3.234 \(\int \frac{(c e+d e x)^3}{(a+b \sin ^{-1}(c+d x))^4} \, dx\)

Optimal. Leaf size=346 \[ -\frac{e^3 \cos \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d}+\frac{4 e^3 \cos \left (\frac{4 a}{b}\right ) \text{CosIntegral}\left (\frac{4 \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d}-\frac{e^3 \sin \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d}+\frac{4 e^3 \sin \left (\frac{4 a}{b}\right ) \text{Si}\left (\frac{4 \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d}+\frac{2 e^3 (c+d x)^4}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac{8 e^3 \sqrt{1-(c+d x)^2} (c+d x)^3}{3 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac{e^3 (c+d x)^2}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac{e^3 \sqrt{1-(c+d x)^2} (c+d x)}{b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac{e^3 \sqrt{1-(c+d x)^2} (c+d x)^3}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.681397, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {4805, 12, 4633, 4719, 4631, 3303, 3299, 3302} \[ -\frac{e^3 \cos \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (\frac{2 a}{b}+2 \sin ^{-1}(c+d x)\right )}{3 b^4 d}+\frac{4 e^3 \cos \left (\frac{4 a}{b}\right ) \text{CosIntegral}\left (\frac{4 a}{b}+4 \sin ^{-1}(c+d x)\right )}{3 b^4 d}-\frac{e^3 \sin \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 a}{b}+2 \sin ^{-1}(c+d x)\right )}{3 b^4 d}+\frac{4 e^3 \sin \left (\frac{4 a}{b}\right ) \text{Si}\left (\frac{4 a}{b}+4 \sin ^{-1}(c+d x)\right )}{3 b^4 d}+\frac{2 e^3 (c+d x)^4}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac{8 e^3 \sqrt{1-(c+d x)^2} (c+d x)^3}{3 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac{e^3 (c+d x)^2}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac{e^3 \sqrt{1-(c+d x)^2} (c+d x)}{b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac{e^3 \sqrt{1-(c+d x)^2} (c+d x)^3}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^3/(a + b*ArcSin[c + d*x])^4,x]

[Out]

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

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]

Rule 12

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

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 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 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 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]

Rubi steps

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

Mathematica [A]  time = 0.993745, size = 320, normalized size = 0.92 \[ \frac{e^3 \left (-\frac{2 b^3 \sqrt{1-(c+d x)^2} (c+d x)^3}{\left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac{b^2 \left (4 (c+d x)^4-3 (c+d x)^2\right )}{\left (a+b \sin ^{-1}(c+d x)\right )^2}+30 \left (\cos \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (2 \left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )\right )+\sin \left (\frac{2 a}{b}\right ) \text{Si}\left (2 \left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )\right )-\log \left (a+b \sin ^{-1}(c+d x)\right )\right )+8 \left (-4 \cos \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (2 \left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )\right )+\cos \left (\frac{4 a}{b}\right ) \text{CosIntegral}\left (4 \left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )\right )-4 \sin \left (\frac{2 a}{b}\right ) \text{Si}\left (2 \left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )\right )+\sin \left (\frac{4 a}{b}\right ) \text{Si}\left (4 \left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )\right )+3 \log \left (a+b \sin ^{-1}(c+d x)\right )\right )+\frac{2 b \sqrt{1-(c+d x)^2} \left (8 (c+d x)^3-3 (c+d x)\right )}{a+b \sin ^{-1}(c+d x)}+6 \log \left (a+b \sin ^{-1}(c+d x)\right )\right )}{6 b^4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^3/(a + b*ArcSin[c + d*x])^4,x]

[Out]

(e^3*((-2*b^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])^3 + (b^2*(-3*(c + d*x)^2 + 4*(c + d*x
)^4))/(a + b*ArcSin[c + d*x])^2 + (2*b*Sqrt[1 - (c + d*x)^2]*(-3*(c + d*x) + 8*(c + d*x)^3))/(a + b*ArcSin[c +
d*x]) + 6*Log[a + b*ArcSin[c + d*x]] + 30*(Cos[(2*a)/b]*CosIntegral[2*(a/b + ArcSin[c + d*x])] - Log[a + b*Ar
cSin[c + d*x]] + Sin[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c + d*x])]) + 8*(-4*Cos[(2*a)/b]*CosIntegral[2*(a/b
+ ArcSin[c + d*x])] + Cos[(4*a)/b]*CosIntegral[4*(a/b + ArcSin[c + d*x])] + 3*Log[a + b*ArcSin[c + d*x]] - 4*S
in[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c + d*x])] + Sin[(4*a)/b]*SinIntegral[4*(a/b + ArcSin[c + d*x])])))/(6
*b^4*d)

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Maple [B]  time = 0.048, size = 782, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24/d*e^3*(32*arcsin(d*x+c)^3*Ci(4*arcsin(d*x+c)+4*a/b)*cos(4*a/b)*b^3-8*arcsin(d*x+c)^3*Si(2*arcsin(d*x+c)+2
*a/b)*sin(2*a/b)*b^3-8*arcsin(d*x+c)^3*Ci(2*arcsin(d*x+c)+2*a/b)*cos(2*a/b)*b^3-16*arcsin(d*x+c)*sin(4*arcsin(
d*x+c))*a*b^2+8*arcsin(d*x+c)*sin(2*arcsin(d*x+c))*a*b^2+96*arcsin(d*x+c)^2*Si(4*arcsin(d*x+c)+4*a/b)*sin(4*a/
b)*a*b^2+96*arcsin(d*x+c)^2*Ci(4*arcsin(d*x+c)+4*a/b)*cos(4*a/b)*a*b^2-24*arcsin(d*x+c)^2*Si(2*arcsin(d*x+c)+2
*a/b)*sin(2*a/b)*a*b^2-24*arcsin(d*x+c)^2*Ci(2*arcsin(d*x+c)+2*a/b)*cos(2*a/b)*a*b^2+96*arcsin(d*x+c)*Si(4*arc
sin(d*x+c)+4*a/b)*sin(4*a/b)*a^2*b+96*arcsin(d*x+c)*Ci(4*arcsin(d*x+c)+4*a/b)*cos(4*a/b)*a^2*b-24*arcsin(d*x+c
)*Si(2*arcsin(d*x+c)+2*a/b)*sin(2*a/b)*a^2*b-24*arcsin(d*x+c)*Ci(2*arcsin(d*x+c)+2*a/b)*cos(2*a/b)*a^2*b+32*ar
csin(d*x+c)^3*Si(4*arcsin(d*x+c)+4*a/b)*sin(4*a/b)*b^3+4*sin(2*arcsin(d*x+c))*a^2*b-2*cos(2*arcsin(d*x+c))*a*b
^2+4*arcsin(d*x+c)^2*sin(2*arcsin(d*x+c))*b^3+2*arcsin(d*x+c)*cos(4*arcsin(d*x+c))*b^3-2*arcsin(d*x+c)*cos(2*a
rcsin(d*x+c))*b^3+32*Si(4*arcsin(d*x+c)+4*a/b)*sin(4*a/b)*a^3+32*Ci(4*arcsin(d*x+c)+4*a/b)*cos(4*a/b)*a^3-8*Si
(2*arcsin(d*x+c)+2*a/b)*sin(2*a/b)*a^3-8*Ci(2*arcsin(d*x+c)+2*a/b)*cos(2*a/b)*a^3-8*sin(4*arcsin(d*x+c))*a^2*b
-8*arcsin(d*x+c)^2*sin(4*arcsin(d*x+c))*b^3+2*cos(4*arcsin(d*x+c))*a*b^2-2*sin(2*arcsin(d*x+c))*b^3+sin(4*arcs
in(d*x+c))*b^3)/(a+b*arcsin(d*x+c))^3/b^4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^3/(a+b*arcsin(d*x+c))^4,x, algorithm="maxima")

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{d^{3} e^{3} x^{3} + 3 \, c d^{2} e^{3} x^{2} + 3 \, c^{2} d e^{3} x + c^{3} e^{3}}{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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^3/(a+b*arcsin(d*x+c))^4,x, algorithm="fricas")

[Out]

integral((d^3*e^3*x^3 + 3*c*d^2*e^3*x^2 + 3*c^2*d*e^3*x + c^3*e^3)/(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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**3/(a+b*asin(d*x+c))**4,x)

[Out]

Timed out

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Giac [B]  time = 2.58533, size = 5392, normalized size = 15.58 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^3/(a+b*arcsin(d*x+c))^4,x, algorithm="giac")

[Out]

32/3*b^3*arcsin(d*x + c)^3*cos(a/b)^4*cos_integral(4*a/b + 4*arcsin(d*x + c))*e^3/(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) + 32/3*b^3*arcsin(d*x + c)^3*cos(a/b)^3*
e^3*sin(a/b)*sin_integral(4*a/b + 4*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) + 32*a*b^2*arcsin(d*x + c)^2*cos(a/b)^4*cos_integral(4*a/b + 4*arcsin
(d*x + c))*e^3/(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) + 32*a*b^2*arcsin(d*x + c)^2*cos(a/b)^3*e^3*sin(a/b)*sin_integral(4*a/b + 4*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) - 32/3*b^3*arcsin(d*x + c)
^3*cos(a/b)^2*cos_integral(4*a/b + 4*arcsin(d*x + c))*e^3/(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) + 32*a^2*b*arcsin(d*x + c)*cos(a/b)^4*cos_integral(4*a/b + 4*arc
sin(d*x + c))*e^3/(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) - 2/3*b^3*arcsin(d*x + c)^3*cos(a/b)^2*cos_integral(2*a/b + 2*arcsin(d*x + c))*e^3/(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) - 16/3*b^3*arcsin(d*x + c)^3*cos(
a/b)*e^3*sin(a/b)*sin_integral(4*a/b + 4*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) + 32*a^2*b*arcsin(d*x + c)*cos(a/b)^3*e^3*sin(a/b)*sin_integral(
4*a/b + 4*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) - 2/3*b^3*arcsin(d*x + c)^3*cos(a/b)*e^3*sin(a/b)*sin_integral(2*a/b + 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) - 32*a*b^2*arcsin
(d*x + c)^2*cos(a/b)^2*cos_integral(4*a/b + 4*arcsin(d*x + c))*e^3/(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) + 32/3*a^3*cos(a/b)^4*cos_integral(4*a/b + 4*arcsin(d*x
+ c))*e^3/(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) -
2*a*b^2*arcsin(d*x + c)^2*cos(a/b)^2*cos_integral(2*a/b + 2*arcsin(d*x + c))*e^3/(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) - 16*a*b^2*arcsin(d*x + c)^2*cos(a/b)*e^
3*sin(a/b)*sin_integral(4*a/b + 4*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) + 32/3*a^3*cos(a/b)^3*e^3*sin(a/b)*sin_integral(4*a/b + 4*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) - 2*a*b^
2*arcsin(d*x + c)^2*cos(a/b)*e^3*sin(a/b)*sin_integral(2*a/b + 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) - 8/3*(-(d*x + c)^2 + 1)^(3/2)*(d*x + c)
*b^3*arcsin(d*x + c)^2*e^3/(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) + 4/3*b^3*arcsin(d*x + c)^3*cos_integral(4*a/b + 4*arcsin(d*x + c))*e^3/(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) - 32*a^2*b*arcsin(d*x + c)*cos(a/b)
^2*cos_integral(4*a/b + 4*arcsin(d*x + c))*e^3/(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*b^3*arcsin(d*x + c)^3*cos_integral(2*a/b + 2*arcsin(d*x + c))*e^3/(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) - 2*a^2*b*arcsi
n(d*x + c)*cos(a/b)^2*cos_integral(2*a/b + 2*arcsin(d*x + c))*e^3/(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) - 16*a^2*b*arcsin(d*x + c)*cos(a/b)*e^3*sin(a/b)*sin_int
egral(4*a/b + 4*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) - 2*a^2*b*arcsin(d*x + c)*cos(a/b)*e^3*sin(a/b)*sin_integral(2*a/b + 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) - 16/3*(-(d*x
+ c)^2 + 1)^(3/2)*(d*x + c)*a*b^2*arcsin(d*x + c)*e^3/(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) + 5/3*sqrt(-(d*x + c)^2 + 1)*(d*x + c)*b^3*arcsin(d*x + c)^2*e^3/(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) + 2/3*((d*x +
c)^2 - 1)^2*b^3*arcsin(d*x + c)*e^3/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsi
n(d*x + c) + a^3*b^4*d) + 4*a*b^2*arcsin(d*x + c)^2*cos_integral(4*a/b + 4*arcsin(d*x + c))*e^3/(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) - 32/3*a^3*cos(a/b)^2*cos_
integral(4*a/b + 4*arcsin(d*x + c))*e^3/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*a
rcsin(d*x + c) + a^3*b^4*d) + a*b^2*arcsin(d*x + c)^2*cos_integral(2*a/b + 2*arcsin(d*x + c))*e^3/(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) - 2/3*a^3*cos(a/b)^2*cos
_integral(2*a/b + 2*arcsin(d*x + c))*e^3/(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) - 16/3*a^3*cos(a/b)*e^3*sin(a/b)*sin_integral(4*a/b + 4*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) - 2/3*a^3*cos(a/b)*e
^3*sin(a/b)*sin_integral(2*a/b + 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) - 8/3*(-(d*x + c)^2 + 1)^(3/2)*(d*x + c)*a^2*b*e^3/(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)*
(d*x + c)*b^3*e^3/(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) + 10/3*sqrt(-(d*x + c)^2 + 1)*(d*x + c)*a*b^2*arcsin(d*x + c)*e^3/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*a
rcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 2/3*((d*x + c)^2 - 1)^2*a*b^2*e^3/(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) + 5/6*((d*x + c)^2 - 1)*b^
3*arcsin(d*x + c)*e^3/(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) + 4*a^2*b*arcsin(d*x + c)*cos_integral(4*a/b + 4*arcsin(d*x + c))*e^3/(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) + a^2*b*arcsin(d*x + c)*cos_integral(2*a/b
+ 2*arcsin(d*x + c))*e^3/(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) + 5/3*sqrt(-(d*x + c)^2 + 1)*(d*x + c)*a^2*b*e^3/(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)*(d*x + c)*b^3*e^3/(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) + 5/6*((d*x + c)^2 - 1)*
a*b^2*e^3/(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*b^3*arcsin(d*x + c)*e^3/(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) + 4/3*a^3*cos_integral(4*a/b + 4*arcsin(d*x + c))*e^3/(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) + 1/3*a^3*cos_integral(2*a/b + 2*arcsin(d*x + c))*e^
3/(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*a*b^
2*e^3/(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)