3.7.68 \(\int \frac {(d+e x^2)^2}{a+b \text {ArcSin}(c x)} \, dx\) [668]

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

[Out]

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

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Rubi [A]
time = 0.53, antiderivative size = 387, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {4757, 4719, 3384, 3380, 3383, 4731, 4491} \begin {gather*} \frac {e^2 \cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right )}{8 b c^5}-\frac {3 e^2 \cos \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 (a+b \text {ArcSin}(c x))}{b}\right )}{16 b c^5}+\frac {e^2 \cos \left (\frac {5 a}{b}\right ) \text {CosIntegral}\left (\frac {5 (a+b \text {ArcSin}(c x))}{b}\right )}{16 b c^5}+\frac {e^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right )}{8 b c^5}-\frac {3 e^2 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \text {ArcSin}(c x))}{b}\right )}{16 b c^5}+\frac {e^2 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \text {ArcSin}(c x))}{b}\right )}{16 b c^5}+\frac {d e \cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right )}{2 b c^3}-\frac {d e \cos \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 (a+b \text {ArcSin}(c x))}{b}\right )}{2 b c^3}+\frac {d e \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right )}{2 b c^3}-\frac {d e \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \text {ArcSin}(c x))}{b}\right )}{2 b c^3}+\frac {d^2 \cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right )}{b c}+\frac {d^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right )}{b c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[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)

Rule 3380

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 3383

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 3384

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 4491

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 4719

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 4731

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

Rule 4757

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])

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 \text {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) \text {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{c^3}+\frac {e^2 \text {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) \text {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 \text {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 ) \text {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 ) \text {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) \text {Subst}\left (\int \frac {\cos (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{2 c^3}-\frac {(d e) \text {Subst}\left (\int \frac {\cos (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{2 c^3}+\frac {e^2 \text {Subst}\left (\int \frac {\cos (5 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 c^5}+\frac {e^2 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.38, size = 253, normalized size = 0.65 \begin {gather*} \frac {2 \left (8 c^4 d^2+4 c^2 d e+e^2\right ) \cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\text {ArcSin}(c x)\right )-e \left (8 c^2 d+3 e\right ) \cos \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (3 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+e^2 \cos \left (\frac {5 a}{b}\right ) \text {CosIntegral}\left (5 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+16 c^4 d^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\text {ArcSin}(c x)\right )+8 c^2 d e \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\text {ArcSin}(c x)\right )+2 e^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\text {ArcSin}(c x)\right )-8 c^2 d e \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )-3 e^2 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+e^2 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )}{16 b c^5} \end {gather*}

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 = 0.18, size = 310, normalized size = 0.80

method	result	size
derivativedivides	\(\frac {16 \sinIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) c^{4} d^{2}+16 \cosineIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) c^{4} d^{2}-8 \sinIntegral \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) c^{2} d e -8 \cosineIntegral \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) c^{2} d e +8 \sinIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) c^{2} d e +8 \cosineIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) c^{2} d e +\sin \left (\frac {5 a}{b}\right ) \sinIntegral \left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) e^{2}+\cos \left (\frac {5 a}{b}\right ) \cosineIntegral \left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) e^{2}-3 \sinIntegral \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) e^{2}-3 \cosineIntegral \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) e^{2}+2 \sinIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) e^{2}+2 \cosineIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) e^{2}}{16 c^{5} b}\)	\(310\)
default	\(\frac {16 \sinIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) c^{4} d^{2}+16 \cosineIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) c^{4} d^{2}-8 \sinIntegral \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) c^{2} d e -8 \cosineIntegral \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) c^{2} d e +8 \sinIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) c^{2} d e +8 \cosineIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) c^{2} d e +\sin \left (\frac {5 a}{b}\right ) \sinIntegral \left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) e^{2}+\cos \left (\frac {5 a}{b}\right ) \cosineIntegral \left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) e^{2}-3 \sinIntegral \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) e^{2}-3 \cosineIntegral \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) e^{2}+2 \sinIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) e^{2}+2 \cosineIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) e^{2}}{16 c^{5} b}\)	\(310\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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((x^4*e^2 + 2*d*x^2*e + d^2)/(b*arcsin(c*x) + a), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x^{2}\right )^{2}}{a + b \operatorname {asin}{\left (c x \right )}}\, dx \end {gather*}

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 = 0.47, size = 633, normalized size = 1.64 \begin {gather*} \frac {e^{2} \cos \left (\frac {a}{b}\right )^{5} \operatorname {Ci}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{b c^{5}} - \frac {2 \, d e \cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{b c^{3}} + \frac {d^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b c} + \frac {e^{2} \cos \left (\frac {a}{b}\right )^{4} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{b c^{5}} - \frac {2 \, d e \cos \left (\frac {a}{b}\right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{b c^{3}} + \frac {d^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b c} - \frac {5 \, e^{2} \cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{4 \, b c^{5}} + \frac {3 \, d e \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{2 \, b c^{3}} - \frac {3 \, e^{2} \cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{4 \, b c^{5}} + \frac {d e \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{2 \, b c^{3}} - \frac {3 \, e^{2} \cos \left (\frac {a}{b}\right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{4 \, b c^{5}} + \frac {d e \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{2 \, b c^{3}} - \frac {3 \, e^{2} \cos \left (\frac {a}{b}\right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{4 \, b c^{5}} + \frac {d e \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{2 \, b c^{3}} + \frac {5 \, e^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{16 \, b c^{5}} + \frac {9 \, e^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{16 \, b c^{5}} + \frac {e^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{8 \, b c^{5}} + \frac {e^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{16 \, b c^{5}} + \frac {3 \, e^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{16 \, b c^{5}} + \frac {e^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{8 \, b c^{5}} \end {gather*}

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]

e^2*cos(a/b)^5*cos_integral(5*a/b + 5*arcsin(c*x))/(b*c^5) - 2*d*e*cos(a/b)^3*cos_integral(3*a/b + 3*arcsin(c*
x))/(b*c^3) + d^2*cos(a/b)*cos_integral(a/b + arcsin(c*x))/(b*c) + e^2*cos(a/b)^4*sin(a/b)*sin_integral(5*a/b
+ 5*arcsin(c*x))/(b*c^5) - 2*d*e*cos(a/b)^2*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) - 5/4*e^2*cos(a/b)^3*cos_integral(5*a/b + 5*arcsin(c*x))/(b*c^5) + 3/2
*d*e*cos(a/b)*cos_integral(3*a/b + 3*arcsin(c*x))/(b*c^3) - 3/4*e^2*cos(a/b)^3*cos_integral(3*a/b + 3*arcsin(c
*x))/(b*c^5) + 1/2*d*e*cos(a/b)*cos_integral(a/b + arcsin(c*x))/(b*c^3) - 3/4*e^2*cos(a/b)^2*sin(a/b)*sin_inte
gral(5*a/b + 5*arcsin(c*x))/(b*c^5) + 1/2*d*e*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b*c^3) - 3/4*e^2*c
os(a/b)^2*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b*c^5) + 1/2*d*e*sin(a/b)*sin_integral(a/b + arcsin(c*
x))/(b*c^3) + 5/16*e^2*cos(a/b)*cos_integral(5*a/b + 5*arcsin(c*x))/(b*c^5) + 9/16*e^2*cos(a/b)*cos_integral(3
*a/b + 3*arcsin(c*x))/(b*c^5) + 1/8*e^2*cos(a/b)*cos_integral(a/b + arcsin(c*x))/(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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (e\,x^2+d\right )}^2}{a+b\,\mathrm {asin}\left (c\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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