3.7.78 \(\int \frac {d+e x^2}{(a+b \text {ArcSin}(c x))^2} \, dx\) [678]

Optimal. Leaf size=249 \[ -\frac {d \sqrt {1-c^2 x^2}}{b c (a+b \text {ArcSin}(c x))}-\frac {e x^2 \sqrt {1-c^2 x^2}}{b c (a+b \text {ArcSin}(c x))}+\frac {d \text {CosIntegral}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{b^2 c}+\frac {e \text {CosIntegral}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{4 b^2 c^3}-\frac {3 e \text {CosIntegral}\left (\frac {3 (a+b \text {ArcSin}(c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{4 b^2 c^3}-\frac {d \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right )}{b^2 c}-\frac {e \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right )}{4 b^2 c^3}+\frac {3 e \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \text {ArcSin}(c x))}{b}\right )}{4 b^2 c^3} \]

[Out]

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

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Rubi [A]
time = 0.27, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {4757, 4717, 4809, 3384, 3380, 3383, 4727} \begin {gather*} \frac {e \sin \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right )}{4 b^2 c^3}-\frac {3 e \sin \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 (a+b \text {ArcSin}(c x))}{b}\right )}{4 b^2 c^3}-\frac {e \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right )}{4 b^2 c^3}+\frac {3 e \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \text {ArcSin}(c x))}{b}\right )}{4 b^2 c^3}+\frac {d \sin \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right )}{b^2 c}-\frac {d \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right )}{b^2 c}-\frac {d \sqrt {1-c^2 x^2}}{b c (a+b \text {ArcSin}(c x))}-\frac {e x^2 \sqrt {1-c^2 x^2}}{b c (a+b \text {ArcSin}(c x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 4717

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 4727

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^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^(n + 1), Sin[
-a/b + x/b]^(m - 1)*(m - (m + 1)*Sin[-a/b + x/b]^2), x], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c}, x]
&& IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

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])

Rule 4809

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(1/(b*c
^(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b]^(2*p + 1), x],
x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m,
0]

Rubi steps

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

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Mathematica [A]
time = 0.63, size = 191, normalized size = 0.77 \begin {gather*} -\frac {\frac {4 b c^2 d \sqrt {1-c^2 x^2}}{a+b \text {ArcSin}(c x)}+\frac {4 b c^2 e x^2 \sqrt {1-c^2 x^2}}{a+b \text {ArcSin}(c x)}-\left (4 c^2 d+e\right ) \text {CosIntegral}\left (\frac {a}{b}+\text {ArcSin}(c x)\right ) \sin \left (\frac {a}{b}\right )+3 e \text {CosIntegral}\left (3 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right ) \sin \left (\frac {3 a}{b}\right )+4 c^2 d \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\text {ArcSin}(c x)\right )+e \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\text {ArcSin}(c x)\right )-3 e \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )}{4 b^2 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.21, size = 367, normalized size = 1.47

method	result	size
derivativedivides	\(\frac {-4 \arcsin \left (c x \right ) \sinIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b \,c^{2} d +4 \arcsin \left (c x \right ) \cosineIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b \,c^{2} d -4 \sinIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a \,c^{2} d +4 \cosineIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a \,c^{2} d +3 \arcsin \left (c x \right ) \sinIntegral \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b e -3 \arcsin \left (c x \right ) \cosineIntegral \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b e -\arcsin \left (c x \right ) \sinIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b e +\arcsin \left (c x \right ) \cosineIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b e -4 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} d +3 \sinIntegral \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a e -3 \cosineIntegral \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a e -\sinIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a e +\cosineIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a e +\cos \left (3 \arcsin \left (c x \right )\right ) b e -\sqrt {-c^{2} x^{2}+1}\, b e}{4 c^{3} \left (a +b \arcsin \left (c x \right )\right ) b^{2}}\)	\(367\)
default	\(\frac {-4 \arcsin \left (c x \right ) \sinIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b \,c^{2} d +4 \arcsin \left (c x \right ) \cosineIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b \,c^{2} d -4 \sinIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a \,c^{2} d +4 \cosineIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a \,c^{2} d +3 \arcsin \left (c x \right ) \sinIntegral \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b e -3 \arcsin \left (c x \right ) \cosineIntegral \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b e -\arcsin \left (c x \right ) \sinIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b e +\arcsin \left (c x \right ) \cosineIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b e -4 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} d +3 \sinIntegral \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a e -3 \cosineIntegral \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a e -\sinIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a e +\cosineIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a e +\cos \left (3 \arcsin \left (c x \right )\right ) b e -\sqrt {-c^{2} x^{2}+1}\, b e}{4 c^{3} \left (a +b \arcsin \left (c x \right )\right ) b^{2}}\)	\(367\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

-((x^2*e + d)*sqrt(c*x + 1)*sqrt(-c*x + 1) - (b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c)*integr
ate((3*c^2*x^3*e + (c^2*d - 2*e)*x)*sqrt(c*x + 1)*sqrt(-c*x + 1)/(a*b*c^3*x^2 - a*b*c + (b^2*c^3*x^2 - b^2*c)*
arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))), 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 = 0.00, 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)/(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 891 vs. \(2 (237) = 474\).
time = 0.46, size = 891, normalized size = 3.58 \begin {gather*} -\frac {3 \, b e \arcsin \left (c x\right ) \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} + \frac {b c^{2} d \arcsin \left (c x\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} + \frac {3 \, b e \arcsin \left (c x\right ) \cos \left (\frac {a}{b}\right )^{3} \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} - \frac {b c^{2} d \arcsin \left (c x\right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} - \frac {3 \, a e \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} + \frac {a c^{2} d \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} + \frac {3 \, a e \cos \left (\frac {a}{b}\right )^{3} \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} - \frac {a c^{2} d \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} + \frac {3 \, b e \arcsin \left (c x\right ) \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{4 \, {\left (b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}\right )}} + \frac {b e \arcsin \left (c x\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{4 \, {\left (b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}\right )}} - \frac {9 \, b e \arcsin \left (c x\right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{4 \, {\left (b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}\right )}} - \frac {b e \arcsin \left (c x\right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{4 \, {\left (b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}\right )}} - \frac {\sqrt {-c^{2} x^{2} + 1} b c^{2} d}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} + \frac {3 \, a e \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{4 \, {\left (b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}\right )}} + \frac {a e \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{4 \, {\left (b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}\right )}} - \frac {9 \, a e \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{4 \, {\left (b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}\right )}} - \frac {a e \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{4 \, {\left (b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}\right )}} + \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} - \frac {\sqrt {-c^{2} x^{2} + 1} b e}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*b*e*arcsin(c*x)*cos(a/b)^2*cos_integral(3*a/b + 3*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) +
b*c^2*d*arcsin(c*x)*cos_integral(a/b + arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 3*b*e*arcsin
(c*x)*cos(a/b)^3*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - b*c^2*d*arcsin(c*x)*c
os(a/b)*sin_integral(a/b + arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 3*a*e*cos(a/b)^2*cos_integral(3*a/
b + 3*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + a*c^2*d*cos_integral(a/b + arcsin(c*x))*sin(a/
b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 3*a*e*cos(a/b)^3*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^3*arcsin(c*
x) + a*b^2*c^3) - a*c^2*d*cos(a/b)*sin_integral(a/b + arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 3/4*b*e
*arcsin(c*x)*cos_integral(3*a/b + 3*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 1/4*b*e*arcsin(c
*x)*cos_integral(a/b + arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 9/4*b*e*arcsin(c*x)*cos(a/b)*
sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 1/4*b*e*arcsin(c*x)*cos(a/b)*sin_integ
ral(a/b + arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - sqrt(-c^2*x^2 + 1)*b*c^2*d/(b^3*c^3*arcsin(c*x) + a
*b^2*c^3) + 3/4*a*e*cos_integral(3*a/b + 3*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 1/4*a*e*c
os_integral(a/b + arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 9/4*a*e*cos(a/b)*sin_integral(3*a/
b + 3*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 1/4*a*e*cos(a/b)*sin_integral(a/b + arcsin(c*x))/(b^3*c
^3*arcsin(c*x) + a*b^2*c^3) + (-c^2*x^2 + 1)^(3/2)*b*e/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - sqrt(-c^2*x^2 + 1)*
b*e/(b^3*c^3*arcsin(c*x) + a*b^2*c^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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