3.3.35 \(\int \frac {(c e+d e x)^2}{(a+b \text {ArcSin}(c+d x))^4} \, dx\) [235]

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

[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 = 0.43, antiderivative size = 337, normalized size of antiderivative = 1.00, 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 = {4889, 12, 4729, 4807, 4727, 3384, 3380, 3383, 4717, 4809} \begin {gather*} -\frac {e^2 \sin \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a+b \text {ArcSin}(c+d x)}{b}\right )}{24 b^4 d}+\frac {9 e^2 \sin \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 (a+b \text {ArcSin}(c+d x))}{b}\right )}{8 b^4 d}+\frac {e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \text {ArcSin}(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 (a+b \text {ArcSin}(c+d x))}{b}\right )}{8 b^4 d}+\frac {3 e^2 \sqrt {1-(c+d x)^2} (c+d x)^2}{2 b^3 d (a+b \text {ArcSin}(c+d x))}-\frac {e^2 \sqrt {1-(c+d x)^2}}{3 b^3 d (a+b \text {ArcSin}(c+d x))}+\frac {e^2 (c+d x)^3}{2 b^2 d (a+b \text {ArcSin}(c+d x))^2}-\frac {e^2 (c+d x)}{3 b^2 d (a+b \text {ArcSin}(c+d x))^2}-\frac {e^2 \sqrt {1-(c+d x)^2} (c+d x)^2}{3 b d (a+b \text {ArcSin}(c+d x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2])/(b*d*(a + b*ArcSin[c + d*x])^3) - (e^2*(c + d*x))/(3*b^2*d*(a + b
*ArcSin[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])/b]*Sin[a/b])/(24*b^4*d) + (9*e^2*CosIntegral[(3*(a + b*ArcSin[c + d*
x]))/b]*Sin[(3*a)/b])/(8*b^4*d) + (e^2*Cos[a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/b])/(24*b^4*d) - (9*e^2*Co
s[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c + d*x]))/b])/(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 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 4729

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 4807

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

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 4889

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 {\text {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 \text {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 ) \text {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 \text {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 \text {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 ) \text {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 \text {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 ) \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+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 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 0.75, size = 264, normalized size = 0.78 \begin {gather*} \frac {e^2 \left (-\frac {8 b^3 (c+d x)^2 \sqrt {1-(c+d x)^2}}{(a+b \text {ArcSin}(c+d x))^3}+\frac {4 b^2 \left (-2 (c+d x)+3 (c+d x)^3\right )}{(a+b \text {ArcSin}(c+d x))^2}+\frac {4 b \sqrt {1-(c+d x)^2} \left (-2+9 (c+d x)^2\right )}{a+b \text {ArcSin}(c+d x)}+80 \left (\text {CosIntegral}\left (\frac {a}{b}+\text {ArcSin}(c+d x)\right ) \sin \left (\frac {a}{b}\right )-\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )\right )+27 \left (-3 \text {CosIntegral}\left (\frac {a}{b}+\text {ArcSin}(c+d x)\right ) \sin \left (\frac {a}{b}\right )+\text {CosIntegral}\left (3 \left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )\right ) \sin \left (\frac {3 a}{b}\right )+3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )-\cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )\right )\right )\right )}{24 b^4 d} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(752\) vs. \(2(313)=626\).
time = 0.27, size = 753, normalized size = 2.23 Too large to display

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,method=_RETURNVERBOSE)

[Out]

1/24/d*e^2*(arcsin(d*x+c)^3*Si(arcsin(d*x+c)+a/b)*cos(a/b)*b^3-arcsin(d*x+c)^3*Ci(arcsin(d*x+c)+a/b)*sin(a/b)*
b^3+2*(1-(d*x+c)^2)^(1/2)*arcsin(d*x+c)*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*a
rcsin(d*x+c)^3*Ci(3*arcsin(d*x+c)+3*a/b)*sin(3*a/b)*b^3-18*cos(3*arcsin(d*x+c))*arcsin(d*x+c)*a*b^2-3*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*ar
csin(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+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+a*b^2*(d*x+c)-2*(1-(d*x+c)^2)^(1/2)*b^3+2*cos(3*arcsin(d*x+c))*b^3+(1-(d*x+c)^2)^(1/2)*a^2*b-3*sin(3
*arcsin(d*x+c))*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*arcsin(d*x+c))*a^2*b-3*sin(3*arcsin(d*x+c))*a*b^2+arcsin(d*x+c)*b^3*(d*x+c)+Si(arcsin(d*x+
c)+a/b)*cos(a/b)*a^3-Ci(arcsin(d*x+c)+a/b)*sin(a/b)*a^3+(1-(d*x+c)^2)^(1/2)*arcsin(d*x+c)^2*b^3-9*cos(3*arcsin
(d*x+c))*arcsin(d*x+c)^2*b^3)/(a+b*arcsin(d*x+c))^3/b^4

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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

[Out]

integral((d^2*x^2 + 2*c*d*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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 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 ) \end {gather*}

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3109 vs. \(2 (313) = 626\).
time = 0.81, size = 3109, normalized size = 9.23 \begin {gather*} \text {Too large to display} \end {gather*}

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*e^2*arcsin(d*x + c)^3*cos(a/b)^2*cos_integral(3*a/b + 3*arcsin(d*x + c))*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*e^2*arcsin(d*x + c)^3*
cos(a/b)^3*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*e^2*arcsin(d*x + c)^2*cos(a/b)^2*cos_integral(3*a/b + 3*ar
csin(d*x + c))*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*e^2*arcsin(d*x + c)^2*cos(a/b)^3*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*e^2*arcsin(d*x
+ c)^3*cos_integral(3*a/b + 3*arcsin(d*x + c))*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*e^2*arcsin(d*x + c)*cos(a/b)^2*cos_integral(3*a/b +
3*arcsin(d*x + c))*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*e^2*arcsin(d*x + c)^3*cos_integral(a/b + arcsin(d*x + c))*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*e^2*arcsin(d*x +
c)^3*cos(a/b)*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*e^2*arcsin(d*x + c)*cos(a/b)^3*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*e^2*arcsin(d*x + c)^3*cos(a/b)*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*e^2*arcsin(d*x + c)^2*cos_i
ntegral(3*a/b + 3*arcsin(d*x + c))*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*e^2*cos(a/b)^2*cos_integral(3*a/b + 3*arcsin(d*x + c))*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*e^2*
arcsin(d*x + c)^2*cos_integral(a/b + arcsin(d*x + c))*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*e^2*arcsin(d*x + c)^2*cos(a/b)*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*e^2*cos(a/b)^3*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*e^2*arcsin(d*x + c)^2*cos(a/b)*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*e^2*arcsin(d*x + c)^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*e^2*
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^2*b*e^2*arcsin(d*x + c)*cos_integral(3*a/b + 3*arcsin(d*x + c))*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*e^2*arcsin(d*x + c)*cos_
integral(a/b + arcsin(d*x + c))*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*e^2*arcsin(d*x + c)*cos(a/b)*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
*e^2*arcsin(d*x + c)*cos(a/b)*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*e^2*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) + 7/6*sqrt(-(
d*x + c)^2 + 1)*b^3*e^2*arcsin(d*x + c)^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*e^2*arcsin(d*x + c)/(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*e^2*cos_in
tegral(3*a/b + 3*arcsin(d*x + c))*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*e^2*cos_integral(a/b + arcsin(d*x + c))*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*e^2*cos(a/b)*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*e^2*cos(a/b)*si...

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

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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