Integrand size = 27, antiderivative size = 210 \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=\frac {b^2 x^2}{4 c^2 d}-\frac {b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^3 d}+\frac {(a+b \arcsin (c x))^2}{4 c^4 d}-\frac {x^2 (a+b \arcsin (c x))^2}{2 c^2 d}+\frac {i (a+b \arcsin (c x))^3}{3 b c^4 d}-\frac {(a+b \arcsin (c x))^2 \log \left (1+e^{2 i \arcsin (c x)}\right )}{c^4 d}+\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{c^4 d}-\frac {b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )}{2 c^4 d} \] Output:
1/4*b^2*x^2/c^2/d-1/2*b*x*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c^3/d+1/4*( a+b*arcsin(c*x))^2/c^4/d-1/2*x^2*(a+b*arcsin(c*x))^2/c^2/d+1/3*I*(a+b*arcs in(c*x))^3/b/c^4/d-(a+b*arcsin(c*x))^2*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)/ c^4/d+I*b*(a+b*arcsin(c*x))*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)/c^4/d -1/2*b^2*polylog(3,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)/c^4/d
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(459\) vs. \(2(210)=420\).
Time = 0.52 (sec) , antiderivative size = 459, normalized size of antiderivative = 2.19 \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=-\frac {12 a^2 c^2 x^2+12 a b c x \sqrt {1-c^2 x^2}+48 i a b \pi \arcsin (c x)+24 a b c^2 x^2 \arcsin (c x)-24 i a b \arcsin (c x)^2-8 i b^2 \arcsin (c x)^3-24 a b \arctan \left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )+3 b^2 \cos (2 \arcsin (c x))-6 b^2 \arcsin (c x)^2 \cos (2 \arcsin (c x))+96 a b \pi \log \left (1+e^{-i \arcsin (c x)}\right )+24 a b \pi \log \left (1-i e^{i \arcsin (c x)}\right )+48 a b \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )-24 a b \pi \log \left (1+i e^{i \arcsin (c x)}\right )+48 a b \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )+24 b^2 \arcsin (c x)^2 \log \left (1+e^{2 i \arcsin (c x)}\right )+12 a^2 \log \left (1-c^2 x^2\right )-96 a b \pi \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )+24 a b \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-24 a b \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-48 i a b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-48 i a b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )-24 i b^2 \arcsin (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )+12 b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )+6 b^2 \arcsin (c x) \sin (2 \arcsin (c x))}{24 c^4 d} \] Input:
Integrate[(x^3*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2),x]
Output:
-1/24*(12*a^2*c^2*x^2 + 12*a*b*c*x*Sqrt[1 - c^2*x^2] + (48*I)*a*b*Pi*ArcSi n[c*x] + 24*a*b*c^2*x^2*ArcSin[c*x] - (24*I)*a*b*ArcSin[c*x]^2 - (8*I)*b^2 *ArcSin[c*x]^3 - 24*a*b*ArcTan[(c*x)/(-1 + Sqrt[1 - c^2*x^2])] + 3*b^2*Cos [2*ArcSin[c*x]] - 6*b^2*ArcSin[c*x]^2*Cos[2*ArcSin[c*x]] + 96*a*b*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 24*a*b*Pi*Log[1 - I*E^(I*ArcSin[c*x])] + 48*a*b *ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 24*a*b*Pi*Log[1 + I*E^(I*ArcSi n[c*x])] + 48*a*b*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 24*b^2*ArcSin [c*x]^2*Log[1 + E^((2*I)*ArcSin[c*x])] + 12*a^2*Log[1 - c^2*x^2] - 96*a*b* Pi*Log[Cos[ArcSin[c*x]/2]] + 24*a*b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] - 24*a*b*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (48*I)*a*b*PolyLog[2, (-I)*E ^(I*ArcSin[c*x])] - (48*I)*a*b*PolyLog[2, I*E^(I*ArcSin[c*x])] - (24*I)*b^ 2*ArcSin[c*x]*PolyLog[2, -E^((2*I)*ArcSin[c*x])] + 12*b^2*PolyLog[3, -E^(( 2*I)*ArcSin[c*x])] + 6*b^2*ArcSin[c*x]*Sin[2*ArcSin[c*x]])/(c^4*d)
Time = 1.61 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.99, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5210, 27, 5180, 3042, 4202, 2620, 3011, 2720, 5210, 15, 5152, 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.
\(\displaystyle \int \frac {x^3 (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx\) |
\(\Big \downarrow \) 5210 |
\(\displaystyle \frac {b \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{c d}+\frac {\int \frac {x (a+b \arcsin (c x))^2}{d \left (1-c^2 x^2\right )}dx}{c^2}-\frac {x^2 (a+b \arcsin (c x))^2}{2 c^2 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{c d}+\frac {\int \frac {x (a+b \arcsin (c x))^2}{1-c^2 x^2}dx}{c^2 d}-\frac {x^2 (a+b \arcsin (c x))^2}{2 c^2 d}\) |
\(\Big \downarrow \) 5180 |
\(\displaystyle \frac {b \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{c d}+\frac {\int \frac {c x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{c^4 d}-\frac {x^2 (a+b \arcsin (c x))^2}{2 c^2 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int (a+b \arcsin (c x))^2 \tan (\arcsin (c x))d\arcsin (c x)}{c^4 d}+\frac {b \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{c d}-\frac {x^2 (a+b \arcsin (c x))^2}{2 c^2 d}\) |
\(\Big \downarrow \) 4202 |
\(\displaystyle \frac {\frac {i (a+b \arcsin (c x))^3}{3 b}-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))^2}{1+e^{2 i \arcsin (c x)}}d\arcsin (c x)}{c^4 d}+\frac {b \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{c d}-\frac {x^2 (a+b \arcsin (c x))^2}{2 c^2 d}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {\frac {i (a+b \arcsin (c x))^3}{3 b}-2 i \left (i b \int (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )d\arcsin (c x)-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{c^4 d}+\frac {b \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{c d}-\frac {x^2 (a+b \arcsin (c x))^2}{2 c^2 d}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {\frac {i (a+b \arcsin (c x))^3}{3 b}-2 i \left (i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \int \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{c^4 d}+\frac {b \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{c d}-\frac {x^2 (a+b \arcsin (c x))^2}{2 c^2 d}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\frac {i (a+b \arcsin (c x))^3}{3 b}-2 i \left (i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{c^4 d}+\frac {b \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{c d}-\frac {x^2 (a+b \arcsin (c x))^2}{2 c^2 d}\) |
\(\Big \downarrow \) 5210 |
\(\displaystyle \frac {\frac {i (a+b \arcsin (c x))^3}{3 b}-2 i \left (i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{c^4 d}+\frac {b \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}+\frac {b \int xdx}{2 c}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}\right )}{c d}-\frac {x^2 (a+b \arcsin (c x))^2}{2 c^2 d}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {\frac {i (a+b \arcsin (c x))^3}{3 b}-2 i \left (i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{c^4 d}+\frac {b \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}+\frac {b x^2}{4 c}\right )}{c d}-\frac {x^2 (a+b \arcsin (c x))^2}{2 c^2 d}\) |
\(\Big \downarrow \) 5152 |
\(\displaystyle \frac {\frac {i (a+b \arcsin (c x))^3}{3 b}-2 i \left (i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{c^4 d}-\frac {x^2 (a+b \arcsin (c x))^2}{2 c^2 d}+\frac {b \left (\frac {(a+b \arcsin (c x))^2}{4 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}+\frac {b x^2}{4 c}\right )}{c d}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {\frac {i (a+b \arcsin (c x))^3}{3 b}-2 i \left (i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )\right )-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{c^4 d}-\frac {x^2 (a+b \arcsin (c x))^2}{2 c^2 d}+\frac {b \left (\frac {(a+b \arcsin (c x))^2}{4 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}+\frac {b x^2}{4 c}\right )}{c d}\) |
Input:
Int[(x^3*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2),x]
Output:
-1/2*(x^2*(a + b*ArcSin[c*x])^2)/(c^2*d) + (b*((b*x^2)/(4*c) - (x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*c^2) + (a + b*ArcSin[c*x])^2/(4*b*c^3))) /(c*d) + (((I/3)*(a + b*ArcSin[c*x])^3)/b - (2*I)*((-1/2*I)*(a + b*ArcSin[ c*x])^2*Log[1 + E^((2*I)*ArcSin[c*x])] + I*b*((I/2)*(a + b*ArcSin[c*x])*Po lyLog[2, -E^((2*I)*ArcSin[c*x])] - (b*PolyLog[3, -E^((2*I)*ArcSin[c*x])])/ 4)))/(c^4*d)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[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[((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[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && NeQ[n, -1]
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_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + S imp[b*f*(n/(c*(m + 2*p + 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] && IGtQ[m , 1] && NeQ[m + 2*p + 1, 0]
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 = 0.57 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.77
method | result | size |
derivativedivides | \(\frac {-\frac {a^{2} \left (\frac {c^{2} x^{2}}{2}+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}+\frac {i a b \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}-\frac {b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x}{2 d}-\frac {b^{2} \arcsin \left (c x \right )^{2} c^{2} x^{2}}{2 d}+\frac {b^{2} \arcsin \left (c x \right )^{2}}{4 d}+\frac {b^{2} c^{2} x^{2}}{4 d}-\frac {b^{2}}{8 d}-\frac {b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {i b^{2} \arcsin \left (c x \right )^{3}}{3 d}-\frac {b^{2} \operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2 d}+\frac {i b^{2} \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}-\frac {a b \sqrt {-c^{2} x^{2}+1}\, c x}{2 d}-\frac {a b \arcsin \left (c x \right ) c^{2} x^{2}}{d}+\frac {a b \arcsin \left (c x \right )}{2 d}-\frac {2 a b \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {i a b \arcsin \left (c x \right )^{2}}{d}}{c^{4}}\) | \(371\) |
default | \(\frac {-\frac {a^{2} \left (\frac {c^{2} x^{2}}{2}+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}+\frac {i a b \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}-\frac {b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x}{2 d}-\frac {b^{2} \arcsin \left (c x \right )^{2} c^{2} x^{2}}{2 d}+\frac {b^{2} \arcsin \left (c x \right )^{2}}{4 d}+\frac {b^{2} c^{2} x^{2}}{4 d}-\frac {b^{2}}{8 d}-\frac {b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {i b^{2} \arcsin \left (c x \right )^{3}}{3 d}-\frac {b^{2} \operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2 d}+\frac {i b^{2} \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}-\frac {a b \sqrt {-c^{2} x^{2}+1}\, c x}{2 d}-\frac {a b \arcsin \left (c x \right ) c^{2} x^{2}}{d}+\frac {a b \arcsin \left (c x \right )}{2 d}-\frac {2 a b \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {i a b \arcsin \left (c x \right )^{2}}{d}}{c^{4}}\) | \(371\) |
parts | \(-\frac {a^{2} x^{2}}{2 d \,c^{2}}-\frac {a^{2} \ln \left (c^{2} x^{2}-1\right )}{2 d \,c^{4}}+\frac {i a b \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d \,c^{4}}-\frac {b^{2} \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x}{2 d \,c^{3}}-\frac {b^{2} \arcsin \left (c x \right )^{2} x^{2}}{2 d \,c^{2}}+\frac {b^{2} x^{2}}{4 c^{2} d}+\frac {b^{2} \arcsin \left (c x \right )^{2}}{4 d \,c^{4}}-\frac {b^{2}}{8 d \,c^{4}}-\frac {b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d \,c^{4}}+\frac {i a b \arcsin \left (c x \right )^{2}}{d \,c^{4}}-\frac {b^{2} \operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2 c^{4} d}+\frac {i b^{2} \arcsin \left (c x \right )^{3}}{3 d \,c^{4}}-\frac {a b \sqrt {-c^{2} x^{2}+1}\, x}{2 d \,c^{3}}-\frac {a b \arcsin \left (c x \right ) x^{2}}{d \,c^{2}}+\frac {a b \arcsin \left (c x \right )}{2 d \,c^{4}}-\frac {2 a b \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d \,c^{4}}+\frac {i b^{2} \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d \,c^{4}}\) | \(403\) |
Input:
int(x^3*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d),x,method=_RETURNVERBOSE)
Output:
1/c^4*(-a^2/d*(1/2*c^2*x^2+1/2*ln(c*x-1)+1/2*ln(c*x+1))+I*a*b/d*polylog(2, -(I*c*x+(-c^2*x^2+1)^(1/2))^2)-1/2*b^2/d*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c* x-1/2*b^2/d*arcsin(c*x)^2*c^2*x^2+1/4*b^2/d*arcsin(c*x)^2+1/4*b^2/d*c^2*x^ 2-1/8*b^2/d-b^2/d*arcsin(c*x)^2*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)+1/3*I*b ^2/d*arcsin(c*x)^3-1/2*b^2*polylog(3,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d+I*b^ 2/d*arcsin(c*x)*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)-1/2*a*b/d*(-c^2*x ^2+1)^(1/2)*c*x-a*b/d*arcsin(c*x)*c^2*x^2+1/2*a*b/d*arcsin(c*x)-2*a*b/d*ar csin(c*x)*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)+I*a*b/d*arcsin(c*x)^2)
\[ \int \frac {x^3 (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{3}}{c^{2} d x^{2} - d} \,d x } \] Input:
integrate(x^3*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d),x, algorithm="fricas")
Output:
integral(-(b^2*x^3*arcsin(c*x)^2 + 2*a*b*x^3*arcsin(c*x) + a^2*x^3)/(c^2*d *x^2 - d), x)
\[ \int \frac {x^3 (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=- \frac {\int \frac {a^{2} x^{3}}{c^{2} x^{2} - 1}\, dx + \int \frac {b^{2} x^{3} \operatorname {asin}^{2}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx + \int \frac {2 a b x^{3} \operatorname {asin}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx}{d} \] Input:
integrate(x**3*(a+b*asin(c*x))**2/(-c**2*d*x**2+d),x)
Output:
-(Integral(a**2*x**3/(c**2*x**2 - 1), x) + Integral(b**2*x**3*asin(c*x)**2 /(c**2*x**2 - 1), x) + Integral(2*a*b*x**3*asin(c*x)/(c**2*x**2 - 1), x))/ d
\[ \int \frac {x^3 (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{3}}{c^{2} d x^{2} - d} \,d x } \] Input:
integrate(x^3*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d),x, algorithm="maxima")
Output:
-1/2*a^2*(x^2/(c^2*d) + log(c^2*x^2 - 1)/(c^4*d)) - 1/2*(b^2*c^2*x^2*arcta n2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*c^4*d*integrate((2*a*b*c^3*x^3 *arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + (b^2*c^2*x^2*arctan2(c*x, sq rt(c*x + 1)*sqrt(-c*x + 1)) + b^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1 ))*log(c*x + 1) + b^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(-c*x + 1))*sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^5*d*x^2 - c^3*d), x) + b^2*arctan2( c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log(c*x + 1) + b^2*arctan2(c*x, sqrt( c*x + 1)*sqrt(-c*x + 1))^2*log(-c*x + 1))/(c^4*d)
Exception generated. \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x^3*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d),x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=\int \frac {x^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{d-c^2\,d\,x^2} \,d x \] Input:
int((x^3*(a + b*asin(c*x))^2)/(d - c^2*d*x^2),x)
Output:
int((x^3*(a + b*asin(c*x))^2)/(d - c^2*d*x^2), x)
\[ \int \frac {x^3 (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=\frac {-2 \mathit {asin} \left (c x \right )^{2} b^{2} c^{2} x^{2}+\mathit {asin} \left (c x \right )^{2} b^{2}-2 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) b^{2} c x -4 \mathit {asin} \left (c x \right ) a b \,c^{2} x^{2}+2 \mathit {asin} \left (c x \right ) a b -2 \sqrt {-c^{2} x^{2}+1}\, a b c x -8 \left (\int \frac {\mathit {asin} \left (c x \right ) x}{c^{2} x^{2}-1}d x \right ) a b \,c^{2}-4 \left (\int \frac {\mathit {asin} \left (c x \right )^{2} x}{c^{2} x^{2}-1}d x \right ) b^{2} c^{2}-2 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2}-2 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2}-2 a^{2} c^{2} x^{2}+b^{2} c^{2} x^{2}-b^{2}}{4 c^{4} d} \] Input:
int(x^3*(a+b*asin(c*x))^2/(-c^2*d*x^2+d),x)
Output:
( - 2*asin(c*x)**2*b**2*c**2*x**2 + asin(c*x)**2*b**2 - 2*sqrt( - c**2*x** 2 + 1)*asin(c*x)*b**2*c*x - 4*asin(c*x)*a*b*c**2*x**2 + 2*asin(c*x)*a*b - 2*sqrt( - c**2*x**2 + 1)*a*b*c*x - 8*int((asin(c*x)*x)/(c**2*x**2 - 1),x)* a*b*c**2 - 4*int((asin(c*x)**2*x)/(c**2*x**2 - 1),x)*b**2*c**2 - 2*log(c** 2*x - c)*a**2 - 2*log(c**2*x + c)*a**2 - 2*a**2*c**2*x**2 + b**2*c**2*x**2 - b**2)/(4*c**4*d)