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

Optimal. Leaf size=282 \[ \frac{\sin \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (\frac{2 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}-\frac{\sin \left (\frac{4 a}{b}\right ) \text{CosIntegral}\left (\frac{4 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{8 b^2 c^3}-\frac{3 \sin \left (\frac{6 a}{b}\right ) \text{CosIntegral}\left (\frac{6 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}-\frac{\sin \left (\frac{8 a}{b}\right ) \text{CosIntegral}\left (\frac{8 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}-\frac{\cos \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}+\frac{\cos \left (\frac{4 a}{b}\right ) \text{Si}\left (\frac{4 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{8 b^2 c^3}+\frac{3 \cos \left (\frac{6 a}{b}\right ) \text{Si}\left (\frac{6 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}+\frac{\cos \left (\frac{8 a}{b}\right ) \text{Si}\left (\frac{8 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}-\frac{x^2 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )} \]

[Out]

-((x^2*(1 - c^2*x^2)^3)/(b*c*(a + b*ArcSin[c*x]))) + (CosIntegral[(2*(a + b*ArcSin[c*x]))/b]*Sin[(2*a)/b])/(16
*b^2*c^3) - (CosIntegral[(4*(a + b*ArcSin[c*x]))/b]*Sin[(4*a)/b])/(8*b^2*c^3) - (3*CosIntegral[(6*(a + b*ArcSi
n[c*x]))/b]*Sin[(6*a)/b])/(16*b^2*c^3) - (CosIntegral[(8*(a + b*ArcSin[c*x]))/b]*Sin[(8*a)/b])/(16*b^2*c^3) -
(Cos[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/(16*b^2*c^3) + (Cos[(4*a)/b]*SinIntegral[(4*(a + b*ArcSi
n[c*x]))/b])/(8*b^2*c^3) + (3*Cos[(6*a)/b]*SinIntegral[(6*(a + b*ArcSin[c*x]))/b])/(16*b^2*c^3) + (Cos[(8*a)/b
]*SinIntegral[(8*(a + b*ArcSin[c*x]))/b])/(16*b^2*c^3)

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Rubi [A]  time = 0.933367, antiderivative size = 282, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {4721, 4723, 4406, 3303, 3299, 3302} \[ \frac{\sin \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (\frac{2 a}{b}+2 \sin ^{-1}(c x)\right )}{16 b^2 c^3}-\frac{\sin \left (\frac{4 a}{b}\right ) \text{CosIntegral}\left (\frac{4 a}{b}+4 \sin ^{-1}(c x)\right )}{8 b^2 c^3}-\frac{3 \sin \left (\frac{6 a}{b}\right ) \text{CosIntegral}\left (\frac{6 a}{b}+6 \sin ^{-1}(c x)\right )}{16 b^2 c^3}-\frac{\sin \left (\frac{8 a}{b}\right ) \text{CosIntegral}\left (\frac{8 a}{b}+8 \sin ^{-1}(c x)\right )}{16 b^2 c^3}-\frac{\cos \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 a}{b}+2 \sin ^{-1}(c x)\right )}{16 b^2 c^3}+\frac{\cos \left (\frac{4 a}{b}\right ) \text{Si}\left (\frac{4 a}{b}+4 \sin ^{-1}(c x)\right )}{8 b^2 c^3}+\frac{3 \cos \left (\frac{6 a}{b}\right ) \text{Si}\left (\frac{6 a}{b}+6 \sin ^{-1}(c x)\right )}{16 b^2 c^3}+\frac{\cos \left (\frac{8 a}{b}\right ) \text{Si}\left (\frac{8 a}{b}+8 \sin ^{-1}(c x)\right )}{16 b^2 c^3}-\frac{x^2 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x])^2,x]

[Out]

-((x^2*(1 - c^2*x^2)^3)/(b*c*(a + b*ArcSin[c*x]))) + (CosIntegral[(2*a)/b + 2*ArcSin[c*x]]*Sin[(2*a)/b])/(16*b
^2*c^3) - (CosIntegral[(4*a)/b + 4*ArcSin[c*x]]*Sin[(4*a)/b])/(8*b^2*c^3) - (3*CosIntegral[(6*a)/b + 6*ArcSin[
c*x]]*Sin[(6*a)/b])/(16*b^2*c^3) - (CosIntegral[(8*a)/b + 8*ArcSin[c*x]]*Sin[(8*a)/b])/(16*b^2*c^3) - (Cos[(2*
a)/b]*SinIntegral[(2*a)/b + 2*ArcSin[c*x]])/(16*b^2*c^3) + (Cos[(4*a)/b]*SinIntegral[(4*a)/b + 4*ArcSin[c*x]])
/(8*b^2*c^3) + (3*Cos[(6*a)/b]*SinIntegral[(6*a)/b + 6*ArcSin[c*x]])/(16*b^2*c^3) + (Cos[(8*a)/b]*SinIntegral[
(8*a)/b + 8*ArcSin[c*x]])/(16*b^2*c^3)

Rule 4721

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[
((f*x)^m*Sqrt[1 - c^2*x^2]*(d + e*x^2)^p*(a + b*ArcSin[c*x])^(n + 1))/(b*c*(n + 1)), x] + (-Dist[(f*m*d^IntPar
t[p]*(d + e*x^2)^FracPart[p])/(b*c*(n + 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] + Dist[(c*(m + 2*p + 1)*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(b*f*(n +
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}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[m, -3] && IGtQ[2*p, 0]

Rule 4723

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

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rubi steps

\begin{align*} \int \frac{x^2 \left (1-c^2 x^2\right )^{5/2}}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=-\frac{x^2 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{2 \int \frac{x \left (1-c^2 x^2\right )^2}{a+b \sin ^{-1}(c x)} \, dx}{b c}-\frac{(8 c) \int \frac{x^3 \left (1-c^2 x^2\right )^2}{a+b \sin ^{-1}(c x)} \, dx}{b}\\ &=-\frac{x^2 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{2 \operatorname{Subst}\left (\int \frac{\cos ^5(x) \sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}-\frac{8 \operatorname{Subst}\left (\int \frac{\cos ^5(x) \sin ^3(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}\\ &=-\frac{x^2 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{2 \operatorname{Subst}\left (\int \left (\frac{5 \sin (2 x)}{32 (a+b x)}+\frac{\sin (4 x)}{8 (a+b x)}+\frac{\sin (6 x)}{32 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}-\frac{8 \operatorname{Subst}\left (\int \left (\frac{3 \sin (2 x)}{64 (a+b x)}+\frac{\sin (4 x)}{64 (a+b x)}-\frac{\sin (6 x)}{64 (a+b x)}-\frac{\sin (8 x)}{128 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}\\ &=-\frac{x^2 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{\sin (6 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^3}+\frac{\operatorname{Subst}\left (\int \frac{\sin (8 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^3}-\frac{\operatorname{Subst}\left (\int \frac{\sin (4 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^3}+\frac{\operatorname{Subst}\left (\int \frac{\sin (6 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^3}+\frac{\operatorname{Subst}\left (\int \frac{\sin (4 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^3}+\frac{5 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^3}-\frac{3 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^3}\\ &=-\frac{x^2 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{\left (5 \cos \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^3}-\frac{\left (3 \cos \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^3}-\frac{\cos \left (\frac{4 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^3}+\frac{\cos \left (\frac{4 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^3}+\frac{\cos \left (\frac{6 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{6 a}{b}+6 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^3}+\frac{\cos \left (\frac{6 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{6 a}{b}+6 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^3}+\frac{\cos \left (\frac{8 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{8 a}{b}+8 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^3}-\frac{\left (5 \sin \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^3}+\frac{\left (3 \sin \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^3}+\frac{\sin \left (\frac{4 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^3}-\frac{\sin \left (\frac{4 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^3}-\frac{\sin \left (\frac{6 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{6 a}{b}+6 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^3}-\frac{\sin \left (\frac{6 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{6 a}{b}+6 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^3}-\frac{\sin \left (\frac{8 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{8 a}{b}+8 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^3}\\ &=-\frac{x^2 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{\text{Ci}\left (\frac{2 a}{b}+2 \sin ^{-1}(c x)\right ) \sin \left (\frac{2 a}{b}\right )}{16 b^2 c^3}-\frac{\text{Ci}\left (\frac{4 a}{b}+4 \sin ^{-1}(c x)\right ) \sin \left (\frac{4 a}{b}\right )}{8 b^2 c^3}-\frac{3 \text{Ci}\left (\frac{6 a}{b}+6 \sin ^{-1}(c x)\right ) \sin \left (\frac{6 a}{b}\right )}{16 b^2 c^3}-\frac{\text{Ci}\left (\frac{8 a}{b}+8 \sin ^{-1}(c x)\right ) \sin \left (\frac{8 a}{b}\right )}{16 b^2 c^3}-\frac{\cos \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 a}{b}+2 \sin ^{-1}(c x)\right )}{16 b^2 c^3}+\frac{\cos \left (\frac{4 a}{b}\right ) \text{Si}\left (\frac{4 a}{b}+4 \sin ^{-1}(c x)\right )}{8 b^2 c^3}+\frac{3 \cos \left (\frac{6 a}{b}\right ) \text{Si}\left (\frac{6 a}{b}+6 \sin ^{-1}(c x)\right )}{16 b^2 c^3}+\frac{\cos \left (\frac{8 a}{b}\right ) \text{Si}\left (\frac{8 a}{b}+8 \sin ^{-1}(c x)\right )}{16 b^2 c^3}\\ \end{align*}

Mathematica [A]  time = 1.08187, size = 414, normalized size = 1.47 \[ \frac{\sin \left (\frac{2 a}{b}\right ) \left (a+b \sin ^{-1}(c x)\right ) \text{CosIntegral}\left (2 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )-2 \sin \left (\frac{4 a}{b}\right ) \left (a+b \sin ^{-1}(c x)\right ) \text{CosIntegral}\left (4 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )-3 a \sin \left (\frac{6 a}{b}\right ) \text{CosIntegral}\left (6 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )-3 b \sin \left (\frac{6 a}{b}\right ) \sin ^{-1}(c x) \text{CosIntegral}\left (6 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )-a \sin \left (\frac{8 a}{b}\right ) \text{CosIntegral}\left (8 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )-b \sin \left (\frac{8 a}{b}\right ) \sin ^{-1}(c x) \text{CosIntegral}\left (8 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )-a \cos \left (\frac{2 a}{b}\right ) \text{Si}\left (2 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )-b \cos \left (\frac{2 a}{b}\right ) \sin ^{-1}(c x) \text{Si}\left (2 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+2 a \cos \left (\frac{4 a}{b}\right ) \text{Si}\left (4 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+2 b \cos \left (\frac{4 a}{b}\right ) \sin ^{-1}(c x) \text{Si}\left (4 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+3 a \cos \left (\frac{6 a}{b}\right ) \text{Si}\left (6 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+3 b \cos \left (\frac{6 a}{b}\right ) \sin ^{-1}(c x) \text{Si}\left (6 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+a \cos \left (\frac{8 a}{b}\right ) \text{Si}\left (8 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+b \cos \left (\frac{8 a}{b}\right ) \sin ^{-1}(c x) \text{Si}\left (8 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+16 b c^8 x^8-48 b c^6 x^6+48 b c^4 x^4-16 b c^2 x^2}{16 b^2 c^3 \left (a+b \sin ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x])^2,x]

[Out]

(-16*b*c^2*x^2 + 48*b*c^4*x^4 - 48*b*c^6*x^6 + 16*b*c^8*x^8 + (a + b*ArcSin[c*x])*CosIntegral[2*(a/b + ArcSin[
c*x])]*Sin[(2*a)/b] - 2*(a + b*ArcSin[c*x])*CosIntegral[4*(a/b + ArcSin[c*x])]*Sin[(4*a)/b] - 3*a*CosIntegral[
6*(a/b + ArcSin[c*x])]*Sin[(6*a)/b] - 3*b*ArcSin[c*x]*CosIntegral[6*(a/b + ArcSin[c*x])]*Sin[(6*a)/b] - a*CosI
ntegral[8*(a/b + ArcSin[c*x])]*Sin[(8*a)/b] - b*ArcSin[c*x]*CosIntegral[8*(a/b + ArcSin[c*x])]*Sin[(8*a)/b] -
a*Cos[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c*x])] - b*ArcSin[c*x]*Cos[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c*x
])] + 2*a*Cos[(4*a)/b]*SinIntegral[4*(a/b + ArcSin[c*x])] + 2*b*ArcSin[c*x]*Cos[(4*a)/b]*SinIntegral[4*(a/b +
ArcSin[c*x])] + 3*a*Cos[(6*a)/b]*SinIntegral[6*(a/b + ArcSin[c*x])] + 3*b*ArcSin[c*x]*Cos[(6*a)/b]*SinIntegral
[6*(a/b + ArcSin[c*x])] + a*Cos[(8*a)/b]*SinIntegral[8*(a/b + ArcSin[c*x])] + b*ArcSin[c*x]*Cos[(8*a)/b]*SinIn
tegral[8*(a/b + ArcSin[c*x])])/(16*b^2*c^3*(a + b*ArcSin[c*x]))

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Maple [A]  time = 0.061, size = 478, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x))^2,x)

[Out]

1/128/c^3*(16*arcsin(c*x)*Si(4*arcsin(c*x)+4*a/b)*cos(4*a/b)*b-16*arcsin(c*x)*Ci(4*arcsin(c*x)+4*a/b)*sin(4*a/
b)*b+8*arcsin(c*x)*Si(8*arcsin(c*x)+8*a/b)*cos(8*a/b)*b-8*arcsin(c*x)*Ci(8*arcsin(c*x)+8*a/b)*sin(8*a/b)*b-8*a
rcsin(c*x)*Si(2*arcsin(c*x)+2*a/b)*cos(2*a/b)*b+8*arcsin(c*x)*Ci(2*arcsin(c*x)+2*a/b)*sin(2*a/b)*b+24*arcsin(c
*x)*Si(6*arcsin(c*x)+6*a/b)*cos(6*a/b)*b-24*arcsin(c*x)*Ci(6*arcsin(c*x)+6*a/b)*sin(6*a/b)*b+16*Si(4*arcsin(c*
x)+4*a/b)*cos(4*a/b)*a-16*Ci(4*arcsin(c*x)+4*a/b)*sin(4*a/b)*a+8*Si(8*arcsin(c*x)+8*a/b)*cos(8*a/b)*a-8*Ci(8*a
rcsin(c*x)+8*a/b)*sin(8*a/b)*a-8*Si(2*arcsin(c*x)+2*a/b)*cos(2*a/b)*a+8*Ci(2*arcsin(c*x)+2*a/b)*sin(2*a/b)*a+2
4*Si(6*arcsin(c*x)+6*a/b)*cos(6*a/b)*a-24*Ci(6*arcsin(c*x)+6*a/b)*sin(6*a/b)*a+4*cos(4*arcsin(c*x))*b+cos(8*ar
csin(c*x))*b-4*cos(2*arcsin(c*x))*b+4*cos(6*arcsin(c*x))*b-5*b)/(a+b*arcsin(c*x))/b^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

(c^6*x^8 - 3*c^4*x^6 + 3*c^2*x^4 - x^2 - (b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c)*integrate(
2*(4*c^6*x^7 - 9*c^4*x^5 + 6*c^2*x^3 - x)/(b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c), x))/(b^2
*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c^{4} x^{6} - 2 \, c^{2} x^{4} + x^{2}\right )} \sqrt{-c^{2} x^{2} + 1}}{b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

integral((c^4*x^6 - 2*c^2*x^4 + x^2)*sqrt(-c^2*x^2 + 1)/(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2), 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(x**2*(-c**2*x**2+1)**(5/2)/(a+b*asin(c*x))**2,x)

[Out]

Timed out

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Giac [B]  time = 1.8185, size = 3322, normalized size = 11.78 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

-8*b*arcsin(c*x)*cos(a/b)^7*cos_integral(8*a/b + 8*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 8
*b*arcsin(c*x)*cos(a/b)^8*sin_integral(8*a/b + 8*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 8*a*cos(a/b)
^7*cos_integral(8*a/b + 8*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 8*a*cos(a/b)^8*sin_integra
l(8*a/b + 8*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 12*b*arcsin(c*x)*cos(a/b)^5*cos_integral(8*a/b +
8*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 6*b*arcsin(c*x)*cos(a/b)^5*cos_integral(6*a/b + 6*
arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 16*b*arcsin(c*x)*cos(a/b)^6*sin_integral(8*a/b + 8*a
rcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 6*b*arcsin(c*x)*cos(a/b)^6*sin_integral(6*a/b + 6*arcsin(c*x))
/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 12*a*cos(a/b)^5*cos_integral(8*a/b + 8*arcsin(c*x))*sin(a/b)/(b^3*c^3*arc
sin(c*x) + a*b^2*c^3) - 6*a*cos(a/b)^5*cos_integral(6*a/b + 6*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b
^2*c^3) - 16*a*cos(a/b)^6*sin_integral(8*a/b + 8*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 6*a*cos(a/b)
^6*sin_integral(6*a/b + 6*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 5*b*arcsin(c*x)*cos(a/b)^3*cos_inte
gral(8*a/b + 8*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 6*b*arcsin(c*x)*cos(a/b)^3*cos_integr
al(6*a/b + 6*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - b*arcsin(c*x)*cos(a/b)^3*cos_integral(4
*a/b + 4*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 10*b*arcsin(c*x)*cos(a/b)^4*sin_integral(8*
a/b + 8*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 9*b*arcsin(c*x)*cos(a/b)^4*sin_integral(6*a/b + 6*arc
sin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + b*arcsin(c*x)*cos(a/b)^4*sin_integral(4*a/b + 4*arcsin(c*x))/(b^
3*c^3*arcsin(c*x) + a*b^2*c^3) - 5*a*cos(a/b)^3*cos_integral(8*a/b + 8*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c
*x) + a*b^2*c^3) + 6*a*cos(a/b)^3*cos_integral(6*a/b + 6*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^
3) - a*cos(a/b)^3*cos_integral(4*a/b + 4*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 10*a*cos(a/
b)^4*sin_integral(8*a/b + 8*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 9*a*cos(a/b)^4*sin_integral(6*a/b
+ 6*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + a*cos(a/b)^4*sin_integral(4*a/b + 4*arcsin(c*x))/(b^3*c^
3*arcsin(c*x) + a*b^2*c^3) + (c^2*x^2 - 1)^4*b/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 1/2*b*arcsin(c*x)*cos(a/b)*
cos_integral(8*a/b + 8*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 9/8*b*arcsin(c*x)*cos(a/b)*co
s_integral(6*a/b + 6*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 1/2*b*arcsin(c*x)*cos(a/b)*cos_
integral(4*a/b + 4*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 1/8*b*arcsin(c*x)*cos(a/b)*cos_in
tegral(2*a/b + 2*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 2*b*arcsin(c*x)*cos(a/b)^2*sin_inte
gral(8*a/b + 8*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 27/8*b*arcsin(c*x)*cos(a/b)^2*sin_integral(6*a
/b + 6*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - b*arcsin(c*x)*cos(a/b)^2*sin_integral(4*a/b + 4*arcsin
(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 1/8*b*arcsin(c*x)*cos(a/b)^2*sin_integral(2*a/b + 2*arcsin(c*x))/(b
^3*c^3*arcsin(c*x) + a*b^2*c^3) + (c^2*x^2 - 1)^3*b/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 1/2*a*cos(a/b)*cos_int
egral(8*a/b + 8*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 9/8*a*cos(a/b)*cos_integral(6*a/b +
6*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 1/2*a*cos(a/b)*cos_integral(4*a/b + 4*arcsin(c*x))
*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 1/8*a*cos(a/b)*cos_integral(2*a/b + 2*arcsin(c*x))*sin(a/b)/(b^3
*c^3*arcsin(c*x) + a*b^2*c^3) - 2*a*cos(a/b)^2*sin_integral(8*a/b + 8*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^
2*c^3) + 27/8*a*cos(a/b)^2*sin_integral(6*a/b + 6*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - a*cos(a/b)^
2*sin_integral(4*a/b + 4*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 1/8*a*cos(a/b)^2*sin_integral(2*a/b
+ 2*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 1/16*b*arcsin(c*x)*sin_integral(8*a/b + 8*arcsin(c*x))/(b
^3*c^3*arcsin(c*x) + a*b^2*c^3) - 3/16*b*arcsin(c*x)*sin_integral(6*a/b + 6*arcsin(c*x))/(b^3*c^3*arcsin(c*x)
+ a*b^2*c^3) + 1/8*b*arcsin(c*x)*sin_integral(4*a/b + 4*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 1/16*
b*arcsin(c*x)*sin_integral(2*a/b + 2*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 1/16*a*sin_integral(8*a/
b + 8*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 3/16*a*sin_integral(6*a/b + 6*arcsin(c*x))/(b^3*c^3*arc
sin(c*x) + a*b^2*c^3) + 1/8*a*sin_integral(4*a/b + 4*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 1/16*a*s
in_integral(2*a/b + 2*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3)