3.54 \(\int \frac{a+b \sin ^{-1}(c x)}{x^4 (d-c^2 d x^2)^3} \, dx\)

Optimal. Leaf size=317 \[ \frac{35 i b c^3 \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{8 d^3}-\frac{35 i b c^3 \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{8 d^3}+\frac{35 c^4 x \left (a+b \sin ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac{35 c^4 x \left (a+b \sin ^{-1}(c x)\right )}{12 d^3 \left (1-c^2 x^2\right )^2}-\frac{7 c^2 \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}-\frac{a+b \sin ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac{35 i c^3 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 d^3}-\frac{29 b c^3}{24 d^3 \sqrt{1-c^2 x^2}}+\frac{b c^3}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{b c}{6 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac{19 b c^3 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{6 d^3} \]

[Out]

(b*c^3)/(12*d^3*(1 - c^2*x^2)^(3/2)) - (b*c)/(6*d^3*x^2*(1 - c^2*x^2)^(3/2)) - (29*b*c^3)/(24*d^3*Sqrt[1 - c^2
*x^2]) - (a + b*ArcSin[c*x])/(3*d^3*x^3*(1 - c^2*x^2)^2) - (7*c^2*(a + b*ArcSin[c*x]))/(3*d^3*x*(1 - c^2*x^2)^
2) + (35*c^4*x*(a + b*ArcSin[c*x]))/(12*d^3*(1 - c^2*x^2)^2) + (35*c^4*x*(a + b*ArcSin[c*x]))/(8*d^3*(1 - c^2*
x^2)) - (((35*I)/4)*c^3*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/d^3 - (19*b*c^3*ArcTanh[Sqrt[1 - c^2*x^
2]])/(6*d^3) + (((35*I)/8)*b*c^3*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/d^3 - (((35*I)/8)*b*c^3*PolyLog[2, I*E^(I
*ArcSin[c*x])])/d^3

________________________________________________________________________________________

Rubi [A]  time = 0.381699, antiderivative size = 369, normalized size of antiderivative = 1.16, number of steps used = 23, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {4701, 4655, 4657, 4181, 2279, 2391, 261, 266, 51, 63, 208} \[ \frac{35 i b c^3 \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{8 d^3}-\frac{35 i b c^3 \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{8 d^3}+\frac{35 c^4 x \left (a+b \sin ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac{35 c^4 x \left (a+b \sin ^{-1}(c x)\right )}{12 d^3 \left (1-c^2 x^2\right )^2}-\frac{7 c^2 \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}-\frac{a+b \sin ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac{35 i c^3 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 d^3}-\frac{49 b c^3}{24 d^3 \sqrt{1-c^2 x^2}}-\frac{7 b c^3}{36 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{5 b c \sqrt{1-c^2 x^2}}{6 d^3 x^2}+\frac{5 b c}{9 d^3 x^2 \sqrt{1-c^2 x^2}}+\frac{b c}{9 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac{19 b c^3 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{6 d^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c*x])/(x^4*(d - c^2*d*x^2)^3),x]

[Out]

(-7*b*c^3)/(36*d^3*(1 - c^2*x^2)^(3/2)) + (b*c)/(9*d^3*x^2*(1 - c^2*x^2)^(3/2)) - (49*b*c^3)/(24*d^3*Sqrt[1 -
c^2*x^2]) + (5*b*c)/(9*d^3*x^2*Sqrt[1 - c^2*x^2]) - (5*b*c*Sqrt[1 - c^2*x^2])/(6*d^3*x^2) - (a + b*ArcSin[c*x]
)/(3*d^3*x^3*(1 - c^2*x^2)^2) - (7*c^2*(a + b*ArcSin[c*x]))/(3*d^3*x*(1 - c^2*x^2)^2) + (35*c^4*x*(a + b*ArcSi
n[c*x]))/(12*d^3*(1 - c^2*x^2)^2) + (35*c^4*x*(a + b*ArcSin[c*x]))/(8*d^3*(1 - c^2*x^2)) - (((35*I)/4)*c^3*(a
+ b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/d^3 - (19*b*c^3*ArcTanh[Sqrt[1 - c^2*x^2]])/(6*d^3) + (((35*I)/8)*
b*c^3*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/d^3 - (((35*I)/8)*b*c^3*PolyLog[2, I*E^(I*ArcSin[c*x])])/d^3

Rule 4701

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] + (Dist[(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] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^F
racPart[p])/(f*(m + 1)*(1 - c^2*x^2)^FracPart[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] && LtQ[m, -1] && Inte
gerQ[m]

Rule 4655

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(x*(d + e*x^2)^(p
+ 1)*(a + b*ArcSin[c*x])^n)/(2*d*(p + 1)), x] + (Dist[(2*p + 3)/(2*d*(p + 1)), Int[(d + e*x^2)^(p + 1)*(a + b*
ArcSin[c*x])^n, x], x] + Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*(p + 1)*(1 - c^2*x^2)^FracPart[p
]), Int[x*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2
*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 4657

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/(c*d), Subst[Int[(a +
b*x)^n*Sec[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]

Rule 4181

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E
^(I*k*Pi)*E^(I*(e + f*x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e,
f}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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]

Rule 2391

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]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{a+b \sin ^{-1}(c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx &=-\frac{a+b \sin ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac{1}{3} \left (7 c^2\right ) \int \frac{a+b \sin ^{-1}(c x)}{x^2 \left (d-c^2 d x^2\right )^3} \, dx+\frac{(b c) \int \frac{1}{x^3 \left (1-c^2 x^2\right )^{5/2}} \, dx}{3 d^3}\\ &=-\frac{a+b \sin ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac{7 c^2 \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac{1}{3} \left (35 c^4\right ) \int \frac{a+b \sin ^{-1}(c x)}{\left (d-c^2 d x^2\right )^3} \, dx+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-c^2 x\right )^{5/2}} \, dx,x,x^2\right )}{6 d^3}+\frac{\left (7 b c^3\right ) \int \frac{1}{x \left (1-c^2 x^2\right )^{5/2}} \, dx}{3 d^3}\\ &=\frac{b c}{9 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac{a+b \sin ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac{7 c^2 \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \sin ^{-1}(c x)\right )}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac{(5 b c) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{18 d^3}+\frac{\left (7 b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1-c^2 x\right )^{5/2}} \, dx,x,x^2\right )}{6 d^3}-\frac{\left (35 b c^5\right ) \int \frac{x}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{12 d^3}+\frac{\left (35 c^4\right ) \int \frac{a+b \sin ^{-1}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx}{4 d}\\ &=-\frac{7 b c^3}{36 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac{b c}{9 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}+\frac{5 b c}{9 d^3 x^2 \sqrt{1-c^2 x^2}}-\frac{a+b \sin ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac{7 c^2 \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \sin ^{-1}(c x)\right )}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \sin ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac{(5 b c) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{6 d^3}+\frac{\left (7 b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1-c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{6 d^3}-\frac{\left (35 b c^5\right ) \int \frac{x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{8 d^3}+\frac{\left (35 c^4\right ) \int \frac{a+b \sin ^{-1}(c x)}{d-c^2 d x^2} \, dx}{8 d^2}\\ &=-\frac{7 b c^3}{36 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac{b c}{9 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac{49 b c^3}{24 d^3 \sqrt{1-c^2 x^2}}+\frac{5 b c}{9 d^3 x^2 \sqrt{1-c^2 x^2}}-\frac{5 b c \sqrt{1-c^2 x^2}}{6 d^3 x^2}-\frac{a+b \sin ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac{7 c^2 \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \sin ^{-1}(c x)\right )}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \sin ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac{\left (35 c^3\right ) \operatorname{Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{8 d^3}+\frac{\left (5 b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{12 d^3}+\frac{\left (7 b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{6 d^3}\\ &=-\frac{7 b c^3}{36 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac{b c}{9 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac{49 b c^3}{24 d^3 \sqrt{1-c^2 x^2}}+\frac{5 b c}{9 d^3 x^2 \sqrt{1-c^2 x^2}}-\frac{5 b c \sqrt{1-c^2 x^2}}{6 d^3 x^2}-\frac{a+b \sin ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac{7 c^2 \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \sin ^{-1}(c x)\right )}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \sin ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}-\frac{35 i c^3 \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 d^3}-\frac{(5 b c) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c^2}-\frac{x^2}{c^2}} \, dx,x,\sqrt{1-c^2 x^2}\right )}{6 d^3}-\frac{(7 b c) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c^2}-\frac{x^2}{c^2}} \, dx,x,\sqrt{1-c^2 x^2}\right )}{3 d^3}-\frac{\left (35 b c^3\right ) \operatorname{Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{8 d^3}+\frac{\left (35 b c^3\right ) \operatorname{Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{8 d^3}\\ &=-\frac{7 b c^3}{36 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac{b c}{9 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac{49 b c^3}{24 d^3 \sqrt{1-c^2 x^2}}+\frac{5 b c}{9 d^3 x^2 \sqrt{1-c^2 x^2}}-\frac{5 b c \sqrt{1-c^2 x^2}}{6 d^3 x^2}-\frac{a+b \sin ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac{7 c^2 \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \sin ^{-1}(c x)\right )}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \sin ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}-\frac{35 i c^3 \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 d^3}-\frac{19 b c^3 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{6 d^3}+\frac{\left (35 i b c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{8 d^3}-\frac{\left (35 i b c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{8 d^3}\\ &=-\frac{7 b c^3}{36 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac{b c}{9 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac{49 b c^3}{24 d^3 \sqrt{1-c^2 x^2}}+\frac{5 b c}{9 d^3 x^2 \sqrt{1-c^2 x^2}}-\frac{5 b c \sqrt{1-c^2 x^2}}{6 d^3 x^2}-\frac{a+b \sin ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac{7 c^2 \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \sin ^{-1}(c x)\right )}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \sin ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}-\frac{35 i c^3 \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 d^3}-\frac{19 b c^3 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{6 d^3}+\frac{35 i b c^3 \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{8 d^3}-\frac{35 i b c^3 \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{8 d^3}\\ \end{align*}

Mathematica [A]  time = 1.52046, size = 587, normalized size = 1.85 \[ -\frac{-210 i b c^3 \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )+210 i b c^3 \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )+\frac{66 a c^4 x}{c^2 x^2-1}-\frac{12 a c^4 x}{\left (c^2 x^2-1\right )^2}+\frac{144 a c^2}{x}+105 a c^3 \log (1-c x)-105 a c^3 \log (c x+1)+\frac{16 a}{x^3}-\frac{b c^4 x \sqrt{1-c^2 x^2}}{(c x-1)^2}+\frac{b c^4 x \sqrt{1-c^2 x^2}}{(c x+1)^2}-\frac{33 b c^3 \sqrt{1-c^2 x^2}}{c x-1}+\frac{33 b c^3 \sqrt{1-c^2 x^2}}{c x+1}+\frac{2 b c^3 \sqrt{1-c^2 x^2}}{(c x-1)^2}+\frac{2 b c^3 \sqrt{1-c^2 x^2}}{(c x+1)^2}+\frac{8 b c \sqrt{1-c^2 x^2}}{x^2}+152 b c^3 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )+\frac{33 b c^3 \sin ^{-1}(c x)}{c x-1}+\frac{33 b c^3 \sin ^{-1}(c x)}{c x+1}-\frac{3 b c^3 \sin ^{-1}(c x)}{(c x-1)^2}+\frac{3 b c^3 \sin ^{-1}(c x)}{(c x+1)^2}+105 i \pi b c^3 \sin ^{-1}(c x)+\frac{144 b c^2 \sin ^{-1}(c x)}{x}-210 b c^3 \sin ^{-1}(c x) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )-105 \pi b c^3 \log \left (1-i e^{i \sin ^{-1}(c x)}\right )+210 b c^3 \sin ^{-1}(c x) \log \left (1+i e^{i \sin ^{-1}(c x)}\right )-105 \pi b c^3 \log \left (1+i e^{i \sin ^{-1}(c x)}\right )+105 \pi b c^3 \log \left (\sin \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )+105 \pi b c^3 \log \left (-\cos \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )+\frac{16 b \sin ^{-1}(c x)}{x^3}}{48 d^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcSin[c*x])/(x^4*(d - c^2*d*x^2)^3),x]

[Out]

-((16*a)/x^3 + (144*a*c^2)/x + (8*b*c*Sqrt[1 - c^2*x^2])/x^2 + (2*b*c^3*Sqrt[1 - c^2*x^2])/(-1 + c*x)^2 - (b*c
^4*x*Sqrt[1 - c^2*x^2])/(-1 + c*x)^2 - (33*b*c^3*Sqrt[1 - c^2*x^2])/(-1 + c*x) + (2*b*c^3*Sqrt[1 - c^2*x^2])/(
1 + c*x)^2 + (b*c^4*x*Sqrt[1 - c^2*x^2])/(1 + c*x)^2 + (33*b*c^3*Sqrt[1 - c^2*x^2])/(1 + c*x) - (12*a*c^4*x)/(
-1 + c^2*x^2)^2 + (66*a*c^4*x)/(-1 + c^2*x^2) + (105*I)*b*c^3*Pi*ArcSin[c*x] + (16*b*ArcSin[c*x])/x^3 + (144*b
*c^2*ArcSin[c*x])/x - (3*b*c^3*ArcSin[c*x])/(-1 + c*x)^2 + (33*b*c^3*ArcSin[c*x])/(-1 + c*x) + (3*b*c^3*ArcSin
[c*x])/(1 + c*x)^2 + (33*b*c^3*ArcSin[c*x])/(1 + c*x) + 152*b*c^3*ArcTanh[Sqrt[1 - c^2*x^2]] - 105*b*c^3*Pi*Lo
g[1 - I*E^(I*ArcSin[c*x])] - 210*b*c^3*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 105*b*c^3*Pi*Log[1 + I*E^(I*
ArcSin[c*x])] + 210*b*c^3*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 105*a*c^3*Log[1 - c*x] - 105*a*c^3*Log[1
+ c*x] + 105*b*c^3*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 105*b*c^3*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (210
*I)*b*c^3*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (210*I)*b*c^3*PolyLog[2, I*E^(I*ArcSin[c*x])])/(48*d^3)

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Maple [A]  time = 0.304, size = 576, normalized size = 1.8 \begin{align*}{\frac{{c}^{3}a}{16\,{d}^{3} \left ( cx-1 \right ) ^{2}}}-{\frac{11\,{c}^{3}a}{16\,{d}^{3} \left ( cx-1 \right ) }}-{\frac{35\,{c}^{3}a\ln \left ( cx-1 \right ) }{16\,{d}^{3}}}-{\frac{{c}^{3}a}{16\,{d}^{3} \left ( cx+1 \right ) ^{2}}}-{\frac{11\,{c}^{3}a}{16\,{d}^{3} \left ( cx+1 \right ) }}+{\frac{35\,{c}^{3}a\ln \left ( cx+1 \right ) }{16\,{d}^{3}}}-{\frac{a}{3\,{d}^{3}{x}^{3}}}-3\,{\frac{{c}^{2}a}{{d}^{3}x}}-{\frac{35\,{c}^{6}b\arcsin \left ( cx \right ){x}^{3}}{8\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }}+{\frac{29\,{c}^{5}b{x}^{2}}{24\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{175\,{c}^{4}b\arcsin \left ( cx \right ) x}{24\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }}-{\frac{9\,b{c}^{3}}{8\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{7\,{c}^{2}b\arcsin \left ( cx \right ) }{3\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) x}}-{\frac{bc}{6\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ){x}^{2}}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{b\arcsin \left ( cx \right ) }{3\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ){x}^{3}}}+{\frac{19\,b{c}^{3}}{6\,{d}^{3}}\ln \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1}-1 \right ) }-{\frac{19\,b{c}^{3}}{6\,{d}^{3}}\ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) }+{\frac{{\frac{35\,i}{8}}{c}^{3}b}{{d}^{3}}{\it dilog} \left ( 1+i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) }-{\frac{{\frac{35\,i}{8}}{c}^{3}b}{{d}^{3}}{\it dilog} \left ( 1-i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) }-{\frac{35\,b{c}^{3}\arcsin \left ( cx \right ) }{8\,{d}^{3}}\ln \left ( 1+i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) }+{\frac{35\,b{c}^{3}\arcsin \left ( cx \right ) }{8\,{d}^{3}}\ln \left ( 1-i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(c*x))/x^4/(-c^2*d*x^2+d)^3,x)

[Out]

1/16*c^3*a/d^3/(c*x-1)^2-11/16*c^3*a/d^3/(c*x-1)-35/16*c^3*a/d^3*ln(c*x-1)-1/16*c^3*a/d^3/(c*x+1)^2-11/16*c^3*
a/d^3/(c*x+1)+35/16*c^3*a/d^3*ln(c*x+1)-1/3*a/d^3/x^3-3*c^2*a/d^3/x-35/8*c^6*b/d^3/(c^4*x^4-2*c^2*x^2+1)*arcsi
n(c*x)*x^3+29/24*c^5*b/d^3/(c^4*x^4-2*c^2*x^2+1)*x^2*(-c^2*x^2+1)^(1/2)+175/24*c^4*b/d^3/(c^4*x^4-2*c^2*x^2+1)
*arcsin(c*x)*x-9/8*c^3*b/d^3/(c^4*x^4-2*c^2*x^2+1)*(-c^2*x^2+1)^(1/2)-7/3*c^2*b/d^3/(c^4*x^4-2*c^2*x^2+1)/x*ar
csin(c*x)-1/6*c*b/d^3/(c^4*x^4-2*c^2*x^2+1)/x^2*(-c^2*x^2+1)^(1/2)-1/3*b/d^3/(c^4*x^4-2*c^2*x^2+1)/x^3*arcsin(
c*x)+19/6*c^3*b/d^3*ln(I*c*x+(-c^2*x^2+1)^(1/2)-1)-19/6*c^3*b/d^3*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+35/8*I*c^3*b/
d^3*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-35/8*I*c^3*b/d^3*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-35/8*c^3*b/d^
3*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+35/8*c^3*b/d^3*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{48} \, a{\left (\frac{105 \, c^{3} \log \left (c x + 1\right )}{d^{3}} - \frac{105 \, c^{3} \log \left (c x - 1\right )}{d^{3}} - \frac{2 \,{\left (105 \, c^{6} x^{6} - 175 \, c^{4} x^{4} + 56 \, c^{2} x^{2} + 8\right )}}{c^{4} d^{3} x^{7} - 2 \, c^{2} d^{3} x^{5} + d^{3} x^{3}}\right )} + \frac{{\left (105 \,{\left (c^{7} x^{7} - 2 \, c^{5} x^{5} + c^{3} x^{3}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) \log \left (c x + 1\right ) - 105 \,{\left (c^{7} x^{7} - 2 \, c^{5} x^{5} + c^{3} x^{3}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) \log \left (-c x + 1\right ) - 2 \,{\left (105 \, c^{6} x^{6} - 175 \, c^{4} x^{4} + 56 \, c^{2} x^{2} + 8\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) -{\left (c^{4} d^{3} x^{7} - 2 \, c^{2} d^{3} x^{5} + d^{3} x^{3}\right )} \int \frac{{\left (210 \, c^{7} x^{6} - 350 \, c^{5} x^{4} + 112 \, c^{3} x^{2} - 105 \,{\left (c^{8} x^{7} - 2 \, c^{6} x^{5} + c^{4} x^{3}\right )} \log \left (c x + 1\right ) + 105 \,{\left (c^{8} x^{7} - 2 \, c^{6} x^{5} + c^{4} x^{3}\right )} \log \left (-c x + 1\right ) + 16 \, c\right )} \sqrt{c x + 1} \sqrt{-c x + 1}}{c^{6} d^{3} x^{9} - 3 \, c^{4} d^{3} x^{7} + 3 \, c^{2} d^{3} x^{5} - d^{3} x^{3}}\,{d x}\right )} b}{48 \,{\left (c^{4} d^{3} x^{7} - 2 \, c^{2} d^{3} x^{5} + d^{3} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))/x^4/(-c^2*d*x^2+d)^3,x, algorithm="maxima")

[Out]

1/48*a*(105*c^3*log(c*x + 1)/d^3 - 105*c^3*log(c*x - 1)/d^3 - 2*(105*c^6*x^6 - 175*c^4*x^4 + 56*c^2*x^2 + 8)/(
c^4*d^3*x^7 - 2*c^2*d^3*x^5 + d^3*x^3)) + 1/48*(105*(c^7*x^7 - 2*c^5*x^5 + c^3*x^3)*arctan2(c*x, sqrt(c*x + 1)
*sqrt(-c*x + 1))*log(c*x + 1) - 105*(c^7*x^7 - 2*c^5*x^5 + c^3*x^3)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))
*log(-c*x + 1) - 2*(105*c^6*x^6 - 175*c^4*x^4 + 56*c^2*x^2 + 8)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + 4
8*(c^4*d^3*x^7 - 2*c^2*d^3*x^5 + d^3*x^3)*integrate(-1/48*(210*c^7*x^6 - 350*c^5*x^4 + 112*c^3*x^2 - 105*(c^8*
x^7 - 2*c^6*x^5 + c^4*x^3)*log(c*x + 1) + 105*(c^8*x^7 - 2*c^6*x^5 + c^4*x^3)*log(-c*x + 1) + 16*c)*sqrt(c*x +
1)*sqrt(-c*x + 1)/(c^6*d^3*x^9 - 3*c^4*d^3*x^7 + 3*c^2*d^3*x^5 - d^3*x^3), x))*b/(c^4*d^3*x^7 - 2*c^2*d^3*x^5
+ d^3*x^3)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{b \arcsin \left (c x\right ) + a}{c^{6} d^{3} x^{10} - 3 \, c^{4} d^{3} x^{8} + 3 \, c^{2} d^{3} x^{6} - d^{3} x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))/x^4/(-c^2*d*x^2+d)^3,x, algorithm="fricas")

[Out]

integral(-(b*arcsin(c*x) + a)/(c^6*d^3*x^10 - 3*c^4*d^3*x^8 + 3*c^2*d^3*x^6 - d^3*x^4), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(c*x))/x**4/(-c**2*d*x**2+d)**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{b \arcsin \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))/x^4/(-c^2*d*x^2+d)^3,x, algorithm="giac")

[Out]

integrate(-(b*arcsin(c*x) + a)/((c^2*d*x^2 - d)^3*x^4), x)