Integrand size = 10, antiderivative size = 276 \[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=-\frac {2 a^2 \arcsin (a x)^2}{x}-\frac {2 a \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 x^2}-\frac {\arcsin (a x)^4}{3 x^3}-8 a^3 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )-\frac {4}{3} a^3 \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )+2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-4 a^3 \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+4 a^3 \arcsin (a x) \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (4,-e^{i \arcsin (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (4,e^{i \arcsin (a x)}\right ) \] Output:
-2*a^2*arcsin(a*x)^2/x-2/3*a*(-a^2*x^2+1)^(1/2)*arcsin(a*x)^3/x^2-1/3*arcs in(a*x)^4/x^3-8*a^3*arcsin(a*x)*arctanh(I*a*x+(-a^2*x^2+1)^(1/2))-4/3*a^3* arcsin(a*x)^3*arctanh(I*a*x+(-a^2*x^2+1)^(1/2))+4*I*a^3*polylog(2,-I*a*x-( -a^2*x^2+1)^(1/2))+2*I*a^3*arcsin(a*x)^2*polylog(2,-I*a*x-(-a^2*x^2+1)^(1/ 2))-4*I*a^3*polylog(2,I*a*x+(-a^2*x^2+1)^(1/2))-2*I*a^3*arcsin(a*x)^2*poly log(2,I*a*x+(-a^2*x^2+1)^(1/2))-4*a^3*arcsin(a*x)*polylog(3,-I*a*x-(-a^2*x ^2+1)^(1/2))+4*a^3*arcsin(a*x)*polylog(3,I*a*x+(-a^2*x^2+1)^(1/2))-4*I*a^3 *polylog(4,-I*a*x-(-a^2*x^2+1)^(1/2))+4*I*a^3*polylog(4,I*a*x+(-a^2*x^2+1) ^(1/2))
Time = 2.79 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.45 \[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=\frac {1}{24} a^3 \left (-2 i \pi ^4+4 i \arcsin (a x)^4-24 \arcsin (a x)^2 \cot \left (\frac {1}{2} \arcsin (a x)\right )-2 \arcsin (a x)^4 \cot \left (\frac {1}{2} \arcsin (a x)\right )-4 \arcsin (a x)^3 \csc ^2\left (\frac {1}{2} \arcsin (a x)\right )-\frac {1}{2} a x \arcsin (a x)^4 \csc ^4\left (\frac {1}{2} \arcsin (a x)\right )+16 \arcsin (a x)^3 \log \left (1-e^{-i \arcsin (a x)}\right )+96 \arcsin (a x) \log \left (1-e^{i \arcsin (a x)}\right )-96 \arcsin (a x) \log \left (1+e^{i \arcsin (a x)}\right )-16 \arcsin (a x)^3 \log \left (1+e^{i \arcsin (a x)}\right )+48 i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{-i \arcsin (a x)}\right )+48 i \left (2+\arcsin (a x)^2\right ) \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-96 i \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )+96 \arcsin (a x) \operatorname {PolyLog}\left (3,e^{-i \arcsin (a x)}\right )-96 \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )-96 i \operatorname {PolyLog}\left (4,e^{-i \arcsin (a x)}\right )-96 i \operatorname {PolyLog}\left (4,-e^{i \arcsin (a x)}\right )+4 \arcsin (a x)^3 \sec ^2\left (\frac {1}{2} \arcsin (a x)\right )-\frac {8 \arcsin (a x)^4 \sin ^4\left (\frac {1}{2} \arcsin (a x)\right )}{a^3 x^3}-24 \arcsin (a x)^2 \tan \left (\frac {1}{2} \arcsin (a x)\right )-2 \arcsin (a x)^4 \tan \left (\frac {1}{2} \arcsin (a x)\right )\right ) \] Input:
Integrate[ArcSin[a*x]^4/x^4,x]
Output:
(a^3*((-2*I)*Pi^4 + (4*I)*ArcSin[a*x]^4 - 24*ArcSin[a*x]^2*Cot[ArcSin[a*x] /2] - 2*ArcSin[a*x]^4*Cot[ArcSin[a*x]/2] - 4*ArcSin[a*x]^3*Csc[ArcSin[a*x] /2]^2 - (a*x*ArcSin[a*x]^4*Csc[ArcSin[a*x]/2]^4)/2 + 16*ArcSin[a*x]^3*Log[ 1 - E^((-I)*ArcSin[a*x])] + 96*ArcSin[a*x]*Log[1 - E^(I*ArcSin[a*x])] - 96 *ArcSin[a*x]*Log[1 + E^(I*ArcSin[a*x])] - 16*ArcSin[a*x]^3*Log[1 + E^(I*Ar cSin[a*x])] + (48*I)*ArcSin[a*x]^2*PolyLog[2, E^((-I)*ArcSin[a*x])] + (48* I)*(2 + ArcSin[a*x]^2)*PolyLog[2, -E^(I*ArcSin[a*x])] - (96*I)*PolyLog[2, E^(I*ArcSin[a*x])] + 96*ArcSin[a*x]*PolyLog[3, E^((-I)*ArcSin[a*x])] - 96* ArcSin[a*x]*PolyLog[3, -E^(I*ArcSin[a*x])] - (96*I)*PolyLog[4, E^((-I)*Arc Sin[a*x])] - (96*I)*PolyLog[4, -E^(I*ArcSin[a*x])] + 4*ArcSin[a*x]^3*Sec[A rcSin[a*x]/2]^2 - (8*ArcSin[a*x]^4*Sin[ArcSin[a*x]/2]^4)/(a^3*x^3) - 24*Ar cSin[a*x]^2*Tan[ArcSin[a*x]/2] - 2*ArcSin[a*x]^4*Tan[ArcSin[a*x]/2]))/24
Time = 1.75 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {5138, 5204, 5138, 5218, 3042, 4671, 2715, 2838, 3011, 7163, 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.
\(\displaystyle \int \frac {\arcsin (a x)^4}{x^4} \, dx\) |
\(\Big \downarrow \) 5138 |
\(\displaystyle \frac {4}{3} a \int \frac {\arcsin (a x)^3}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {\arcsin (a x)^4}{3 x^3}\) |
\(\Big \downarrow \) 5204 |
\(\displaystyle \frac {4}{3} a \left (\frac {1}{2} a^2 \int \frac {\arcsin (a x)^3}{x \sqrt {1-a^2 x^2}}dx+\frac {3}{2} a \int \frac {\arcsin (a x)^2}{x^2}dx-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 x^2}\right )-\frac {\arcsin (a x)^4}{3 x^3}\) |
\(\Big \downarrow \) 5138 |
\(\displaystyle \frac {4}{3} a \left (\frac {3}{2} a \left (2 a \int \frac {\arcsin (a x)}{x \sqrt {1-a^2 x^2}}dx-\frac {\arcsin (a x)^2}{x}\right )+\frac {1}{2} a^2 \int \frac {\arcsin (a x)^3}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 x^2}\right )-\frac {\arcsin (a x)^4}{3 x^3}\) |
\(\Big \downarrow \) 5218 |
\(\displaystyle \frac {4}{3} a \left (\frac {1}{2} a^2 \int \frac {\arcsin (a x)^3}{a x}d\arcsin (a x)+\frac {3}{2} a \left (2 a \int \frac {\arcsin (a x)}{a x}d\arcsin (a x)-\frac {\arcsin (a x)^2}{x}\right )-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 x^2}\right )-\frac {\arcsin (a x)^4}{3 x^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4}{3} a \left (\frac {1}{2} a^2 \int \arcsin (a x)^3 \csc (\arcsin (a x))d\arcsin (a x)+\frac {3}{2} a \left (2 a \int \arcsin (a x) \csc (\arcsin (a x))d\arcsin (a x)-\frac {\arcsin (a x)^2}{x}\right )-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 x^2}\right )-\frac {\arcsin (a x)^4}{3 x^3}\) |
\(\Big \downarrow \) 4671 |
\(\displaystyle -\frac {\arcsin (a x)^4}{3 x^3}+\frac {4}{3} a \left (\frac {1}{2} a^2 \left (-3 \int \arcsin (a x)^2 \log \left (1-e^{i \arcsin (a x)}\right )d\arcsin (a x)+3 \int \arcsin (a x)^2 \log \left (1+e^{i \arcsin (a x)}\right )d\arcsin (a x)-2 \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )\right )+\frac {3}{2} a \left (-\frac {\arcsin (a x)^2}{x}+2 a \left (-\int \log \left (1-e^{i \arcsin (a x)}\right )d\arcsin (a x)+\int \log \left (1+e^{i \arcsin (a x)}\right )d\arcsin (a x)-2 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )\right )\right )-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 x^2}\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {\arcsin (a x)^4}{3 x^3}+\frac {4}{3} a \left (\frac {1}{2} a^2 \left (-3 \int \arcsin (a x)^2 \log \left (1-e^{i \arcsin (a x)}\right )d\arcsin (a x)+3 \int \arcsin (a x)^2 \log \left (1+e^{i \arcsin (a x)}\right )d\arcsin (a x)-2 \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )\right )+\frac {3}{2} a \left (-\frac {\arcsin (a x)^2}{x}+2 a \left (i \int e^{-i \arcsin (a x)} \log \left (1-e^{i \arcsin (a x)}\right )de^{i \arcsin (a x)}-i \int e^{-i \arcsin (a x)} \log \left (1+e^{i \arcsin (a x)}\right )de^{i \arcsin (a x)}-2 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )\right )\right )-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 x^2}\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {\arcsin (a x)^4}{3 x^3}+\frac {4}{3} a \left (\frac {1}{2} a^2 \left (-3 \int \arcsin (a x)^2 \log \left (1-e^{i \arcsin (a x)}\right )d\arcsin (a x)+3 \int \arcsin (a x)^2 \log \left (1+e^{i \arcsin (a x)}\right )d\arcsin (a x)-2 \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )\right )-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 x^2}+\frac {3}{2} a \left (-\frac {\arcsin (a x)^2}{x}+2 a \left (-2 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )\right )\right )\right )\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -\frac {\arcsin (a x)^4}{3 x^3}+\frac {4}{3} a \left (\frac {1}{2} a^2 \left (3 \left (i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-2 i \int \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )d\arcsin (a x)\right )-3 \left (i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-2 i \int \arcsin (a x) \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )d\arcsin (a x)\right )-2 \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )\right )-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 x^2}+\frac {3}{2} a \left (-\frac {\arcsin (a x)^2}{x}+2 a \left (-2 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )\right )\right )\right )\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle -\frac {\arcsin (a x)^4}{3 x^3}+\frac {4}{3} a \left (\frac {1}{2} a^2 \left (3 \left (i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-2 i \left (i \int \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )d\arcsin (a x)-i \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )\right )\right )-3 \left (i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-2 i \left (i \int \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )d\arcsin (a x)-i \arcsin (a x) \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )\right )\right )-2 \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )\right )-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 x^2}+\frac {3}{2} a \left (-\frac {\arcsin (a x)^2}{x}+2 a \left (-2 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )\right )\right )\right )\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -\frac {\arcsin (a x)^4}{3 x^3}+\frac {4}{3} a \left (\frac {1}{2} a^2 \left (3 \left (i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-2 i \left (\int e^{-i \arcsin (a x)} \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )de^{i \arcsin (a x)}-i \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )\right )\right )-3 \left (i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-2 i \left (\int e^{-i \arcsin (a x)} \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )de^{i \arcsin (a x)}-i \arcsin (a x) \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )\right )\right )-2 \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )\right )-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 x^2}+\frac {3}{2} a \left (-\frac {\arcsin (a x)^2}{x}+2 a \left (-2 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )\right )\right )\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle -\frac {\arcsin (a x)^4}{3 x^3}+\frac {4}{3} a \left (\frac {1}{2} a^2 \left (-2 \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+3 \left (i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-2 i \left (\operatorname {PolyLog}\left (4,-e^{i \arcsin (a x)}\right )-i \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )\right )\right )-3 \left (i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-2 i \left (\operatorname {PolyLog}\left (4,e^{i \arcsin (a x)}\right )-i \arcsin (a x) \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )\right )\right )\right )-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 x^2}+\frac {3}{2} a \left (-\frac {\arcsin (a x)^2}{x}+2 a \left (-2 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )\right )\right )\right )\) |
Input:
Int[ArcSin[a*x]^4/x^4,x]
Output:
-1/3*ArcSin[a*x]^4/x^3 + (4*a*(-1/2*(Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/x^2 + (3*a*(-(ArcSin[a*x]^2/x) + 2*a*(-2*ArcSin[a*x]*ArcTanh[E^(I*ArcSin[a*x]) ] + I*PolyLog[2, -E^(I*ArcSin[a*x])] - I*PolyLog[2, E^(I*ArcSin[a*x])])))/ 2 + (a^2*(-2*ArcSin[a*x]^3*ArcTanh[E^(I*ArcSin[a*x])] + 3*(I*ArcSin[a*x]^2 *PolyLog[2, -E^(I*ArcSin[a*x])] - (2*I)*((-I)*ArcSin[a*x]*PolyLog[3, -E^(I *ArcSin[a*x])] + PolyLog[4, -E^(I*ArcSin[a*x])])) - 3*(I*ArcSin[a*x]^2*Pol yLog[2, E^(I*ArcSin[a*x])] - (2*I)*((-I)*ArcSin[a*x]*PolyLog[3, E^(I*ArcSi n[a*x])] + PolyLog[4, E^(I*ArcSin[a*x])]))))/2))/3
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[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[(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[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[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]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
Time = 0.34 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.37
method | result | size |
derivativedivides | \(a^{3} \left (-\frac {\arcsin \left (a x \right )^{2} \left (2 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x +\arcsin \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{3 a^{3} x^{3}}-\frac {2 \arcsin \left (a x \right )^{3} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )}{3}+2 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 i \operatorname {polylog}\left (4, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+\frac {2 \arcsin \left (a x \right )^{3} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )}{3}-2 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 i \operatorname {polylog}\left (4, i a x +\sqrt {-a^{2} x^{2}+1}\right )-4 \arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 i \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+4 \arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 i \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )\) | \(377\) |
default | \(a^{3} \left (-\frac {\arcsin \left (a x \right )^{2} \left (2 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x +\arcsin \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{3 a^{3} x^{3}}-\frac {2 \arcsin \left (a x \right )^{3} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )}{3}+2 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 i \operatorname {polylog}\left (4, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+\frac {2 \arcsin \left (a x \right )^{3} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )}{3}-2 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 i \operatorname {polylog}\left (4, i a x +\sqrt {-a^{2} x^{2}+1}\right )-4 \arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 i \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+4 \arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 i \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )\) | \(377\) |
Input:
int(arcsin(a*x)^4/x^4,x,method=_RETURNVERBOSE)
Output:
a^3*(-1/3/a^3/x^3*arcsin(a*x)^2*(2*arcsin(a*x)*(-a^2*x^2+1)^(1/2)*a*x+arcs in(a*x)^2+6*a^2*x^2)-2/3*arcsin(a*x)^3*ln(1+I*a*x+(-a^2*x^2+1)^(1/2))+2*I* arcsin(a*x)^2*polylog(2,-I*a*x-(-a^2*x^2+1)^(1/2))-4*arcsin(a*x)*polylog(3 ,-I*a*x-(-a^2*x^2+1)^(1/2))-4*I*polylog(4,-I*a*x-(-a^2*x^2+1)^(1/2))+2/3*a rcsin(a*x)^3*ln(1-I*a*x-(-a^2*x^2+1)^(1/2))-2*I*arcsin(a*x)^2*polylog(2,I* a*x+(-a^2*x^2+1)^(1/2))+4*arcsin(a*x)*polylog(3,I*a*x+(-a^2*x^2+1)^(1/2))+ 4*I*polylog(4,I*a*x+(-a^2*x^2+1)^(1/2))-4*arcsin(a*x)*ln(1+I*a*x+(-a^2*x^2 +1)^(1/2))+4*I*polylog(2,-I*a*x-(-a^2*x^2+1)^(1/2))+4*arcsin(a*x)*ln(1-I*a *x-(-a^2*x^2+1)^(1/2))-4*I*polylog(2,I*a*x+(-a^2*x^2+1)^(1/2)))
\[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=\int { \frac {\arcsin \left (a x\right )^{4}}{x^{4}} \,d x } \] Input:
integrate(arcsin(a*x)^4/x^4,x, algorithm="fricas")
Output:
integral(arcsin(a*x)^4/x^4, x)
\[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=\int \frac {\operatorname {asin}^{4}{\left (a x \right )}}{x^{4}}\, dx \] Input:
integrate(asin(a*x)**4/x**4,x)
Output:
Integral(asin(a*x)**4/x**4, x)
\[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=\int { \frac {\arcsin \left (a x\right )^{4}}{x^{4}} \,d x } \] Input:
integrate(arcsin(a*x)^4/x^4,x, algorithm="maxima")
Output:
-1/3*(12*a*x^3*integrate(1/3*sqrt(a*x + 1)*sqrt(-a*x + 1)*arctan2(a*x, sqr t(a*x + 1)*sqrt(-a*x + 1))^3/(a^2*x^5 - x^3), x) + arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^4)/x^3
\[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=\int { \frac {\arcsin \left (a x\right )^{4}}{x^{4}} \,d x } \] Input:
integrate(arcsin(a*x)^4/x^4,x, algorithm="giac")
Output:
integrate(arcsin(a*x)^4/x^4, x)
Timed out. \[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=\int \frac {{\mathrm {asin}\left (a\,x\right )}^4}{x^4} \,d x \] Input:
int(asin(a*x)^4/x^4,x)
Output:
int(asin(a*x)^4/x^4, x)
\[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=\int \frac {\mathit {asin} \left (a x \right )^{4}}{x^{4}}d x \] Input:
int(asin(a*x)^4/x^4,x)
Output:
int(asin(a*x)**4/x**4,x)