\(\int \frac {(d+e x)^2}{(a+b \arcsin (c x))^2} \, dx\) [22]

Optimal result

Integrand size = 18, antiderivative size = 362 \[ \int \frac {(d+e x)^2}{(a+b \arcsin (c x))^2} \, dx=-\frac {d^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {2 d e x \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {e^2 x^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}+\frac {2 d e \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}+\frac {d^2 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{b^2 c}+\frac {e^2 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{4 b^2 c^3}-\frac {3 e^2 \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{4 b^2 c^3}-\frac {d^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c}-\frac {e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b^2 c^3}+\frac {2 d e \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}+\frac {3 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b^2 c^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {4829, 4717, 4809, 3384, 3380, 3383, 4727} \[ \int \frac {(d+e x)^2}{(a+b \arcsin (c x))^2} \, dx=\frac {e^2 \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b^2 c^3}-\frac {3 e^2 \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b^2 c^3}-\frac {e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b^2 c^3}+\frac {3 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b^2 c^3}+\frac {2 d e \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}+\frac {2 d e \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}+\frac {d^2 \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c}-\frac {d^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c}-\frac {d^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {2 d e x \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {e^2 x^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))} \]

[In]

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

Rule 4829

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(d + e
*x)^m*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 0] && LtQ[n, -1]

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

Mathematica [A] (verified)

Time = 1.63 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.80 \[ \int \frac {(d+e x)^2}{(a+b \arcsin (c x))^2} \, dx=-\frac {\frac {4 b c^2 d^2 \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}+\frac {8 b c^2 d e x \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}+\frac {4 b c^2 e^2 x^2 \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}-8 c d e \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-\left (4 c^2 d^2+e^2\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right ) \sin \left (\frac {a}{b}\right )+3 e^2 \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {3 a}{b}\right )+4 c^2 d^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )+e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )-8 c d e \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-3 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{4 b^2 c^3} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.45

method	result	size
derivativedivides	\(-\frac {4 \arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b \,c^{2} d^{2}-4 \arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b \,c^{2} d^{2}-8 \arcsin \left (c x \right ) \operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) b c d e -8 \arcsin \left (c x \right ) \operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b c d e +4 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a \,c^{2} d^{2}-4 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a \,c^{2} d^{2}+4 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} d^{2}+\arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b \,e^{2}-\arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b \,e^{2}-3 \arcsin \left (c x \right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b \,e^{2}+3 \arcsin \left (c x \right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b \,e^{2}-8 \,\operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a c d e -8 \,\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a c d e +\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a \,e^{2}-\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a \,e^{2}+4 \sin \left (2 \arcsin \left (c x \right )\right ) b c d e -3 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a \,e^{2}+3 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a \,e^{2}+\sqrt {-c^{2} x^{2}+1}\, b \,e^{2}-\cos \left (3 \arcsin \left (c x \right )\right ) b \,e^{2}}{4 c^{3} \left (a +b \arcsin \left (c x \right )\right ) b^{2}}\)	\(526\)
default	\(-\frac {4 \arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b \,c^{2} d^{2}-4 \arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b \,c^{2} d^{2}-8 \arcsin \left (c x \right ) \operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) b c d e -8 \arcsin \left (c x \right ) \operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b c d e +4 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a \,c^{2} d^{2}-4 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a \,c^{2} d^{2}+4 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} d^{2}+\arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b \,e^{2}-\arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b \,e^{2}-3 \arcsin \left (c x \right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b \,e^{2}+3 \arcsin \left (c x \right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b \,e^{2}-8 \,\operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a c d e -8 \,\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a c d e +\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a \,e^{2}-\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a \,e^{2}+4 \sin \left (2 \arcsin \left (c x \right )\right ) b c d e -3 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a \,e^{2}+3 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a \,e^{2}+\sqrt {-c^{2} x^{2}+1}\, b \,e^{2}-\cos \left (3 \arcsin \left (c x \right )\right ) b \,e^{2}}{4 c^{3} \left (a +b \arcsin \left (c x \right )\right ) b^{2}}\)	\(526\)

[In]

int((e*x+d)^2/(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^2-4*arcsin(c*x)*Ci(arcsin(c*x)+a/b)*sin(a/b)*b*c^
2*d^2-8*arcsin(c*x)*Si(2*arcsin(c*x)+2*a/b)*sin(2*a/b)*b*c*d*e-8*arcsin(c*x)*Ci(2*arcsin(c*x)+2*a/b)*cos(2*a/b
)*b*c*d*e+4*Si(arcsin(c*x)+a/b)*cos(a/b)*a*c^2*d^2-4*Ci(arcsin(c*x)+a/b)*sin(a/b)*a*c^2*d^2+4*(-c^2*x^2+1)^(1/
2)*b*c^2*d^2+arcsin(c*x)*Si(arcsin(c*x)+a/b)*cos(a/b)*b*e^2-arcsin(c*x)*Ci(arcsin(c*x)+a/b)*sin(a/b)*b*e^2-3*a
rcsin(c*x)*Si(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*b*e^2+3*arcsin(c*x)*Ci(3*arcsin(c*x)+3*a/b)*sin(3*a/b)*b*e^2-8*S
i(2*arcsin(c*x)+2*a/b)*sin(2*a/b)*a*c*d*e-8*Ci(2*arcsin(c*x)+2*a/b)*cos(2*a/b)*a*c*d*e+Si(arcsin(c*x)+a/b)*cos
(a/b)*a*e^2-Ci(arcsin(c*x)+a/b)*sin(a/b)*a*e^2+4*sin(2*arcsin(c*x))*b*c*d*e-3*Si(3*arcsin(c*x)+3*a/b)*cos(3*a/
b)*a*e^2+3*Ci(3*arcsin(c*x)+3*a/b)*sin(3*a/b)*a*e^2+(-c^2*x^2+1)^(1/2)*b*e^2-cos(3*arcsin(c*x))*b*e^2)/(a+b*ar
csin(c*x))/b^2

Fricas [F]

\[ \int \frac {(d+e x)^2}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {(d+e x)^2}{(a+b \arcsin (c x))^2} \, dx=\int \frac {\left (d + e x\right )^{2}}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {(d+e x)^2}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

-((e^2*x^2 + 2*d*e*x + d^2)*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)*integrate((3*c^2*e^2*x^3 + 4*c^2*d*e*x^2 - 2*d*e + (c^2*d^2 - 2*e^2)*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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1276 vs. \(2 (348) = 696\).

Time = 0.38 (sec) , antiderivative size = 1276, normalized size of antiderivative = 3.52 \[ \int \frac {(d+e x)^2}{(a+b \arcsin (c x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

4*b*c*d*e*arcsin(c*x)*cos(a/b)^2*cos_integral(2*a/b + 2*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 3*b*e
^2*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^2*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^2*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) + 4*b*c*d*e*arcsin(c*x)
*cos(a/b)*sin(a/b)*sin_integral(2*a/b + 2*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - b*c^2*d^2*arcsin(c*
x)*cos(a/b)*sin_integral(a/b + arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 4*a*c*d*e*cos(a/b)^2*cos_integ
ral(2*a/b + 2*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 3*a*e^2*cos(a/b)^2*cos_integral(3*a/b + 3*arcsi
n(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + a*c^2*d^2*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^2*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) + 4*a*c*d*e*cos(a/b)*sin(a/b)*sin_integral(2*a/b + 2*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) -
a*c^2*d^2*cos(a/b)*sin_integral(a/b + arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 2*sqrt(-c^2*x^2 + 1)*b
*c^2*d*e*x/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 2*b*c*d*e*arcsin(c*x)*cos_integral(2*a/b + 2*arcsin(c*x))/(b^3*
c^3*arcsin(c*x) + a*b^2*c^3) + 3/4*b*e^2*arcsin(c*x)*cos_integral(3*a/b + 3*arcsin(c*x))*sin(a/b)/(b^3*c^3*arc
sin(c*x) + a*b^2*c^3) + 1/4*b*e^2*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^2*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^2*arcsin(c*x)*cos(a/b)*sin_integral(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^2/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 2*a*c*d*e*cos_integral(2*a/b + 2*arcsin(c*x))/(b^3
*c^3*arcsin(c*x) + a*b^2*c^3) + 3/4*a*e^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) + 1/4*a*e^2*cos_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^2*
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^2*cos(a/b)*sin_integr
al(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^2/(b^3*c^3*arcsin(c*x) + a*
b^2*c^3) - sqrt(-c^2*x^2 + 1)*b*e^2/(b^3*c^3*arcsin(c*x) + a*b^2*c^3)

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^2}{(a+b \arcsin (c x))^2} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \]

[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)