2,-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))-polylog(3,-I*(d*x+c)-(1-(d*x+c)^2) ^(1/2))-2*arctanh(I*(d*x+c)+(1-(d*x+c)^2)^(1/2)))+3*a*b^2/e^4*(-1/3*(arcsi n(d*x+c)*(1-(d*x+c)^2)^(1/2)*(d*x+c)+arcsin(d*x+c)^2+(d*x+c)^2)/(d*x+c)^3+ 1/3*arcsin(d*x+c)*ln(1-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))-1/3*I*polylog(2,I*(d *x+c)+(1-(d*x+c)^2)^(1/2))-1/3*arcsin(d*x+c)*ln(1+I*(d*x+c)+(1-(d*x+c)^2)^ (1/2))+1/3*I*polylog(2,-I*(d*x+c)-(1-(d*x+c)^2)^(1/2)))+3*a^2*b/e^4*(-1/3/ (d*x+c)^3*arcsin(d*x+c)-1/6/(d*x+c)^2*(1-(d*x+c)^2)^(1/2)-1/6*arctanh(1/(1 -(d*x+c)^2)^(1/2))))
\[ \int \frac {(a+b \arcsin (c+d x))^3}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{4}} \,d x } \]
integral((b^3*arcsin(d*x + c)^3 + 3*a*b^2*arcsin(d*x + c)^2 + 3*a^2*b*arcs in(d*x + c) + a^3)/(d^4*e^4*x^4 + 4*c*d^3*e^4*x^3 + 6*c^2*d^2*e^4*x^2 + 4* c^3*d*e^4*x + c^4*e^4), x)
\[ \int \frac {(a+b \arcsin (c+d x))^3}{(c e+d e x)^4} \, dx=\frac {\int \frac {a^{3}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {3 a b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {3 a^{2} b \operatorname {asin}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx}{e^{4}} \]
(Integral(a**3/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d** 4*x**4), x) + Integral(b**3*asin(c + d*x)**3/(c**4 + 4*c**3*d*x + 6*c**2*d **2*x**2 + 4*c*d**3*x**3 + d**4*x**4), x) + Integral(3*a*b**2*asin(c + d*x )**2/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4), x ) + Integral(3*a**2*b*asin(c + d*x)/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4), x))/e**4
\[ \int \frac {(a+b \arcsin (c+d x))^3}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{4}} \,d x } \]
-1/3*a^3/(d^4*e^4*x^3 + 3*c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x + c^3*d*e^4) - 1 /3*(b^3*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^3 + 3*(d^4* e^4*x^3 + 3*c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x + c^3*d*e^4)*integrate(((b^3*d *x + b^3*c)*sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^2 - 3*(a*b^2*d^2*x^2 + 2*a*b^2*c*d*x + a*b^2 *c^2 - a*b^2)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^2 - 3 *(a^2*b*d^2*x^2 + 2*a^2*b*c*d*x + a^2*b*c^2 - a^2*b)*arctan2(d*x + c, sqrt (d*x + c + 1)*sqrt(-d*x - c + 1)))/(d^6*e^4*x^6 + 6*c*d^5*e^4*x^5 + (15*c^ 2 - 1)*d^4*e^4*x^4 + 4*(5*c^3 - c)*d^3*e^4*x^3 + 3*(5*c^4 - 2*c^2)*d^2*e^4 *x^2 + 2*(3*c^5 - 2*c^3)*d*e^4*x + (c^6 - c^4)*e^4), x))/(d^4*e^4*x^3 + 3* c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x + c^3*d*e^4)
\[ \int \frac {(a+b \arcsin (c+d x))^3}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{4}} \,d x } \]
Timed out. \[ \int \frac {(a+b \arcsin (c+d x))^3}{(c e+d e x)^4} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3}{{\left (c\,e+d\,e\,x\right )}^4} \,d x \]