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

Optimal. Leaf size=278 \[ \frac{3 \cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{128 b^2 c^4}+\frac{3 \cos \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 b^2 c^4}-\frac{21 \cos \left (\frac{7 a}{b}\right ) \text{CosIntegral}\left (\frac{7 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{256 b^2 c^4}-\frac{9 \cos \left (\frac{9 a}{b}\right ) \text{CosIntegral}\left (\frac{9 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{256 b^2 c^4}+\frac{3 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{128 b^2 c^4}+\frac{3 \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 b^2 c^4}-\frac{21 \sin \left (\frac{7 a}{b}\right ) \text{Si}\left (\frac{7 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{256 b^2 c^4}-\frac{9 \sin \left (\frac{9 a}{b}\right ) \text{Si}\left (\frac{9 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{256 b^2 c^4}-\frac{x^3 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )} \]

[Out]

-((x^3*(1 - c^2*x^2)^3)/(b*c*(a + b*ArcSin[c*x]))) + (3*Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(128*b^2*
c^4) + (3*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c*x]))/b])/(32*b^2*c^4) - (21*Cos[(7*a)/b]*CosIntegral[(7*
(a + b*ArcSin[c*x]))/b])/(256*b^2*c^4) - (9*Cos[(9*a)/b]*CosIntegral[(9*(a + b*ArcSin[c*x]))/b])/(256*b^2*c^4)
+ (3*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(128*b^2*c^4) + (3*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSi
n[c*x]))/b])/(32*b^2*c^4) - (21*Sin[(7*a)/b]*SinIntegral[(7*(a + b*ArcSin[c*x]))/b])/(256*b^2*c^4) - (9*Sin[(9
*a)/b]*SinIntegral[(9*(a + b*ArcSin[c*x]))/b])/(256*b^2*c^4)

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Rubi [A]  time = 1.15499, antiderivative size = 274, normalized size of antiderivative = 0.99, number of steps used = 34, 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} \[ \frac{3 \cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{128 b^2 c^4}+\frac{3 \cos \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{32 b^2 c^4}-\frac{21 \cos \left (\frac{7 a}{b}\right ) \text{CosIntegral}\left (\frac{7 a}{b}+7 \sin ^{-1}(c x)\right )}{256 b^2 c^4}-\frac{9 \cos \left (\frac{9 a}{b}\right ) \text{CosIntegral}\left (\frac{9 a}{b}+9 \sin ^{-1}(c x)\right )}{256 b^2 c^4}+\frac{3 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{128 b^2 c^4}+\frac{3 \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{32 b^2 c^4}-\frac{21 \sin \left (\frac{7 a}{b}\right ) \text{Si}\left (\frac{7 a}{b}+7 \sin ^{-1}(c x)\right )}{256 b^2 c^4}-\frac{9 \sin \left (\frac{9 a}{b}\right ) \text{Si}\left (\frac{9 a}{b}+9 \sin ^{-1}(c x)\right )}{256 b^2 c^4}-\frac{x^3 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x])^2,x]

[Out]

-((x^3*(1 - c^2*x^2)^3)/(b*c*(a + b*ArcSin[c*x]))) + (3*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]])/(128*b^2*c^4)
+ (3*Cos[(3*a)/b]*CosIntegral[(3*a)/b + 3*ArcSin[c*x]])/(32*b^2*c^4) - (21*Cos[(7*a)/b]*CosIntegral[(7*a)/b +
7*ArcSin[c*x]])/(256*b^2*c^4) - (9*Cos[(9*a)/b]*CosIntegral[(9*a)/b + 9*ArcSin[c*x]])/(256*b^2*c^4) + (3*Sin[
a/b]*SinIntegral[a/b + ArcSin[c*x]])/(128*b^2*c^4) + (3*Sin[(3*a)/b]*SinIntegral[(3*a)/b + 3*ArcSin[c*x]])/(32
*b^2*c^4) - (21*Sin[(7*a)/b]*SinIntegral[(7*a)/b + 7*ArcSin[c*x]])/(256*b^2*c^4) - (9*Sin[(9*a)/b]*SinIntegral
[(9*a)/b + 9*ArcSin[c*x]])/(256*b^2*c^4)

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^3 \left (1-c^2 x^2\right )^{5/2}}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=-\frac{x^3 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{3 \int \frac{x^2 \left (1-c^2 x^2\right )^2}{a+b \sin ^{-1}(c x)} \, dx}{b c}-\frac{(9 c) \int \frac{x^4 \left (1-c^2 x^2\right )^2}{a+b \sin ^{-1}(c x)} \, dx}{b}\\ &=-\frac{x^3 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{3 \operatorname{Subst}\left (\int \frac{\cos ^5(x) \sin ^2(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c^4}-\frac{9 \operatorname{Subst}\left (\int \frac{\cos ^5(x) \sin ^4(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c^4}\\ &=-\frac{x^3 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{3 \operatorname{Subst}\left (\int \left (\frac{5 \cos (x)}{64 (a+b x)}-\frac{\cos (3 x)}{64 (a+b x)}-\frac{3 \cos (5 x)}{64 (a+b x)}-\frac{\cos (7 x)}{64 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^4}-\frac{9 \operatorname{Subst}\left (\int \left (\frac{3 \cos (x)}{128 (a+b x)}-\frac{\cos (3 x)}{64 (a+b x)}-\frac{\cos (5 x)}{64 (a+b x)}+\frac{\cos (7 x)}{256 (a+b x)}+\frac{\cos (9 x)}{256 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^4}\\ &=-\frac{x^3 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{9 \operatorname{Subst}\left (\int \frac{\cos (7 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{256 b c^4}-\frac{9 \operatorname{Subst}\left (\int \frac{\cos (9 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{256 b c^4}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos (7 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}+\frac{9 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}-\frac{27 \operatorname{Subst}\left (\int \frac{\cos (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{128 b c^4}+\frac{15 \operatorname{Subst}\left (\int \frac{\cos (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}\\ &=-\frac{x^3 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{\left (27 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{128 b c^4}+\frac{\left (15 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}-\frac{\left (3 \cos \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}+\frac{\left (9 \cos \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}-\frac{\left (9 \cos \left (\frac{7 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{256 b c^4}-\frac{\left (3 \cos \left (\frac{7 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}-\frac{\left (9 \cos \left (\frac{9 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{9 a}{b}+9 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{256 b c^4}-\frac{\left (27 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{128 b c^4}+\frac{\left (15 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}-\frac{\left (3 \sin \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}+\frac{\left (9 \sin \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}-\frac{\left (9 \sin \left (\frac{7 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{256 b c^4}-\frac{\left (3 \sin \left (\frac{7 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}-\frac{\left (9 \sin \left (\frac{9 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{9 a}{b}+9 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{256 b c^4}\\ &=-\frac{x^3 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{3 \cos \left (\frac{a}{b}\right ) \text{Ci}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{128 b^2 c^4}+\frac{3 \cos \left (\frac{3 a}{b}\right ) \text{Ci}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{32 b^2 c^4}-\frac{21 \cos \left (\frac{7 a}{b}\right ) \text{Ci}\left (\frac{7 a}{b}+7 \sin ^{-1}(c x)\right )}{256 b^2 c^4}-\frac{9 \cos \left (\frac{9 a}{b}\right ) \text{Ci}\left (\frac{9 a}{b}+9 \sin ^{-1}(c x)\right )}{256 b^2 c^4}+\frac{3 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{128 b^2 c^4}+\frac{3 \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{32 b^2 c^4}-\frac{21 \sin \left (\frac{7 a}{b}\right ) \text{Si}\left (\frac{7 a}{b}+7 \sin ^{-1}(c x)\right )}{256 b^2 c^4}-\frac{9 \sin \left (\frac{9 a}{b}\right ) \text{Si}\left (\frac{9 a}{b}+9 \sin ^{-1}(c x)\right )}{256 b^2 c^4}\\ \end{align*}

Mathematica [A]  time = 1.50773, size = 408, normalized size = 1.47 \[ -\frac{-6 \cos \left (\frac{a}{b}\right ) \left (a+b \sin ^{-1}(c x)\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )-24 \cos \left (\frac{3 a}{b}\right ) \left (a+b \sin ^{-1}(c x)\right ) \text{CosIntegral}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+21 a \cos \left (\frac{7 a}{b}\right ) \text{CosIntegral}\left (7 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+21 b \cos \left (\frac{7 a}{b}\right ) \sin ^{-1}(c x) \text{CosIntegral}\left (7 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+9 a \cos \left (\frac{9 a}{b}\right ) \text{CosIntegral}\left (9 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+9 b \cos \left (\frac{9 a}{b}\right ) \sin ^{-1}(c x) \text{CosIntegral}\left (9 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )-6 a \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )-6 b \sin \left (\frac{a}{b}\right ) \sin ^{-1}(c x) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )-24 a \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )-24 b \sin \left (\frac{3 a}{b}\right ) \sin ^{-1}(c x) \text{Si}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+21 a \sin \left (\frac{7 a}{b}\right ) \text{Si}\left (7 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+21 b \sin \left (\frac{7 a}{b}\right ) \sin ^{-1}(c x) \text{Si}\left (7 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+9 a \sin \left (\frac{9 a}{b}\right ) \text{Si}\left (9 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+9 b \sin \left (\frac{9 a}{b}\right ) \sin ^{-1}(c x) \text{Si}\left (9 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )-256 b c^9 x^9+768 b c^7 x^7-768 b c^5 x^5+256 b c^3 x^3}{256 b^2 c^4 \left (a+b \sin ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x])^2,x]

[Out]

-(256*b*c^3*x^3 - 768*b*c^5*x^5 + 768*b*c^7*x^7 - 256*b*c^9*x^9 - 6*(a + b*ArcSin[c*x])*Cos[a/b]*CosIntegral[a
/b + ArcSin[c*x]] - 24*(a + b*ArcSin[c*x])*Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c*x])] + 21*a*Cos[(7*a)/b]
*CosIntegral[7*(a/b + ArcSin[c*x])] + 21*b*ArcSin[c*x]*Cos[(7*a)/b]*CosIntegral[7*(a/b + ArcSin[c*x])] + 9*a*C
os[(9*a)/b]*CosIntegral[9*(a/b + ArcSin[c*x])] + 9*b*ArcSin[c*x]*Cos[(9*a)/b]*CosIntegral[9*(a/b + ArcSin[c*x]
)] - 6*a*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] - 6*b*ArcSin[c*x]*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] - 2
4*a*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] - 24*b*ArcSin[c*x]*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSi
n[c*x])] + 21*a*Sin[(7*a)/b]*SinIntegral[7*(a/b + ArcSin[c*x])] + 21*b*ArcSin[c*x]*Sin[(7*a)/b]*SinIntegral[7*
(a/b + ArcSin[c*x])] + 9*a*Sin[(9*a)/b]*SinIntegral[9*(a/b + ArcSin[c*x])] + 9*b*ArcSin[c*x]*Sin[(9*a)/b]*SinI
ntegral[9*(a/b + ArcSin[c*x])])/(256*b^2*c^4*(a + b*ArcSin[c*x]))

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Maple [A]  time = 0.06, size = 455, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x))^2,x)

[Out]

-1/256/c^4*(21*arcsin(c*x)*Ci(7*arcsin(c*x)+7*a/b)*cos(7*a/b)*b-6*arcsin(c*x)*Ci(arcsin(c*x)+a/b)*cos(a/b)*b+9
*arcsin(c*x)*Si(9*arcsin(c*x)+9*a/b)*sin(9*a/b)*b+9*arcsin(c*x)*Ci(9*arcsin(c*x)+9*a/b)*cos(9*a/b)*b-24*arcsin
(c*x)*Si(3*arcsin(c*x)+3*a/b)*sin(3*a/b)*b-24*arcsin(c*x)*Ci(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*b+21*arcsin(c*x)*
Si(7*arcsin(c*x)+7*a/b)*sin(7*a/b)*b-6*arcsin(c*x)*Si(arcsin(c*x)+a/b)*sin(a/b)*b+21*Ci(7*arcsin(c*x)+7*a/b)*c
os(7*a/b)*a-6*Ci(arcsin(c*x)+a/b)*cos(a/b)*a+9*Si(9*arcsin(c*x)+9*a/b)*sin(9*a/b)*a+9*Ci(9*arcsin(c*x)+9*a/b)*
cos(9*a/b)*a-24*Si(3*arcsin(c*x)+3*a/b)*sin(3*a/b)*a-24*Ci(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*a+21*Si(7*arcsin(c*
x)+7*a/b)*sin(7*a/b)*a-6*Si(arcsin(c*x)+a/b)*sin(a/b)*a+6*x*b*c-sin(9*arcsin(c*x))*b+8*sin(3*arcsin(c*x))*b-3*
sin(7*arcsin(c*x))*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^{9} - 3 \, c^{4} x^{7} + 3 \, c^{2} x^{5} - x^{3} - 3 \,{\left (b^{2} c \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) + a b c\right )} \int \frac{3 \, c^{6} x^{8} - 7 \, c^{4} x^{6} + 5 \, c^{2} x^{4} - x^{2}}{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^3*(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

(c^6*x^9 - 3*c^4*x^7 + 3*c^2*x^5 - x^3 - (b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c)*integrate(
3*(3*c^6*x^8 - 7*c^4*x^6 + 5*c^2*x^4 - x^2)/(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^{7} - 2 \, c^{2} x^{5} + x^{3}\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^3*(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

integral((c^4*x^7 - 2*c^2*x^5 + x^3)*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**3*(-c**2*x**2+1)**(5/2)/(a+b*asin(c*x))**2,x)

[Out]

Timed out

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Giac [B]  time = 1.7738, size = 3347, normalized size = 12.04 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

-9*b*arcsin(c*x)*cos(a/b)^9*cos_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 9*b*arcsin
(c*x)*cos(a/b)^8*sin(a/b)*sin_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 9*a*cos(a/b)
^9*cos_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 9*a*cos(a/b)^8*sin(a/b)*sin_integra
l(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 81/4*b*arcsin(c*x)*cos(a/b)^7*cos_integral(9*a/b
+ 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 21/4*b*arcsin(c*x)*cos(a/b)^7*cos_integral(7*a/b + 7*arcs
in(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 63/4*b*arcsin(c*x)*cos(a/b)^6*sin(a/b)*sin_integral(9*a/b + 9*arc
sin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 21/4*b*arcsin(c*x)*cos(a/b)^6*sin(a/b)*sin_integral(7*a/b + 7*ar
csin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 81/4*a*cos(a/b)^7*cos_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*
arcsin(c*x) + a*b^2*c^4) - 21/4*a*cos(a/b)^7*cos_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*
c^4) + 63/4*a*cos(a/b)^6*sin(a/b)*sin_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 21/4
*a*cos(a/b)^6*sin(a/b)*sin_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 243/16*b*arcsin
(c*x)*cos(a/b)^5*cos_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 147/16*b*arcsin(c*x)*
cos(a/b)^5*cos_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 135/16*b*arcsin(c*x)*cos(a/
b)^4*sin(a/b)*sin_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 105/16*b*arcsin(c*x)*cos
(a/b)^4*sin(a/b)*sin_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + (c^2*x^2 - 1)^4*b*c*x
/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 243/16*a*cos(a/b)^5*cos_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c
*x) + a*b^2*c^4) + 147/16*a*cos(a/b)^5*cos_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) -
135/16*a*cos(a/b)^4*sin(a/b)*sin_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 105/16*a
*cos(a/b)^4*sin(a/b)*sin_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + (c^2*x^2 - 1)^3*b
*c*x/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 135/32*b*arcsin(c*x)*cos(a/b)^3*cos_integral(9*a/b + 9*arcsin(c*x))/(
b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 147/32*b*arcsin(c*x)*cos(a/b)^3*cos_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^
4*arcsin(c*x) + a*b^2*c^4) + 3/8*b*arcsin(c*x)*cos(a/b)^3*cos_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^4*arcsin(
c*x) + a*b^2*c^4) + 45/32*b*arcsin(c*x)*cos(a/b)^2*sin(a/b)*sin_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsi
n(c*x) + a*b^2*c^4) - 63/32*b*arcsin(c*x)*cos(a/b)^2*sin(a/b)*sin_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^4*arc
sin(c*x) + a*b^2*c^4) + 3/8*b*arcsin(c*x)*cos(a/b)^2*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^4*arc
sin(c*x) + a*b^2*c^4) + 135/32*a*cos(a/b)^3*cos_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c
^4) - 147/32*a*cos(a/b)^3*cos_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 3/8*a*cos(a/
b)^3*cos_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 45/32*a*cos(a/b)^2*sin(a/b)*sin_i
ntegral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 63/32*a*cos(a/b)^2*sin(a/b)*sin_integral(7*
a/b + 7*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 3/8*a*cos(a/b)^2*sin(a/b)*sin_integral(3*a/b + 3*arcs
in(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 81/256*b*arcsin(c*x)*cos(a/b)*cos_integral(9*a/b + 9*arcsin(c*x))
/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 147/256*b*arcsin(c*x)*cos(a/b)*cos_integral(7*a/b + 7*arcsin(c*x))/(b^3*c
^4*arcsin(c*x) + a*b^2*c^4) - 9/32*b*arcsin(c*x)*cos(a/b)*cos_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^4*arcsin(
c*x) + a*b^2*c^4) + 3/128*b*arcsin(c*x)*cos(a/b)*cos_integral(a/b + arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*
c^4) - 9/256*b*arcsin(c*x)*sin(a/b)*sin_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 21
/256*b*arcsin(c*x)*sin(a/b)*sin_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 3/32*b*arc
sin(c*x)*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 3/128*b*arcsin(c*x)*
sin(a/b)*sin_integral(a/b + arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 81/256*a*cos(a/b)*cos_integral(9*
a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 147/256*a*cos(a/b)*cos_integral(7*a/b + 7*arcsin(c*x)
)/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 9/32*a*cos(a/b)*cos_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^4*arcsin(c*x)
+ a*b^2*c^4) + 3/128*a*cos(a/b)*cos_integral(a/b + arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 9/256*a*s
in(a/b)*sin_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 21/256*a*sin(a/b)*sin_integral
(7*a/b + 7*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 3/32*a*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x)
)/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 3/128*a*sin(a/b)*sin_integral(a/b + arcsin(c*x))/(b^3*c^4*arcsin(c*x) +
a*b^2*c^4)