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

Optimal. Leaf size=214 \[ \frac{\cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{8 b^2 c^2}+\frac{9 \cos \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^2}+\frac{5 \cos \left (\frac{5 a}{b}\right ) \text{CosIntegral}\left (\frac{5 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^2}+\frac{\sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{8 b^2 c^2}+\frac{9 \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^2}+\frac{5 \sin \left (\frac{5 a}{b}\right ) \text{Si}\left (\frac{5 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^2}-\frac{x \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )} \]

[Out]

-((x*(1 - c^2*x^2)^2)/(b*c*(a + b*ArcSin[c*x]))) + (Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(8*b^2*c^2) +
(9*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c*x]))/b])/(16*b^2*c^2) + (5*Cos[(5*a)/b]*CosIntegral[(5*(a + b*
ArcSin[c*x]))/b])/(16*b^2*c^2) + (Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(8*b^2*c^2) + (9*Sin[(3*a)/b]*S
inIntegral[(3*(a + b*ArcSin[c*x]))/b])/(16*b^2*c^2) + (5*Sin[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c*x]))/b])/
(16*b^2*c^2)

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Rubi [A]  time = 0.666335, antiderivative size = 210, normalized size of antiderivative = 0.98, number of steps used = 22, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {4721, 4661, 3312, 3303, 3299, 3302, 4723, 4406} \[ \frac{\cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{8 b^2 c^2}+\frac{9 \cos \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{16 b^2 c^2}+\frac{5 \cos \left (\frac{5 a}{b}\right ) \text{CosIntegral}\left (\frac{5 a}{b}+5 \sin ^{-1}(c x)\right )}{16 b^2 c^2}+\frac{\sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{8 b^2 c^2}+\frac{9 \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{16 b^2 c^2}+\frac{5 \sin \left (\frac{5 a}{b}\right ) \text{Si}\left (\frac{5 a}{b}+5 \sin ^{-1}(c x)\right )}{16 b^2 c^2}-\frac{x \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((x*(1 - c^2*x^2)^2)/(b*c*(a + b*ArcSin[c*x]))) + (Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]])/(8*b^2*c^2) + (9*
Cos[(3*a)/b]*CosIntegral[(3*a)/b + 3*ArcSin[c*x]])/(16*b^2*c^2) + (5*Cos[(5*a)/b]*CosIntegral[(5*a)/b + 5*ArcS
in[c*x]])/(16*b^2*c^2) + (Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]])/(8*b^2*c^2) + (9*Sin[(3*a)/b]*SinIntegral[(
3*a)/b + 3*ArcSin[c*x]])/(16*b^2*c^2) + (5*Sin[(5*a)/b]*SinIntegral[(5*a)/b + 5*ArcSin[c*x]])/(16*b^2*c^2)

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 4661

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^p/c, Subst[Int[(
a + b*x)^n*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] && I
GtQ[2*p, 0] && (IntegerQ[p] || GtQ[d, 0])

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

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]

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]

Rubi steps

\begin{align*} \int \frac{x \left (1-c^2 x^2\right )^{3/2}}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=-\frac{x \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{\int \frac{1-c^2 x^2}{a+b \sin ^{-1}(c x)} \, dx}{b c}-\frac{(5 c) \int \frac{x^2 \left (1-c^2 x^2\right )}{a+b \sin ^{-1}(c x)} \, dx}{b}\\ &=-\frac{x \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{\cos ^3(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c^2}-\frac{5 \operatorname{Subst}\left (\int \frac{\cos ^3(x) \sin ^2(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c^2}\\ &=-\frac{x \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{\operatorname{Subst}\left (\int \left (\frac{3 \cos (x)}{4 (a+b x)}+\frac{\cos (3 x)}{4 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^2}-\frac{5 \operatorname{Subst}\left (\int \left (\frac{\cos (x)}{8 (a+b x)}-\frac{\cos (3 x)}{16 (a+b x)}-\frac{\cos (5 x)}{16 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^2}\\ &=-\frac{x \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{\cos (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^2}+\frac{5 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^2}+\frac{5 \operatorname{Subst}\left (\int \frac{\cos (5 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^2}-\frac{5 \operatorname{Subst}\left (\int \frac{\cos (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^2}+\frac{3 \operatorname{Subst}\left (\int \frac{\cos (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^2}\\ &=-\frac{x \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{\left (5 \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 )}{8 b c^2}+\frac{\left (3 \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 )}{4 b c^2}+\frac{\cos \left (\frac{3 a}{b}\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 )}{4 b c^2}+\frac{\left (5 \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 )}{16 b c^2}+\frac{\left (5 \cos \left (\frac{5 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^2}-\frac{\left (5 \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 )}{8 b c^2}+\frac{\left (3 \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 )}{4 b c^2}+\frac{\sin \left (\frac{3 a}{b}\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 )}{4 b c^2}+\frac{\left (5 \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 )}{16 b c^2}+\frac{\left (5 \sin \left (\frac{5 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^2}\\ &=-\frac{x \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{\cos \left (\frac{a}{b}\right ) \text{Ci}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{8 b^2 c^2}+\frac{9 \cos \left (\frac{3 a}{b}\right ) \text{Ci}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{16 b^2 c^2}+\frac{5 \cos \left (\frac{5 a}{b}\right ) \text{Ci}\left (\frac{5 a}{b}+5 \sin ^{-1}(c x)\right )}{16 b^2 c^2}+\frac{\sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{8 b^2 c^2}+\frac{9 \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{16 b^2 c^2}+\frac{5 \sin \left (\frac{5 a}{b}\right ) \text{Si}\left (\frac{5 a}{b}+5 \sin ^{-1}(c x)\right )}{16 b^2 c^2}\\ \end{align*}

Mathematica [A]  time = 0.516958, size = 295, normalized size = 1.38 \[ \frac{2 \cos \left (\frac{a}{b}\right ) \left (a+b \sin ^{-1}(c x)\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )+9 \cos \left (\frac{3 a}{b}\right ) \left (a+b \sin ^{-1}(c x)\right ) \text{CosIntegral}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+5 a \cos \left (\frac{5 a}{b}\right ) \text{CosIntegral}\left (5 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+5 b \cos \left (\frac{5 a}{b}\right ) \sin ^{-1}(c x) \text{CosIntegral}\left (5 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+2 a \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )+2 b \sin \left (\frac{a}{b}\right ) \sin ^{-1}(c x) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )+9 a \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+9 b \sin \left (\frac{3 a}{b}\right ) \sin ^{-1}(c x) \text{Si}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+5 a \sin \left (\frac{5 a}{b}\right ) \text{Si}\left (5 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+5 b \sin \left (\frac{5 a}{b}\right ) \sin ^{-1}(c x) \text{Si}\left (5 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )-16 b c^5 x^5+32 b c^3 x^3-16 b c x}{16 b^2 c^2 \left (a+b \sin ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-16*b*c*x + 32*b*c^3*x^3 - 16*b*c^5*x^5 + 2*(a + b*ArcSin[c*x])*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]] + 9*(
a + b*ArcSin[c*x])*Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c*x])] + 5*a*Cos[(5*a)/b]*CosIntegral[5*(a/b + Arc
Sin[c*x])] + 5*b*ArcSin[c*x]*Cos[(5*a)/b]*CosIntegral[5*(a/b + ArcSin[c*x])] + 2*a*Sin[a/b]*SinIntegral[a/b +
ArcSin[c*x]] + 2*b*ArcSin[c*x]*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 9*a*Sin[(3*a)/b]*SinIntegral[3*(a/b +
ArcSin[c*x])] + 9*b*ArcSin[c*x]*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] + 5*a*Sin[(5*a)/b]*SinIntegra
l[5*(a/b + ArcSin[c*x])] + 5*b*ArcSin[c*x]*Sin[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c*x])])/(16*b^2*c^2*(a + b
*ArcSin[c*x]))

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Maple [A]  time = 0.051, size = 341, normalized size = 1.6 \begin{align*}{\frac{1}{16\,{c}^{2} \left ( a+b\arcsin \left ( cx \right ) \right ){b}^{2}} \left ( 9\,\arcsin \left ( cx \right ){\it Si} \left ( 3\,\arcsin \left ( cx \right ) +3\,{\frac{a}{b}} \right ) \sin \left ( 3\,{\frac{a}{b}} \right ) b+9\,\arcsin \left ( cx \right ){\it Ci} \left ( 3\,\arcsin \left ( cx \right ) +3\,{\frac{a}{b}} \right ) \cos \left ( 3\,{\frac{a}{b}} \right ) b+2\,\arcsin \left ( cx \right ){\it Si} \left ( \arcsin \left ( cx \right ) +{\frac{a}{b}} \right ) \sin \left ({\frac{a}{b}} \right ) b+2\,\arcsin \left ( cx \right ){\it Ci} \left ( \arcsin \left ( cx \right ) +{\frac{a}{b}} \right ) \cos \left ({\frac{a}{b}} \right ) b+5\,\arcsin \left ( cx \right ) \sin \left ( 5\,{\frac{a}{b}} \right ){\it Si} \left ( 5\,\arcsin \left ( cx \right ) +5\,{\frac{a}{b}} \right ) b+5\,\arcsin \left ( cx \right ){\it Ci} \left ( 5\,\arcsin \left ( cx \right ) +5\,{\frac{a}{b}} \right ) \cos \left ( 5\,{\frac{a}{b}} \right ) b+9\,{\it Si} \left ( 3\,\arcsin \left ( cx \right ) +3\,{\frac{a}{b}} \right ) \sin \left ( 3\,{\frac{a}{b}} \right ) a+9\,{\it Ci} \left ( 3\,\arcsin \left ( cx \right ) +3\,{\frac{a}{b}} \right ) \cos \left ( 3\,{\frac{a}{b}} \right ) a+2\,{\it Si} \left ( \arcsin \left ( cx \right ) +{\frac{a}{b}} \right ) \sin \left ({\frac{a}{b}} \right ) a+2\,{\it Ci} \left ( \arcsin \left ( cx \right ) +{\frac{a}{b}} \right ) \cos \left ({\frac{a}{b}} \right ) a+5\,\sin \left ( 5\,{\frac{a}{b}} \right ){\it Si} \left ( 5\,\arcsin \left ( cx \right ) +5\,{\frac{a}{b}} \right ) a+5\,{\it Ci} \left ( 5\,\arcsin \left ( cx \right ) +5\,{\frac{a}{b}} \right ) \cos \left ( 5\,{\frac{a}{b}} \right ) a-2\,xbc-3\,\sin \left ( 3\,\arcsin \left ( cx \right ) \right ) b-\sin \left ( 5\,\arcsin \left ( cx \right ) \right ) b \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16/c^2*(9*arcsin(c*x)*Si(3*arcsin(c*x)+3*a/b)*sin(3*a/b)*b+9*arcsin(c*x)*Ci(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*
b+2*arcsin(c*x)*Si(arcsin(c*x)+a/b)*sin(a/b)*b+2*arcsin(c*x)*Ci(arcsin(c*x)+a/b)*cos(a/b)*b+5*arcsin(c*x)*sin(
5*a/b)*Si(5*arcsin(c*x)+5*a/b)*b+5*arcsin(c*x)*Ci(5*arcsin(c*x)+5*a/b)*cos(5*a/b)*b+9*Si(3*arcsin(c*x)+3*a/b)*
sin(3*a/b)*a+9*Ci(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*a+2*Si(arcsin(c*x)+a/b)*sin(a/b)*a+2*Ci(arcsin(c*x)+a/b)*cos
(a/b)*a+5*sin(5*a/b)*Si(5*arcsin(c*x)+5*a/b)*a+5*Ci(5*arcsin(c*x)+5*a/b)*cos(5*a/b)*a-2*x*b*c-3*sin(3*arcsin(c
*x))*b-sin(5*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^{4} x^{5} - 2 \, c^{2} x^{3} + x - \frac{{\left (5 \, c^{4} \int \frac{x^{4}}{b \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) + a}\,{d x} - 6 \, c^{2} \int \frac{x^{2}}{b \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) + a}\,{d x} + \int \frac{1}{b \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) + a}\,{d x}\right )}{\left (b^{2} c \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) + a b c\right )}}{b c}}{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*(-c^2*x^2+1)^(3/2)/(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

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

[Out]

integral(-(c^2*x^3 - x)*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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{2}}}{\left (a + b \operatorname{asin}{\left (c x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-c**2*x**2+1)**(3/2)/(a+b*asin(c*x))**2,x)

[Out]

Integral(x*(-(c*x - 1)*(c*x + 1))**(3/2)/(a + b*asin(c*x))**2, x)

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Giac [B]  time = 1.70941, size = 1640, normalized size = 7.66 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*b*arcsin(c*x)*cos(a/b)^5*cos_integral(5*a/b + 5*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 5*b*arcsin(
c*x)*cos(a/b)^4*sin(a/b)*sin_integral(5*a/b + 5*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 5*a*cos(a/b)^
5*cos_integral(5*a/b + 5*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 5*a*cos(a/b)^4*sin(a/b)*sin_integral
(5*a/b + 5*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) - 25/4*b*arcsin(c*x)*cos(a/b)^3*cos_integral(5*a/b +
5*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 9/4*b*arcsin(c*x)*cos(a/b)^3*cos_integral(3*a/b + 3*arcsin
(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) - 15/4*b*arcsin(c*x)*cos(a/b)^2*sin(a/b)*sin_integral(5*a/b + 5*arcsi
n(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 9/4*b*arcsin(c*x)*cos(a/b)^2*sin(a/b)*sin_integral(3*a/b + 3*arcsi
n(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) - (c^2*x^2 - 1)^2*b*c*x/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) - 25/4*a*c
os(a/b)^3*cos_integral(5*a/b + 5*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 9/4*a*cos(a/b)^3*cos_integra
l(3*a/b + 3*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) - 15/4*a*cos(a/b)^2*sin(a/b)*sin_integral(5*a/b + 5
*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 9/4*a*cos(a/b)^2*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x)
)/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 25/16*b*arcsin(c*x)*cos(a/b)*cos_integral(5*a/b + 5*arcsin(c*x))/(b^3*c^
2*arcsin(c*x) + a*b^2*c^2) - 27/16*b*arcsin(c*x)*cos(a/b)*cos_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^2*arcsin(
c*x) + a*b^2*c^2) + 1/8*b*arcsin(c*x)*cos(a/b)*cos_integral(a/b + arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^
2) + 5/16*b*arcsin(c*x)*sin(a/b)*sin_integral(5*a/b + 5*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) - 9/16*
b*arcsin(c*x)*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 1/8*b*arcsin(c*
x)*sin(a/b)*sin_integral(a/b + arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 25/16*a*cos(a/b)*cos_integral(
5*a/b + 5*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) - 27/16*a*cos(a/b)*cos_integral(3*a/b + 3*arcsin(c*x)
)/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 1/8*a*cos(a/b)*cos_integral(a/b + arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*
b^2*c^2) + 5/16*a*sin(a/b)*sin_integral(5*a/b + 5*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) - 9/16*a*sin(
a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 1/8*a*sin(a/b)*sin_integral(a/b +
arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2)