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

Optimal. Leaf size=387 \[ \frac{d e \cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{2 b c^3}-\frac{d e \cos \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{2 b c^3}+\frac{e^2 \cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{8 b c^5}-\frac{3 e^2 \cos \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b c^5}+\frac{e^2 \cos \left (\frac{5 a}{b}\right ) \text{CosIntegral}\left (\frac{5 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b c^5}+\frac{d e \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{2 b c^3}-\frac{d e \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{2 b c^3}+\frac{e^2 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{8 b c^5}-\frac{3 e^2 \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b c^5}+\frac{e^2 \sin \left (\frac{5 a}{b}\right ) \text{Si}\left (\frac{5 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b c^5}+\frac{d^2 \cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{b c}+\frac{d^2 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{b c} \]

[Out]

(d^2*Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(b*c) + (d*e*Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(2
*b*c^3) + (e^2*Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(8*b*c^5) - (d*e*Cos[(3*a)/b]*CosIntegral[(3*(a +
b*ArcSin[c*x]))/b])/(2*b*c^3) - (3*e^2*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c*x]))/b])/(16*b*c^5) + (e^2*
Cos[(5*a)/b]*CosIntegral[(5*(a + b*ArcSin[c*x]))/b])/(16*b*c^5) + (d^2*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x]
)/b])/(b*c) + (d*e*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(2*b*c^3) + (e^2*Sin[a/b]*SinIntegral[(a + b*A
rcSin[c*x])/b])/(8*b*c^5) - (d*e*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(2*b*c^3) - (3*e^2*Sin[(
3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(16*b*c^5) + (e^2*Sin[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c*
x]))/b])/(16*b*c^5)

________________________________________________________________________________________

Rubi [A]  time = 0.770322, antiderivative size = 379, normalized size of antiderivative = 0.98, number of steps used = 27, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {4667, 4623, 3303, 3299, 3302, 4635, 4406} \[ \frac{d e \cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{2 b c^3}-\frac{d e \cos \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{2 b c^3}+\frac{e^2 \cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{8 b c^5}-\frac{3 e^2 \cos \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{16 b c^5}+\frac{e^2 \cos \left (\frac{5 a}{b}\right ) \text{CosIntegral}\left (\frac{5 a}{b}+5 \sin ^{-1}(c x)\right )}{16 b c^5}+\frac{d e \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{2 b c^3}-\frac{d e \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{2 b c^3}+\frac{e^2 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{8 b c^5}-\frac{3 e^2 \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{16 b c^5}+\frac{e^2 \sin \left (\frac{5 a}{b}\right ) \text{Si}\left (\frac{5 a}{b}+5 \sin ^{-1}(c x)\right )}{16 b c^5}+\frac{d^2 \cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{b c}+\frac{d^2 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*e*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]])/(2*b*c^3) + (e^2*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]])/(8*b*c
^5) - (d*e*Cos[(3*a)/b]*CosIntegral[(3*a)/b + 3*ArcSin[c*x]])/(2*b*c^3) - (3*e^2*Cos[(3*a)/b]*CosIntegral[(3*a
)/b + 3*ArcSin[c*x]])/(16*b*c^5) + (e^2*Cos[(5*a)/b]*CosIntegral[(5*a)/b + 5*ArcSin[c*x]])/(16*b*c^5) + (d^2*C
os[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(b*c) + (d*e*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]])/(2*b*c^3) +
(e^2*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]])/(8*b*c^5) - (d*e*Sin[(3*a)/b]*SinIntegral[(3*a)/b + 3*ArcSin[c*x
]])/(2*b*c^3) - (3*e^2*Sin[(3*a)/b]*SinIntegral[(3*a)/b + 3*ArcSin[c*x]])/(16*b*c^5) + (e^2*Sin[(5*a)/b]*SinIn
tegral[(5*a)/b + 5*ArcSin[c*x]])/(16*b*c^5) + (d^2*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(b*c)

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 4623

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Cos[a/b - x/b], x], x, a
+ b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x]

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 4635

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*S
in[x]^m*Cos[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 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{\left (d+e x^2\right )^2}{a+b \sin ^{-1}(c x)} \, dx &=\int \left (\frac{d^2}{a+b \sin ^{-1}(c x)}+\frac{2 d e x^2}{a+b \sin ^{-1}(c x)}+\frac{e^2 x^4}{a+b \sin ^{-1}(c x)}\right ) \, dx\\ &=d^2 \int \frac{1}{a+b \sin ^{-1}(c x)} \, dx+(2 d e) \int \frac{x^2}{a+b \sin ^{-1}(c x)} \, dx+e^2 \int \frac{x^4}{a+b \sin ^{-1}(c x)} \, dx\\ &=\frac{d^2 \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}-\frac{x}{b}\right )}{x} \, dx,x,a+b \sin ^{-1}(c x)\right )}{b c}+\frac{(2 d e) \operatorname{Subst}\left (\int \frac{\cos (x) \sin ^2(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{c^3}+\frac{e^2 \operatorname{Subst}\left (\int \frac{\cos (x) \sin ^4(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{c^5}\\ &=\frac{(2 d e) \operatorname{Subst}\left (\int \left (\frac{\cos (x)}{4 (a+b x)}-\frac{\cos (3 x)}{4 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^3}+\frac{e^2 \operatorname{Subst}\left (\int \left (\frac{\cos (x)}{8 (a+b x)}-\frac{3 \cos (3 x)}{16 (a+b x)}+\frac{\cos (5 x)}{16 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^5}+\frac{\left (d^2 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{x}{b}\right )}{x} \, dx,x,a+b \sin ^{-1}(c x)\right )}{b c}+\frac{\left (d^2 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{x}{b}\right )}{x} \, dx,x,a+b \sin ^{-1}(c x)\right )}{b c}\\ &=\frac{d^2 \cos \left (\frac{a}{b}\right ) \text{Ci}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{b c}+\frac{d^2 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{b c}+\frac{(d e) \operatorname{Subst}\left (\int \frac{\cos (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{2 c^3}-\frac{(d e) \operatorname{Subst}\left (\int \frac{\cos (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{2 c^3}+\frac{e^2 \operatorname{Subst}\left (\int \frac{\cos (5 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 c^5}+\frac{e^2 \operatorname{Subst}\left (\int \frac{\cos (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 c^5}-\frac{\left (3 e^2\right ) \operatorname{Subst}\left (\int \frac{\cos (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 c^5}\\ &=\frac{d^2 \cos \left (\frac{a}{b}\right ) \text{Ci}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{b c}+\frac{d^2 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{b c}+\frac{\left (d e \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 )}{2 c^3}+\frac{\left (e^2 \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 c^5}-\frac{\left (d e \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 )}{2 c^3}-\frac{\left (3 e^2 \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 c^5}+\frac{\left (e^2 \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 c^5}+\frac{\left (d e \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 )}{2 c^3}+\frac{\left (e^2 \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 c^5}-\frac{\left (d e \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 )}{2 c^3}-\frac{\left (3 e^2 \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 c^5}+\frac{\left (e^2 \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 c^5}\\ &=\frac{d e \cos \left (\frac{a}{b}\right ) \text{Ci}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{2 b c^3}+\frac{e^2 \cos \left (\frac{a}{b}\right ) \text{Ci}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{8 b c^5}-\frac{d e \cos \left (\frac{3 a}{b}\right ) \text{Ci}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{2 b c^3}-\frac{3 e^2 \cos \left (\frac{3 a}{b}\right ) \text{Ci}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{16 b c^5}+\frac{e^2 \cos \left (\frac{5 a}{b}\right ) \text{Ci}\left (\frac{5 a}{b}+5 \sin ^{-1}(c x)\right )}{16 b c^5}+\frac{d^2 \cos \left (\frac{a}{b}\right ) \text{Ci}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{b c}+\frac{d e \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{2 b c^3}+\frac{e^2 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{8 b c^5}-\frac{d e \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{2 b c^3}-\frac{3 e^2 \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{16 b c^5}+\frac{e^2 \sin \left (\frac{5 a}{b}\right ) \text{Si}\left (\frac{5 a}{b}+5 \sin ^{-1}(c x)\right )}{16 b c^5}+\frac{d^2 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{b c}\\ \end{align*}

Mathematica [A]  time = 0.614831, size = 253, normalized size = 0.65 \[ \frac{2 \cos \left (\frac{a}{b}\right ) \left (8 c^4 d^2+4 c^2 d e+e^2\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )-e \cos \left (\frac{3 a}{b}\right ) \left (8 c^2 d+3 e\right ) \text{CosIntegral}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+16 c^4 d^2 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )+8 c^2 d e \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )-8 c^2 d e \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+e^2 \cos \left (\frac{5 a}{b}\right ) \text{CosIntegral}\left (5 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+2 e^2 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )-3 e^2 \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+e^2 \sin \left (\frac{5 a}{b}\right ) \text{Si}\left (5 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )}{16 b c^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(8*c^4*d^2 + 4*c^2*d*e + e^2)*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]] - e*(8*c^2*d + 3*e)*Cos[(3*a)/b]*CosI
ntegral[3*(a/b + ArcSin[c*x])] + e^2*Cos[(5*a)/b]*CosIntegral[5*(a/b + ArcSin[c*x])] + 16*c^4*d^2*Sin[a/b]*Sin
Integral[a/b + ArcSin[c*x]] + 8*c^2*d*e*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 2*e^2*Sin[a/b]*SinIntegral[a
/b + ArcSin[c*x]] - 8*c^2*d*e*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] - 3*e^2*Sin[(3*a)/b]*SinIntegral
[3*(a/b + ArcSin[c*x])] + e^2*Sin[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c*x])])/(16*b*c^5)

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Maple [A]  time = 0.046, size = 310, normalized size = 0.8 \begin{align*}{\frac{1}{16\,{c}^{5}b} \left ( 16\,{\it Si} \left ( \arcsin \left ( cx \right ) +{\frac{a}{b}} \right ) \sin \left ({\frac{a}{b}} \right ){c}^{4}{d}^{2}+16\,{\it Ci} \left ( \arcsin \left ( cx \right ) +{\frac{a}{b}} \right ) \cos \left ({\frac{a}{b}} \right ){c}^{4}{d}^{2}-8\,{\it Si} \left ( 3\,\arcsin \left ( cx \right ) +3\,{\frac{a}{b}} \right ) \sin \left ( 3\,{\frac{a}{b}} \right ){c}^{2}de-8\,{\it Ci} \left ( 3\,\arcsin \left ( cx \right ) +3\,{\frac{a}{b}} \right ) \cos \left ( 3\,{\frac{a}{b}} \right ){c}^{2}de+8\,{\it Si} \left ( \arcsin \left ( cx \right ) +{\frac{a}{b}} \right ) \sin \left ({\frac{a}{b}} \right ){c}^{2}de+8\,{\it Ci} \left ( \arcsin \left ( cx \right ) +{\frac{a}{b}} \right ) \cos \left ({\frac{a}{b}} \right ){c}^{2}de+\sin \left ( 5\,{\frac{a}{b}} \right ){\it Si} \left ( 5\,\arcsin \left ( cx \right ) +5\,{\frac{a}{b}} \right ){e}^{2}+\cos \left ( 5\,{\frac{a}{b}} \right ){\it Ci} \left ( 5\,\arcsin \left ( cx \right ) +5\,{\frac{a}{b}} \right ){e}^{2}-3\,{\it Si} \left ( 3\,\arcsin \left ( cx \right ) +3\,{\frac{a}{b}} \right ) \sin \left ( 3\,{\frac{a}{b}} \right ){e}^{2}-3\,{\it Ci} \left ( 3\,\arcsin \left ( cx \right ) +3\,{\frac{a}{b}} \right ) \cos \left ( 3\,{\frac{a}{b}} \right ){e}^{2}+2\,{\it Si} \left ( \arcsin \left ( cx \right ) +{\frac{a}{b}} \right ) \sin \left ({\frac{a}{b}} \right ){e}^{2}+2\,{\it Ci} \left ( \arcsin \left ( cx \right ) +{\frac{a}{b}} \right ) \cos \left ({\frac{a}{b}} \right ){e}^{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^2/(a+b*arcsin(c*x)),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d\right )}^{2}}{b \arcsin \left (c x\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 \arcsin \left (c x\right ) + a}, 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)),x, algorithm="fricas")

[Out]

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

[Out]

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

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Giac [A]  time = 1.39394, size = 846, normalized size = 2.19 \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)),x, algorithm="giac")

[Out]

d^2*cos(a/b)*cos_integral(a/b + arcsin(c*x))/(b*c) - 2*d*cos(a/b)^3*cos_integral(3*a/b + 3*arcsin(c*x))*e/(b*c
^3) - 2*d*cos(a/b)^2*e*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b*c^3) + d^2*sin(a/b)*sin_integral(a/b +
arcsin(c*x))/(b*c) + cos(a/b)^5*cos_integral(5*a/b + 5*arcsin(c*x))*e^2/(b*c^5) + cos(a/b)^4*e^2*sin(a/b)*sin_
integral(5*a/b + 5*arcsin(c*x))/(b*c^5) + 3/2*d*cos(a/b)*cos_integral(3*a/b + 3*arcsin(c*x))*e/(b*c^3) + 1/2*d
*cos(a/b)*cos_integral(a/b + arcsin(c*x))*e/(b*c^3) + 1/2*d*e*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b*
c^3) + 1/2*d*e*sin(a/b)*sin_integral(a/b + arcsin(c*x))/(b*c^3) - 5/4*cos(a/b)^3*cos_integral(5*a/b + 5*arcsin
(c*x))*e^2/(b*c^5) - 3/4*cos(a/b)^3*cos_integral(3*a/b + 3*arcsin(c*x))*e^2/(b*c^5) - 3/4*cos(a/b)^2*e^2*sin(a
/b)*sin_integral(5*a/b + 5*arcsin(c*x))/(b*c^5) - 3/4*cos(a/b)^2*e^2*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*
x))/(b*c^5) + 5/16*cos(a/b)*cos_integral(5*a/b + 5*arcsin(c*x))*e^2/(b*c^5) + 9/16*cos(a/b)*cos_integral(3*a/b
+ 3*arcsin(c*x))*e^2/(b*c^5) + 1/8*cos(a/b)*cos_integral(a/b + arcsin(c*x))*e^2/(b*c^5) + 1/16*e^2*sin(a/b)*s
in_integral(5*a/b + 5*arcsin(c*x))/(b*c^5) + 3/16*e^2*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b*c^5) + 1
/8*e^2*sin(a/b)*sin_integral(a/b + arcsin(c*x))/(b*c^5)