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

Optimal. Leaf size=498 \[ \frac{d e \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{2 b^2 c^3}-\frac{3 d e \sin \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{2 b^2 c^3}+\frac{e^2 \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{8 b^2 c^5}-\frac{9 e^2 \sin \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^5}+\frac{5 e^2 \sin \left (\frac{5 a}{b}\right ) \text{CosIntegral}\left (\frac{5 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^5}-\frac{d e \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{2 b^2 c^3}+\frac{3 d e \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{2 b^2 c^3}-\frac{e^2 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{8 b^2 c^5}+\frac{9 e^2 \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^5}-\frac{5 e^2 \cos \left (\frac{5 a}{b}\right ) \text{Si}\left (\frac{5 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^5}+\frac{d^2 \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{b^2 c}-\frac{d^2 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{b^2 c}-\frac{d^2 \sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{2 d e x^2 \sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{e^2 x^4 \sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )} \]

[Out]

-((d^2*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x]))) - (2*d*e*x^2*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x]))
- (e^2*x^4*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x])) + (d^2*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(
b^2*c) + (d*e*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(2*b^2*c^3) + (e^2*CosIntegral[(a + b*ArcSin[c*x])/
b]*Sin[a/b])/(8*b^2*c^5) - (3*d*e*CosIntegral[(3*(a + b*ArcSin[c*x]))/b]*Sin[(3*a)/b])/(2*b^2*c^3) - (9*e^2*Co
sIntegral[(3*(a + b*ArcSin[c*x]))/b]*Sin[(3*a)/b])/(16*b^2*c^5) + (5*e^2*CosIntegral[(5*(a + b*ArcSin[c*x]))/b
]*Sin[(5*a)/b])/(16*b^2*c^5) - (d^2*Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(b^2*c) - (d*e*Cos[a/b]*SinIn
tegral[(a + b*ArcSin[c*x])/b])/(2*b^2*c^3) - (e^2*Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(8*b^2*c^5) + (
3*d*e*Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(2*b^2*c^3) + (9*e^2*Cos[(3*a)/b]*SinIntegral[(3*(a
+ b*ArcSin[c*x]))/b])/(16*b^2*c^5) - (5*e^2*Cos[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c*x]))/b])/(16*b^2*c^5)

________________________________________________________________________________________

Rubi [A]  time = 0.761288, antiderivative size = 486, normalized size of antiderivative = 0.98, number of steps used = 26, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {4667, 4621, 4723, 3303, 3299, 3302, 4631} \[ \frac{d e \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{2 b^2 c^3}-\frac{3 d e \sin \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{2 b^2 c^3}+\frac{e^2 \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{8 b^2 c^5}-\frac{9 e^2 \sin \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{16 b^2 c^5}+\frac{5 e^2 \sin \left (\frac{5 a}{b}\right ) \text{CosIntegral}\left (\frac{5 a}{b}+5 \sin ^{-1}(c x)\right )}{16 b^2 c^5}-\frac{d e \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{2 b^2 c^3}+\frac{3 d e \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{2 b^2 c^3}-\frac{e^2 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{8 b^2 c^5}+\frac{9 e^2 \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{16 b^2 c^5}-\frac{5 e^2 \cos \left (\frac{5 a}{b}\right ) \text{Si}\left (\frac{5 a}{b}+5 \sin ^{-1}(c x)\right )}{16 b^2 c^5}+\frac{d^2 \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{b^2 c}-\frac{d^2 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{b^2 c}-\frac{d^2 \sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{2 d e x^2 \sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{e^2 x^4 \sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((d^2*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x]))) - (2*d*e*x^2*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x]))
- (e^2*x^4*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x])) + (d^2*CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b])/(b^2*
c) + (d*e*CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b])/(2*b^2*c^3) + (e^2*CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b])
/(8*b^2*c^5) - (3*d*e*CosIntegral[(3*a)/b + 3*ArcSin[c*x]]*Sin[(3*a)/b])/(2*b^2*c^3) - (9*e^2*CosIntegral[(3*a
)/b + 3*ArcSin[c*x]]*Sin[(3*a)/b])/(16*b^2*c^5) + (5*e^2*CosIntegral[(5*a)/b + 5*ArcSin[c*x]]*Sin[(5*a)/b])/(1
6*b^2*c^5) - (d^2*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]])/(b^2*c) - (d*e*Cos[a/b]*SinIntegral[a/b + ArcSin[c*
x]])/(2*b^2*c^3) - (e^2*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]])/(8*b^2*c^5) + (3*d*e*Cos[(3*a)/b]*SinIntegral
[(3*a)/b + 3*ArcSin[c*x]])/(2*b^2*c^3) + (9*e^2*Cos[(3*a)/b]*SinIntegral[(3*a)/b + 3*ArcSin[c*x]])/(16*b^2*c^5
) - (5*e^2*Cos[(5*a)/b]*SinIntegral[(5*a)/b + 5*ArcSin[c*x]])/(16*b^2*c^5)

Rule 4667

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcSin[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p]
&& (GtQ[p, 0] || IGtQ[n, 0])

Rule 4621

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(n + 1))
/(b*c*(n + 1)), x] + Dist[c/(b*(n + 1)), Int[(x*(a + b*ArcSin[c*x])^(n + 1))/Sqrt[1 - c^2*x^2], x], x] /; Free
Q[{a, b, c}, x] && LtQ[n, -1]

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 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 4631

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^m*Sqrt[1 - c^2*x^2]*(a + b*ArcSin
[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[1/(b*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a + b*x)^(n + 1)
, Sin[x]^(m - 1)*(m - (m + 1)*Sin[x]^2), x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && G
eQ[n, -2] && LtQ[n, -1]

Rubi steps

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

Mathematica [A]  time = 2.081, size = 359, normalized size = 0.72 \[ -\frac{-2 \sin \left (\frac{a}{b}\right ) \left (8 c^4 d^2+4 c^2 d e+e^2\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )+3 e \sin \left (\frac{3 a}{b}\right ) \left (8 c^2 d+3 e\right ) \text{CosIntegral}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+16 c^4 d^2 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )+\frac{16 b c^4 d^2 \sqrt{1-c^2 x^2}}{a+b \sin ^{-1}(c x)}+8 c^2 d e \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )-24 c^2 d e \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+\frac{32 b c^4 d e x^2 \sqrt{1-c^2 x^2}}{a+b \sin ^{-1}(c x)}+\frac{16 b c^4 e^2 x^4 \sqrt{1-c^2 x^2}}{a+b \sin ^{-1}(c x)}-5 e^2 \sin \left (\frac{5 a}{b}\right ) \text{CosIntegral}\left (5 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+2 e^2 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )-9 e^2 \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+5 e^2 \cos \left (\frac{5 a}{b}\right ) \text{Si}\left (5 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )}{16 b^2 c^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((16*b*c^4*d^2*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]) + (32*b*c^4*d*e*x^2*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*
x]) + (16*b*c^4*e^2*x^4*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]) - 2*(8*c^4*d^2 + 4*c^2*d*e + e^2)*CosIntegral[a
/b + ArcSin[c*x]]*Sin[a/b] + 3*e*(8*c^2*d + 3*e)*CosIntegral[3*(a/b + ArcSin[c*x])]*Sin[(3*a)/b] - 5*e^2*CosIn
tegral[5*(a/b + ArcSin[c*x])]*Sin[(5*a)/b] + 16*c^4*d^2*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 8*c^2*d*e*Co
s[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 2*e^2*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]] - 24*c^2*d*e*Cos[(3*a)/b
]*SinIntegral[3*(a/b + ArcSin[c*x])] - 9*e^2*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] + 5*e^2*Cos[(5*a)
/b]*SinIntegral[5*(a/b + ArcSin[c*x])])/(16*b^2*c^5)

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Maple [A]  time = 0.092, size = 795, 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((e*x^2+d)^2/(a+b*arcsin(c*x))^2,x)

[Out]

-1/16/c^5*(5*Si(5*arcsin(c*x)+5*a/b)*cos(5*a/b)*a*e^2-5*Ci(5*arcsin(c*x)+5*a/b)*sin(5*a/b)*a*e^2-9*Si(3*arcsin
(c*x)+3*a/b)*cos(3*a/b)*a*e^2+9*Ci(3*arcsin(c*x)+3*a/b)*sin(3*a/b)*a*e^2+2*Si(arcsin(c*x)+a/b)*cos(a/b)*a*e^2-
2*Ci(arcsin(c*x)+a/b)*sin(a/b)*a*e^2+16*(-c^2*x^2+1)^(1/2)*b*c^4*d^2+cos(5*arcsin(c*x))*b*e^2+24*Ci(3*arcsin(c
*x)+3*a/b)*sin(3*a/b)*a*c^2*d*e-3*cos(3*arcsin(c*x))*b*e^2+2*(-c^2*x^2+1)^(1/2)*b*e^2-8*cos(3*arcsin(c*x))*b*c
^2*d*e+16*Si(arcsin(c*x)+a/b)*cos(a/b)*a*c^4*d^2-16*Ci(arcsin(c*x)+a/b)*sin(a/b)*a*c^4*d^2+8*(-c^2*x^2+1)^(1/2
)*b*c^2*d*e+5*arcsin(c*x)*Si(5*arcsin(c*x)+5*a/b)*cos(5*a/b)*b*e^2-5*arcsin(c*x)*Ci(5*arcsin(c*x)+5*a/b)*sin(5
*a/b)*b*e^2-9*arcsin(c*x)*Si(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*b*e^2+9*arcsin(c*x)*Ci(3*arcsin(c*x)+3*a/b)*sin(3
*a/b)*b*e^2+2*arcsin(c*x)*Si(arcsin(c*x)+a/b)*cos(a/b)*b*e^2-2*arcsin(c*x)*Ci(arcsin(c*x)+a/b)*sin(a/b)*b*e^2+
16*arcsin(c*x)*Si(arcsin(c*x)+a/b)*cos(a/b)*b*c^4*d^2-16*arcsin(c*x)*Ci(arcsin(c*x)+a/b)*sin(a/b)*b*c^4*d^2-24
*Si(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*a*c^2*d*e-24*arcsin(c*x)*Si(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*b*c^2*d*e+24*a
rcsin(c*x)*Ci(3*arcsin(c*x)+3*a/b)*sin(3*a/b)*b*c^2*d*e+8*arcsin(c*x)*Si(arcsin(c*x)+a/b)*cos(a/b)*b*c^2*d*e-8
*arcsin(c*x)*Ci(arcsin(c*x)+a/b)*sin(a/b)*b*c^2*d*e+8*Si(arcsin(c*x)+a/b)*cos(a/b)*a*c^2*d*e-8*Ci(arcsin(c*x)+
a/b)*sin(a/b)*a*c^2*d*e)/(a+b*arcsin(c*x))/b^2

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Maxima [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((e*x^2+d)^2/(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}{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((e*x^2+d)^2/(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

integral((e^2*x^4 + 2*d*e*x^2 + d^2)/(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{\left (d + e x^{2}\right )^{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((e*x**2+d)**2/(a+b*asin(c*x))**2,x)

[Out]

Integral((d + e*x**2)**2/(a + b*asin(c*x))**2, x)

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Giac [B]  time = 1.67164, size = 3137, normalized size = 6.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*c^4*d^2*arcsin(c*x)*cos_integral(a/b + arcsin(c*x))*sin(a/b)/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) - 6*b*c^2*d*a
rcsin(c*x)*cos(a/b)^2*cos_integral(3*a/b + 3*arcsin(c*x))*e*sin(a/b)/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) + 6*b*c
^2*d*arcsin(c*x)*cos(a/b)^3*e*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) - b*c^4*d^
2*arcsin(c*x)*cos(a/b)*sin_integral(a/b + arcsin(c*x))/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) + a*c^4*d^2*cos_integ
ral(a/b + arcsin(c*x))*sin(a/b)/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) + 5*b*arcsin(c*x)*cos(a/b)^4*cos_integral(5*
a/b + 5*arcsin(c*x))*e^2*sin(a/b)/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) - 6*a*c^2*d*cos(a/b)^2*cos_integral(3*a/b
+ 3*arcsin(c*x))*e*sin(a/b)/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) - 5*b*arcsin(c*x)*cos(a/b)^5*e^2*sin_integral(5*
a/b + 5*arcsin(c*x))/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) + 6*a*c^2*d*cos(a/b)^3*e*sin_integral(3*a/b + 3*arcsin(
c*x))/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) - a*c^4*d^2*cos(a/b)*sin_integral(a/b + arcsin(c*x))/(b^3*c^5*arcsin(c
*x) + a*b^2*c^5) + 5*a*cos(a/b)^4*cos_integral(5*a/b + 5*arcsin(c*x))*e^2*sin(a/b)/(b^3*c^5*arcsin(c*x) + a*b^
2*c^5) + 3/2*b*c^2*d*arcsin(c*x)*cos_integral(3*a/b + 3*arcsin(c*x))*e*sin(a/b)/(b^3*c^5*arcsin(c*x) + a*b^2*c
^5) + 1/2*b*c^2*d*arcsin(c*x)*cos_integral(a/b + arcsin(c*x))*e*sin(a/b)/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) - 5
*a*cos(a/b)^5*e^2*sin_integral(5*a/b + 5*arcsin(c*x))/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) - 9/2*b*c^2*d*arcsin(c
*x)*cos(a/b)*e*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) - 1/2*b*c^2*d*arcsin(c*x)
*cos(a/b)*e*sin_integral(a/b + arcsin(c*x))/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) - sqrt(-c^2*x^2 + 1)*b*c^4*d^2/(
b^3*c^5*arcsin(c*x) + a*b^2*c^5) - 15/4*b*arcsin(c*x)*cos(a/b)^2*cos_integral(5*a/b + 5*arcsin(c*x))*e^2*sin(a
/b)/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) - 9/4*b*arcsin(c*x)*cos(a/b)^2*cos_integral(3*a/b + 3*arcsin(c*x))*e^2*s
in(a/b)/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) + 3/2*a*c^2*d*cos_integral(3*a/b + 3*arcsin(c*x))*e*sin(a/b)/(b^3*c^
5*arcsin(c*x) + a*b^2*c^5) + 1/2*a*c^2*d*cos_integral(a/b + arcsin(c*x))*e*sin(a/b)/(b^3*c^5*arcsin(c*x) + a*b
^2*c^5) + 25/4*b*arcsin(c*x)*cos(a/b)^3*e^2*sin_integral(5*a/b + 5*arcsin(c*x))/(b^3*c^5*arcsin(c*x) + a*b^2*c
^5) + 9/4*b*arcsin(c*x)*cos(a/b)^3*e^2*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) -
9/2*a*c^2*d*cos(a/b)*e*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) - 1/2*a*c^2*d*co
s(a/b)*e*sin_integral(a/b + arcsin(c*x))/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) + 2*(-c^2*x^2 + 1)^(3/2)*b*c^2*d*e/
(b^3*c^5*arcsin(c*x) + a*b^2*c^5) - 15/4*a*cos(a/b)^2*cos_integral(5*a/b + 5*arcsin(c*x))*e^2*sin(a/b)/(b^3*c^
5*arcsin(c*x) + a*b^2*c^5) - 9/4*a*cos(a/b)^2*cos_integral(3*a/b + 3*arcsin(c*x))*e^2*sin(a/b)/(b^3*c^5*arcsin
(c*x) + a*b^2*c^5) + 25/4*a*cos(a/b)^3*e^2*sin_integral(5*a/b + 5*arcsin(c*x))/(b^3*c^5*arcsin(c*x) + a*b^2*c^
5) + 9/4*a*cos(a/b)^3*e^2*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) - 2*sqrt(-c^2*
x^2 + 1)*b*c^2*d*e/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) + 5/16*b*arcsin(c*x)*cos_integral(5*a/b + 5*arcsin(c*x))*
e^2*sin(a/b)/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) + 9/16*b*arcsin(c*x)*cos_integral(3*a/b + 3*arcsin(c*x))*e^2*si
n(a/b)/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) + 1/8*b*arcsin(c*x)*cos_integral(a/b + arcsin(c*x))*e^2*sin(a/b)/(b^3
*c^5*arcsin(c*x) + a*b^2*c^5) - 25/16*b*arcsin(c*x)*cos(a/b)*e^2*sin_integral(5*a/b + 5*arcsin(c*x))/(b^3*c^5*
arcsin(c*x) + a*b^2*c^5) - 27/16*b*arcsin(c*x)*cos(a/b)*e^2*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^5*arcsi
n(c*x) + a*b^2*c^5) - 1/8*b*arcsin(c*x)*cos(a/b)*e^2*sin_integral(a/b + arcsin(c*x))/(b^3*c^5*arcsin(c*x) + a*
b^2*c^5) - (c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*e^2/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) + 5/16*a*cos_integral(5*
a/b + 5*arcsin(c*x))*e^2*sin(a/b)/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) + 9/16*a*cos_integral(3*a/b + 3*arcsin(c*x
))*e^2*sin(a/b)/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) + 1/8*a*cos_integral(a/b + arcsin(c*x))*e^2*sin(a/b)/(b^3*c^
5*arcsin(c*x) + a*b^2*c^5) - 25/16*a*cos(a/b)*e^2*sin_integral(5*a/b + 5*arcsin(c*x))/(b^3*c^5*arcsin(c*x) + a
*b^2*c^5) - 27/16*a*cos(a/b)*e^2*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) - 1/8*a
*cos(a/b)*e^2*sin_integral(a/b + arcsin(c*x))/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) + 2*(-c^2*x^2 + 1)^(3/2)*b*e^2
/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) - sqrt(-c^2*x^2 + 1)*b*e^2/(b^3*c^5*arcsin(c*x) + a*b^2*c^5)