3.138 \(\int \frac{a+b \sin ^{-1}(c x)}{x^3 (d-c^2 d x^2)^{5/2}} \, dx\)

Optimal. Leaf size=433 \[ \frac{5 i b c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 i b c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}-\frac{5 b c^3 x}{12 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{3 b c \sqrt{1-c^2 x^2}}{4 d^2 x \sqrt{d-c^2 d x^2}}+\frac{b c}{4 d^2 x \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{13 b c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}(c x)}{6 d^2 \sqrt{d-c^2 d x^2}} \]

[Out]

(b*c)/(4*d^2*x*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) - (5*b*c^3*x)/(12*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x
^2]) - (3*b*c*Sqrt[1 - c^2*x^2])/(4*d^2*x*Sqrt[d - c^2*d*x^2]) + (5*c^2*(a + b*ArcSin[c*x]))/(6*d*(d - c^2*d*x
^2)^(3/2)) - (a + b*ArcSin[c*x])/(2*d*x^2*(d - c^2*d*x^2)^(3/2)) + (5*c^2*(a + b*ArcSin[c*x]))/(2*d^2*Sqrt[d -
c^2*d*x^2]) - (5*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^
2]) - (13*b*c^2*Sqrt[1 - c^2*x^2]*ArcTanh[c*x])/(6*d^2*Sqrt[d - c^2*d*x^2]) + (((5*I)/2)*b*c^2*Sqrt[1 - c^2*x^
2]*PolyLog[2, -E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (((5*I)/2)*b*c^2*Sqrt[1 - c^2*x^2]*PolyLog[2, E
^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.582206, antiderivative size = 433, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.407, Rules used = {4701, 4705, 4713, 4709, 4183, 2279, 2391, 206, 199, 290, 325} \[ \frac{5 i b c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 i b c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}-\frac{5 b c^3 x}{12 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{3 b c \sqrt{1-c^2 x^2}}{4 d^2 x \sqrt{d-c^2 d x^2}}+\frac{b c}{4 d^2 x \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{13 b c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}(c x)}{6 d^2 \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c*x])/(x^3*(d - c^2*d*x^2)^(5/2)),x]

[Out]

(b*c)/(4*d^2*x*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) - (5*b*c^3*x)/(12*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x
^2]) - (3*b*c*Sqrt[1 - c^2*x^2])/(4*d^2*x*Sqrt[d - c^2*d*x^2]) + (5*c^2*(a + b*ArcSin[c*x]))/(6*d*(d - c^2*d*x
^2)^(3/2)) - (a + b*ArcSin[c*x])/(2*d*x^2*(d - c^2*d*x^2)^(3/2)) + (5*c^2*(a + b*ArcSin[c*x]))/(2*d^2*Sqrt[d -
c^2*d*x^2]) - (5*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^
2]) - (13*b*c^2*Sqrt[1 - c^2*x^2]*ArcTanh[c*x])/(6*d^2*Sqrt[d - c^2*d*x^2]) + (((5*I)/2)*b*c^2*Sqrt[1 - c^2*x^
2]*PolyLog[2, -E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (((5*I)/2)*b*c^2*Sqrt[1 - c^2*x^2]*PolyLog[2, E
^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2])

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 4705

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)/(2*d*f*(p + 1)), x] + (Dist[(m + 2*p + 3)/(2*d*(p +
1)), Int[(f*x)^m*(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)^Frac
Part[p])/(2*f*(p + 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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] &&  !GtQ
[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])

Rule 4713

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[
Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2], Int[((f*x)^m*(a + b*ArcSin[c*x])^n)/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a,
b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] &&  !GtQ[d, 0] && (IntegerQ[m] || EqQ[n, 1])

Rule 4709

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[1/(c^(m
+ 1)*Sqrt[d]), 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] && GtQ[d, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 4183

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] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[(d*m)/f, Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && 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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rubi steps

\begin{align*} \int \frac{a+b \sin ^{-1}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx &=-\frac{a+b \sin ^{-1}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac{1}{2} \left (5 c^2\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx+\frac{\left (b c \sqrt{1-c^2 x^2}\right ) \int \frac{1}{x^2 \left (1-c^2 x^2\right )^2} \, dx}{2 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b c}{4 d^2 x \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac{\left (5 c^2\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx}{2 d}+\frac{\left (3 b c \sqrt{1-c^2 x^2}\right ) \int \frac{1}{x^2 \left (1-c^2 x^2\right )} \, dx}{4 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 b c^3 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{\left (1-c^2 x^2\right )^2} \, dx}{6 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b c}{4 d^2 x \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{5 b c^3 x}{12 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{3 b c \sqrt{1-c^2 x^2}}{4 d^2 x \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 c^2\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \sqrt{d-c^2 d x^2}} \, dx}{2 d^2}-\frac{\left (5 b c^3 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{1-c^2 x^2} \, dx}{12 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (3 b c^3 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{1-c^2 x^2} \, dx}{4 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 b c^3 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{1-c^2 x^2} \, dx}{2 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b c}{4 d^2 x \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{5 b c^3 x}{12 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{3 b c \sqrt{1-c^2 x^2}}{4 d^2 x \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{13 b c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}(c x)}{6 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 c^2 \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \sqrt{1-c^2 x^2}} \, dx}{2 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b c}{4 d^2 x \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{5 b c^3 x}{12 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{3 b c \sqrt{1-c^2 x^2}}{4 d^2 x \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{13 b c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}(c x)}{6 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b c}{4 d^2 x \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{5 b c^3 x}{12 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{3 b c \sqrt{1-c^2 x^2}}{4 d^2 x \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 c^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{13 b c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}(c x)}{6 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 b c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 b c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b c}{4 d^2 x \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{5 b c^3 x}{12 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{3 b c \sqrt{1-c^2 x^2}}{4 d^2 x \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 c^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{13 b c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}(c x)}{6 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 i b c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 i b c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b c}{4 d^2 x \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{5 b c^3 x}{12 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{3 b c \sqrt{1-c^2 x^2}}{4 d^2 x \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 c^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{13 b c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}(c x)}{6 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 i b c^2 \sqrt{1-c^2 x^2} \text{Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 i b c^2 \sqrt{1-c^2 x^2} \text{Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{2 d^2 \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 7.48286, size = 537, normalized size = 1.24 \[ \frac{b c^2 \sqrt{1-c^2 x^2} \left (60 i \left (\text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )-\text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )\right )-\frac{2 \left (\sin ^{-1}(c x)-1\right )}{c x-1}+52 \sin ^{-1}(c x)+60 \sin ^{-1}(c x) \left (\log \left (1-e^{i \sin ^{-1}(c x)}\right )-\log \left (1+e^{i \sin ^{-1}(c x)}\right )\right )-6 \tan \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\frac{52 \sin ^{-1}(c x) \sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )}{\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )}+\frac{4 \sin ^{-1}(c x) \sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )}{\left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )^3}-\frac{52 \sin ^{-1}(c x) \sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )}{\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )}+\frac{2 \left (\sin ^{-1}(c x)+1\right )}{\left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )^2}-\frac{4 \sin ^{-1}(c x) \sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )}{\left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )^3}-6 \cot \left (\frac{1}{2} \sin ^{-1}(c x)\right )-3 \sin ^{-1}(c x) \csc ^2\left (\frac{1}{2} \sin ^{-1}(c x)\right )+3 \sin ^{-1}(c x) \sec ^2\left (\frac{1}{2} \sin ^{-1}(c x)\right )+52 \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )-52 \log \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )\right )}{24 d^2 \sqrt{d \left (1-c^2 x^2\right )}}+\sqrt{-d \left (c^2 x^2-1\right )} \left (-\frac{2 a c^2}{d^3 \left (c^2 x^2-1\right )}+\frac{a c^2}{3 d^3 \left (c^2 x^2-1\right )^2}-\frac{a}{2 d^3 x^2}\right )-\frac{5 a c^2 \log \left (\sqrt{d} \sqrt{-d \left (c^2 x^2-1\right )}+d\right )}{2 d^{5/2}}+\frac{5 a c^2 \log (x)}{2 d^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcSin[c*x])/(x^3*(d - c^2*d*x^2)^(5/2)),x]

[Out]

Sqrt[-(d*(-1 + c^2*x^2))]*(-a/(2*d^3*x^2) + (a*c^2)/(3*d^3*(-1 + c^2*x^2)^2) - (2*a*c^2)/(d^3*(-1 + c^2*x^2)))
+ (5*a*c^2*Log[x])/(2*d^(5/2)) - (5*a*c^2*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/(2*d^(5/2)) + (b*c^2*Sq
rt[1 - c^2*x^2]*((-2*(-1 + ArcSin[c*x]))/(-1 + c*x) + 52*ArcSin[c*x] - 6*Cot[ArcSin[c*x]/2] - 3*ArcSin[c*x]*Cs
c[ArcSin[c*x]/2]^2 + 60*ArcSin[c*x]*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]) + 52*Log[Cos[Arc
Sin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 52*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + (60*I)*(PolyLog[2, -E^(I
*ArcSin[c*x])] - PolyLog[2, E^(I*ArcSin[c*x])]) + 3*ArcSin[c*x]*Sec[ArcSin[c*x]/2]^2 + (4*ArcSin[c*x]*Sin[ArcS
in[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3 + (52*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]
/2] - Sin[ArcSin[c*x]/2]) - (4*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^3 + (
2*(1 + ArcSin[c*x]))/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2 - (52*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(Cos[Ar
cSin[c*x]/2] + Sin[ArcSin[c*x]/2]) - 6*Tan[ArcSin[c*x]/2]))/(24*d^2*Sqrt[d*(1 - c^2*x^2)])

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Maple [A]  time = 0.279, size = 624, normalized size = 1.4 \begin{align*} -{\frac{a}{2\,d{x}^{2}} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}+{\frac{5\,a{c}^{2}}{6\,d} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}+{\frac{5\,a{c}^{2}}{2\,{d}^{2}}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}}-{\frac{5\,a{c}^{2}}{2}\ln \left ({\frac{1}{x} \left ( 2\,d+2\,\sqrt{d}\sqrt{-{c}^{2}d{x}^{2}+d} \right ) } \right ){d}^{-{\frac{5}{2}}}}-{\frac{5\,b{x}^{2}\arcsin \left ( cx \right ){c}^{4}}{2\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{bx{c}^{3}}{3\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{10\,b\arcsin \left ( cx \right ){c}^{2}}{3\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{bc}{2\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) x}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{b\arcsin \left ( cx \right ) }{2\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ){x}^{2}}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{5\,b\arcsin \left ( cx \right ){c}^{2}}{2\,{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}\ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) }-{\frac{{\frac{13\,i}{3}}b{c}^{2}}{{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}\arctan \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) }-{\frac{{\frac{5\,i}{2}}b{c}^{2}}{{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}{\it dilog} \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) }-{\frac{{\frac{5\,i}{2}}b{c}^{2}}{{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}{\it dilog} \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(c*x))/x^3/(-c^2*d*x^2+d)^(5/2),x)

[Out]

-1/2*a/d/x^2/(-c^2*d*x^2+d)^(3/2)+5/6*a*c^2/d/(-c^2*d*x^2+d)^(3/2)+5/2*a*c^2/d^2/(-c^2*d*x^2+d)^(1/2)-5/2*a*c^
2/d^(5/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)-5/2*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)*x^
2*arcsin(c*x)*c^4+1/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)*x*c^3*(-c^2*x^2+1)^(1/2)+10/3*b*(-d*(
c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)*arcsin(c*x)*c^2-1/2*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^
2+1)/x*(-c^2*x^2+1)^(1/2)*c-1/2*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)/x^2*arcsin(c*x)+5/2*b*(-c^2
*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)*c^2*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-13/3*I*b*(
-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)*c^2*arctan(I*c*x+(-c^2*x^2+1)^(1/2))-5/2*I*b*(-c^2*x^
2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)*c^2*dilog(I*c*x+(-c^2*x^2+1)^(1/2))-5/2*I*b*(-c^2*x^2+1)^(1/
2)*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)*c^2*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))/x^3/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))/x^3/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-c^2*d*x^2 + d)*(b*arcsin(c*x) + a)/(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)

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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**3/(-c**2*d*x**2+d)**(5/2),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 )}^{\frac{5}{2}} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))/x^3/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arcsin(c*x) + a)/((-c^2*d*x^2 + d)^(5/2)*x^3), x)