3.235 \(\int \frac {(c e+d e x)^2}{(a+b \sin ^{-1}(c+d x))^4} \, dx\)

Optimal. Leaf size=337 \[ -\frac {e^2 \sin \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c+d x)}{b}\right )}{24 b^4 d}+\frac {9 e^2 \sin \left (\frac {3 a}{b}\right ) \text {Ci}\left (\frac {3 \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{8 b^4 d}+\frac {e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c+d x)}{b}\right )}{24 b^4 d}-\frac {9 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{8 b^4 d}+\frac {3 e^2 \sqrt {1-(c+d x)^2} (c+d x)^2}{2 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac {e^2 \sqrt {1-(c+d x)^2}}{3 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {e^2 (c+d x)^3}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {e^2 (c+d x)}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {e^2 \sqrt {1-(c+d x)^2} (c+d x)^2}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3} \]

[Out]

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

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Rubi [A]  time = 0.67, antiderivative size = 333, normalized size of antiderivative = 0.99, number of steps used = 18, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {4805, 12, 4633, 4719, 4631, 3303, 3299, 3302, 4621, 4723} \[ -\frac {e^2 \sin \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )}{24 b^4 d}+\frac {9 e^2 \sin \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 a}{b}+3 \sin ^{-1}(c+d x)\right )}{8 b^4 d}+\frac {e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )}{24 b^4 d}-\frac {9 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c+d x)\right )}{8 b^4 d}+\frac {e^2 (c+d x)^3}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {3 e^2 \sqrt {1-(c+d x)^2} (c+d x)^2}{2 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac {e^2 (c+d x)}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {e^2 \sqrt {1-(c+d x)^2}}{3 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac {e^2 \sqrt {1-(c+d x)^2} (c+d x)^2}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^2/(a + b*ArcSin[c + d*x])^4,x]

[Out]

-(e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2])/(3*b*d*(a + b*ArcSin[c + d*x])^3) - (e^2*(c + d*x))/(3*b^2*d*(a + b*A
rcSin[c + d*x])^2) + (e^2*(c + d*x)^3)/(2*b^2*d*(a + b*ArcSin[c + d*x])^2) - (e^2*Sqrt[1 - (c + d*x)^2])/(3*b^
3*d*(a + b*ArcSin[c + d*x])) + (3*e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2])/(2*b^3*d*(a + b*ArcSin[c + d*x])) - (
e^2*CosIntegral[a/b + ArcSin[c + d*x]]*Sin[a/b])/(24*b^4*d) + (9*e^2*CosIntegral[(3*a)/b + 3*ArcSin[c + d*x]]*
Sin[(3*a)/b])/(8*b^4*d) + (e^2*Cos[a/b]*SinIntegral[a/b + ArcSin[c + d*x]])/(24*b^4*d) - (9*e^2*Cos[(3*a)/b]*S
inIntegral[(3*a)/b + 3*ArcSin[c + d*x]])/(8*b^4*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 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 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]

Rule 4633

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

Rule 4719

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
((f*x)^m*(a + b*ArcSin[c*x])^(n + 1))/(b*c*Sqrt[d]*(n + 1)), x] - Dist[(f*m)/(b*c*Sqrt[d]*(n + 1)), Int[(f*x)^
(m - 1)*(a + b*ArcSin[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[n,
-1] && GtQ[d, 0]

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 4805

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]

Rubi steps

\begin {align*} \int \frac {(c e+d e x)^2}{\left (a+b \sin ^{-1}(c+d x)\right )^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {e^2 x^2}{\left (a+b \sin ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \operatorname {Subst}\left (\int \frac {x^2}{\left (a+b \sin ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac {\left (2 e^2\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \left (a+b \sin ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{3 b d}-\frac {e^2 \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {1-x^2} \left (a+b \sin ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{b d}\\ &=-\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}-\frac {e^2 (c+d x)}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {e^2 (c+d x)^3}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {e^2 \operatorname {Subst}\left (\int \frac {1}{\left (a+b \sin ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{3 b^2 d}-\frac {\left (3 e^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (a+b \sin ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{2 b^2 d}\\ &=-\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}-\frac {e^2 (c+d x)}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {e^2 (c+d x)^3}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {e^2 \sqrt {1-(c+d x)^2}}{3 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{2 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac {e^2 \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \left (a+b \sin ^{-1}(x)\right )} \, dx,x,c+d x\right )}{3 b^3 d}-\frac {\left (3 e^2\right ) \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+d x)\right )}{2 b^3 d}\\ &=-\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}-\frac {e^2 (c+d x)}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {e^2 (c+d x)^3}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {e^2 \sqrt {1-(c+d x)^2}}{3 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{2 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac {e^2 \operatorname {Subst}\left (\int \frac {\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{3 b^3 d}+\frac {\left (3 e^2\right ) \operatorname {Subst}\left (\int \frac {\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 b^3 d}-\frac {\left (9 e^2\right ) \operatorname {Subst}\left (\int \frac {\sin (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 b^3 d}\\ &=-\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}-\frac {e^2 (c+d x)}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {e^2 (c+d x)^3}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {e^2 \sqrt {1-(c+d x)^2}}{3 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{2 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\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+d x)\right )}{3 b^3 d}+\frac {\left (3 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+d x)\right )}{8 b^3 d}-\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+d x)\right )}{8 b^3 d}+\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+d x)\right )}{3 b^3 d}-\frac {\left (3 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+d x)\right )}{8 b^3 d}+\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+d x)\right )}{8 b^3 d}\\ &=-\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}-\frac {e^2 (c+d x)}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {e^2 (c+d x)^3}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {e^2 \sqrt {1-(c+d x)^2}}{3 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{2 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac {e^2 \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right ) \sin \left (\frac {a}{b}\right )}{24 b^4 d}+\frac {9 e^2 \text {Ci}\left (\frac {3 a}{b}+3 \sin ^{-1}(c+d x)\right ) \sin \left (\frac {3 a}{b}\right )}{8 b^4 d}+\frac {e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )}{24 b^4 d}-\frac {9 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c+d x)\right )}{8 b^4 d}\\ \end {align*}

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Mathematica [A]  time = 1.15, size = 264, normalized size = 0.78 \[ \frac {e^2 \left (-\frac {8 b^3 (c+d x)^2 \sqrt {1-(c+d x)^2}}{\left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac {4 b^2 \left (3 (c+d x)^3-2 (c+d x)\right )}{\left (a+b \sin ^{-1}(c+d x)\right )^2}+80 \left (\sin \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )-\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )\right )+27 \left (-3 \sin \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )+\sin \left (\frac {3 a}{b}\right ) \text {Ci}\left (3 \left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )\right )+3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )-\cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )\right )\right )+\frac {4 b \sqrt {1-(c+d x)^2} \left (9 (c+d x)^2-2\right )}{a+b \sin ^{-1}(c+d x)}\right )}{24 b^4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^2/(a + b*ArcSin[c + d*x])^4,x]

[Out]

(e^2*((-8*b^3*(c + d*x)^2*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])^3 + (4*b^2*(-2*(c + d*x) + 3*(c + d*x
)^3))/(a + b*ArcSin[c + d*x])^2 + (4*b*Sqrt[1 - (c + d*x)^2]*(-2 + 9*(c + d*x)^2))/(a + b*ArcSin[c + d*x]) + 8
0*(CosIntegral[a/b + ArcSin[c + d*x]]*Sin[a/b] - Cos[a/b]*SinIntegral[a/b + ArcSin[c + d*x]]) + 27*(-3*CosInte
gral[a/b + ArcSin[c + d*x]]*Sin[a/b] + CosIntegral[3*(a/b + ArcSin[c + d*x])]*Sin[(3*a)/b] + 3*Cos[a/b]*SinInt
egral[a/b + ArcSin[c + d*x]] - Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c + d*x])])))/(24*b^4*d)

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fricas [F]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}{b^{4} \arcsin \left (d x + c\right )^{4} + 4 \, a b^{3} \arcsin \left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \arcsin \left (d x + c\right )^{2} + 4 \, a^{3} b \arcsin \left (d x + c\right ) + a^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^2/(a+b*arcsin(d*x+c))^4,x, algorithm="fricas")

[Out]

integral((d^2*e^2*x^2 + 2*c*d*e^2*x + c^2*e^2)/(b^4*arcsin(d*x + c)^4 + 4*a*b^3*arcsin(d*x + c)^3 + 6*a^2*b^2*
arcsin(d*x + c)^2 + 4*a^3*b*arcsin(d*x + c) + a^4), x)

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giac [B]  time = 1.19, size = 3073, normalized size = 9.12 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^2/(a+b*arcsin(d*x+c))^4,x, algorithm="giac")

[Out]

9/2*b^3*arcsin(d*x + c)^3*cos(a/b)^2*cos_integral(3*a/b + 3*arcsin(d*x + c))*e^2*sin(a/b)/(b^7*d*arcsin(d*x +
c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 9/2*b^3*arcsin(d*x + c)^3*cos(
a/b)^3*e^2*sin_integral(3*a/b + 3*arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*
a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 27/2*a*b^2*arcsin(d*x + c)^2*cos(a/b)^2*cos_integral(3*a/b + 3*arcsin
(d*x + c))*e^2*sin(a/b)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) +
a^3*b^4*d) - 27/2*a*b^2*arcsin(d*x + c)^2*cos(a/b)^3*e^2*sin_integral(3*a/b + 3*arcsin(d*x + c))/(b^7*d*arcsi
n(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 9/8*b^3*arcsin(d*x + c
)^3*cos_integral(3*a/b + 3*arcsin(d*x + c))*e^2*sin(a/b)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^
2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 27/2*a^2*b*arcsin(d*x + c)*cos(a/b)^2*cos_integral(3*a/b + 3*ar
csin(d*x + c))*e^2*sin(a/b)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x +
c) + a^3*b^4*d) - 1/24*b^3*arcsin(d*x + c)^3*cos_integral(a/b + arcsin(d*x + c))*e^2*sin(a/b)/(b^7*d*arcsin(d*
x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 27/8*b^3*arcsin(d*x + c)^3
*cos(a/b)*e^2*sin_integral(3*a/b + 3*arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 +
3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 27/2*a^2*b*arcsin(d*x + c)*cos(a/b)^3*e^2*sin_integral(3*a/b + 3*a
rcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4
*d) + 1/24*b^3*arcsin(d*x + c)^3*cos(a/b)*e^2*sin_integral(a/b + arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3
*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 27/8*a*b^2*arcsin(d*x + c)^2*cos_integ
ral(3*a/b + 3*arcsin(d*x + c))*e^2*sin(a/b)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5
*d*arcsin(d*x + c) + a^3*b^4*d) + 9/2*a^3*cos(a/b)^2*cos_integral(3*a/b + 3*arcsin(d*x + c))*e^2*sin(a/b)/(b^7
*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 1/8*a*b^2*arcs
in(d*x + c)^2*cos_integral(a/b + arcsin(d*x + c))*e^2*sin(a/b)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x
+ c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 81/8*a*b^2*arcsin(d*x + c)^2*cos(a/b)*e^2*sin_integral(3*
a/b + 3*arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c)
+ a^3*b^4*d) - 9/2*a^3*cos(a/b)^3*e^2*sin_integral(3*a/b + 3*arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b
^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 1/8*a*b^2*arcsin(d*x + c)^2*cos(a/b)*e^2*s
in_integral(a/b + arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin
(d*x + c) + a^3*b^4*d) - 3/2*(-(d*x + c)^2 + 1)^(3/2)*b^3*arcsin(d*x + c)^2*e^2/(b^7*d*arcsin(d*x + c)^3 + 3*a
*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 1/2*((d*x + c)^2 - 1)*(d*x + c)*b^3*arcs
in(d*x + c)*e^2/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4
*d) - 27/8*a^2*b*arcsin(d*x + c)*cos_integral(3*a/b + 3*arcsin(d*x + c))*e^2*sin(a/b)/(b^7*d*arcsin(d*x + c)^3
+ 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 1/8*a^2*b*arcsin(d*x + c)*cos_inte
gral(a/b + arcsin(d*x + c))*e^2*sin(a/b)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*
arcsin(d*x + c) + a^3*b^4*d) + 81/8*a^2*b*arcsin(d*x + c)*cos(a/b)*e^2*sin_integral(3*a/b + 3*arcsin(d*x + c))
/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 1/8*a^2*b
*arcsin(d*x + c)*cos(a/b)*e^2*sin_integral(a/b + arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(
d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 3*(-(d*x + c)^2 + 1)^(3/2)*a*b^2*arcsin(d*x + c)*e^2/(
b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 7/6*sqrt(-(
d*x + c)^2 + 1)*b^3*arcsin(d*x + c)^2*e^2/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d
*arcsin(d*x + c) + a^3*b^4*d) + 1/2*((d*x + c)^2 - 1)*(d*x + c)*a*b^2*e^2/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d
*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 1/6*(d*x + c)*b^3*arcsin(d*x + c)*e^2/(b^7*d*a
rcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 9/8*a^3*cos_integr
al(3*a/b + 3*arcsin(d*x + c))*e^2*sin(a/b)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*
d*arcsin(d*x + c) + a^3*b^4*d) - 1/24*a^3*cos_integral(a/b + arcsin(d*x + c))*e^2*sin(a/b)/(b^7*d*arcsin(d*x +
c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 27/8*a^3*cos(a/b)*e^2*sin_int
egral(3*a/b + 3*arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d
*x + c) + a^3*b^4*d) + 1/24*a^3*cos(a/b)*e^2*sin_integral(a/b + arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*
a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 3/2*(-(d*x + c)^2 + 1)^(3/2)*a^2*b*e^2/
(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 1/3*(-(d*x
+ c)^2 + 1)^(3/2)*b^3*e^2/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c
) + a^3*b^4*d) + 7/3*sqrt(-(d*x + c)^2 + 1)*a*b^2*arcsin(d*x + c)*e^2/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arc
sin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 1/6*(d*x + c)*a*b^2*e^2/(b^7*d*arcsin(d*x + c)^3 +
3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 7/6*sqrt(-(d*x + c)^2 + 1)*a^2*b*e^2
/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 1/3*sqrt(
-(d*x + c)^2 + 1)*b^3*e^2/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c)
+ a^3*b^4*d)

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maple [B]  time = 0.22, size = 753, normalized size = 2.23 \[ \frac {e^{2} \left (\Si \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a^{3}-\Ci \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a^{3}-27 \Si \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a^{3}+27 \Ci \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a^{3}+\arcsin \left (d x +c \right ) \left (d x +c \right ) b^{3}+\arcsin \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}\, b^{3}+\sqrt {1-\left (d x +c \right )^{2}}\, a^{2} b +\left (d x +c \right ) a \,b^{2}-3 \arcsin \left (d x +c \right ) \sin \left (3 \arcsin \left (d x +c \right )\right ) b^{3}-9 \cos \left (3 \arcsin \left (d x +c \right )\right ) a^{2} b -3 \sin \left (3 \arcsin \left (d x +c \right )\right ) a \,b^{2}-9 \arcsin \left (d x +c \right )^{2} \cos \left (3 \arcsin \left (d x +c \right )\right ) b^{3}+81 \arcsin \left (d x +c \right ) \Ci \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a^{2} b +3 \arcsin \left (d x +c \right )^{2} \Si \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a \,b^{2}-3 \arcsin \left (d x +c \right )^{2} \Ci \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a \,b^{2}+3 \arcsin \left (d x +c \right ) \Si \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a^{2} b -3 \arcsin \left (d x +c \right ) \Ci \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a^{2} b -81 \arcsin \left (d x +c \right )^{2} \Si \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a \,b^{2}+81 \arcsin \left (d x +c \right )^{2} \Ci \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a \,b^{2}-81 \arcsin \left (d x +c \right ) \Si \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a^{2} b +2 \cos \left (3 \arcsin \left (d x +c \right )\right ) b^{3}-2 \sqrt {1-\left (d x +c \right )^{2}}\, b^{3}+2 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\, a \,b^{2}-18 \arcsin \left (d x +c \right ) \cos \left (3 \arcsin \left (d x +c \right )\right ) a \,b^{2}-27 \arcsin \left (d x +c \right )^{3} \Si \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b^{3}+27 \arcsin \left (d x +c \right )^{3} \Ci \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b^{3}+\arcsin \left (d x +c \right )^{3} \Si \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b^{3}-\arcsin \left (d x +c \right )^{3} \Ci \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b^{3}\right )}{24 d \left (a +b \arcsin \left (d x +c \right )\right )^{3} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24/d*e^2*(arcsin(d*x+c)*(d*x+c)*b^3+Si(arcsin(d*x+c)+a/b)*cos(a/b)*a^3-Ci(arcsin(d*x+c)+a/b)*sin(a/b)*a^3+ar
csin(d*x+c)^2*(1-(d*x+c)^2)^(1/2)*b^3+(1-(d*x+c)^2)^(1/2)*a^2*b+(d*x+c)*a*b^2-3*arcsin(d*x+c)*sin(3*arcsin(d*x
+c))*b^3-27*Si(3*arcsin(d*x+c)+3*a/b)*cos(3*a/b)*a^3+27*Ci(3*arcsin(d*x+c)+3*a/b)*sin(3*a/b)*a^3-9*cos(3*arcsi
n(d*x+c))*a^2*b-3*sin(3*arcsin(d*x+c))*a*b^2-9*arcsin(d*x+c)^2*cos(3*arcsin(d*x+c))*b^3+81*arcsin(d*x+c)*Ci(3*
arcsin(d*x+c)+3*a/b)*sin(3*a/b)*a^2*b+3*arcsin(d*x+c)^2*Si(arcsin(d*x+c)+a/b)*cos(a/b)*a*b^2-3*arcsin(d*x+c)^2
*Ci(arcsin(d*x+c)+a/b)*sin(a/b)*a*b^2+3*arcsin(d*x+c)*Si(arcsin(d*x+c)+a/b)*cos(a/b)*a^2*b-3*arcsin(d*x+c)*Ci(
arcsin(d*x+c)+a/b)*sin(a/b)*a^2*b-81*arcsin(d*x+c)^2*Si(3*arcsin(d*x+c)+3*a/b)*cos(3*a/b)*a*b^2+81*arcsin(d*x+
c)^2*Ci(3*arcsin(d*x+c)+3*a/b)*sin(3*a/b)*a*b^2-81*arcsin(d*x+c)*Si(3*arcsin(d*x+c)+3*a/b)*cos(3*a/b)*a^2*b+2*
cos(3*arcsin(d*x+c))*b^3-2*(1-(d*x+c)^2)^(1/2)*b^3+2*arcsin(d*x+c)*(1-(d*x+c)^2)^(1/2)*a*b^2-27*arcsin(d*x+c)^
3*Si(3*arcsin(d*x+c)+3*a/b)*cos(3*a/b)*b^3+27*arcsin(d*x+c)^3*Ci(3*arcsin(d*x+c)+3*a/b)*sin(3*a/b)*b^3-18*arcs
in(d*x+c)*cos(3*arcsin(d*x+c))*a*b^2+arcsin(d*x+c)^3*Si(arcsin(d*x+c)+a/b)*cos(a/b)*b^3-arcsin(d*x+c)^3*Ci(arc
sin(d*x+c)+a/b)*sin(a/b)*b^3)/(a+b*arcsin(d*x+c))^3/b^4

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^2/(a+b*arcsin(d*x+c))^4,x, algorithm="maxima")

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,e+d\,e\,x\right )}^2}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ e^{2} \left (\int \frac {c^{2}}{a^{4} + 4 a^{3} b \operatorname {asin}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asin}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asin}^{4}{\left (c + d x \right )}}\, dx + \int \frac {d^{2} x^{2}}{a^{4} + 4 a^{3} b \operatorname {asin}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asin}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asin}^{4}{\left (c + d x \right )}}\, dx + \int \frac {2 c d x}{a^{4} + 4 a^{3} b \operatorname {asin}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asin}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asin}^{4}{\left (c + d x \right )}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**2/(a+b*asin(d*x+c))**4,x)

[Out]

e**2*(Integral(c**2/(a**4 + 4*a**3*b*asin(c + d*x) + 6*a**2*b**2*asin(c + d*x)**2 + 4*a*b**3*asin(c + d*x)**3
+ b**4*asin(c + d*x)**4), x) + Integral(d**2*x**2/(a**4 + 4*a**3*b*asin(c + d*x) + 6*a**2*b**2*asin(c + d*x)**
2 + 4*a*b**3*asin(c + d*x)**3 + b**4*asin(c + d*x)**4), x) + Integral(2*c*d*x/(a**4 + 4*a**3*b*asin(c + d*x) +
6*a**2*b**2*asin(c + d*x)**2 + 4*a*b**3*asin(c + d*x)**3 + b**4*asin(c + d*x)**4), x))

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