3.3.34 \(\int \frac {(c e+d e x)^3}{(a+b \text {ArcSin}(c+d x))^4} \, dx\) [234]

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

[Out]

-1/2*e^3*(d*x+c)^2/b^2/d/(a+b*arcsin(d*x+c))^2+2/3*e^3*(d*x+c)^4/b^2/d/(a+b*arcsin(d*x+c))^2-1/3*e^3*Ci(2*(a+b
*arcsin(d*x+c))/b)*cos(2*a/b)/b^4/d+4/3*e^3*Ci(4*(a+b*arcsin(d*x+c))/b)*cos(4*a/b)/b^4/d-1/3*e^3*Si(2*(a+b*arc
sin(d*x+c))/b)*sin(2*a/b)/b^4/d+4/3*e^3*Si(4*(a+b*arcsin(d*x+c))/b)*sin(4*a/b)/b^4/d-1/3*e^3*(d*x+c)^3*(1-(d*x
+c)^2)^(1/2)/b/d/(a+b*arcsin(d*x+c))^3-e^3*(d*x+c)*(1-(d*x+c)^2)^(1/2)/b^3/d/(a+b*arcsin(d*x+c))+8/3*e^3*(d*x+
c)^3*(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 = 346, normalized size of antiderivative = 1.00, 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 = {4889, 12, 4729, 4807, 4727, 3384, 3380, 3383} \begin {gather*} -\frac {e^3 \cos \left (\frac {2 a}{b}\right ) \text {CosIntegral}\left (\frac {2 (a+b \text {ArcSin}(c+d x))}{b}\right )}{3 b^4 d}+\frac {4 e^3 \cos \left (\frac {4 a}{b}\right ) \text {CosIntegral}\left (\frac {4 (a+b \text {ArcSin}(c+d x))}{b}\right )}{3 b^4 d}-\frac {e^3 \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \text {ArcSin}(c+d x))}{b}\right )}{3 b^4 d}+\frac {4 e^3 \sin \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \text {ArcSin}(c+d x))}{b}\right )}{3 b^4 d}+\frac {8 e^3 \sqrt {1-(c+d x)^2} (c+d x)^3}{3 b^3 d (a+b \text {ArcSin}(c+d x))}-\frac {e^3 \sqrt {1-(c+d x)^2} (c+d x)}{b^3 d (a+b \text {ArcSin}(c+d x))}+\frac {2 e^3 (c+d x)^4}{3 b^2 d (a+b \text {ArcSin}(c+d x))^2}-\frac {e^3 (c+d x)^2}{2 b^2 d (a+b \text {ArcSin}(c+d x))^2}-\frac {e^3 \sqrt {1-(c+d x)^2} (c+d x)^3}{3 b d (a+b \text {ArcSin}(c+d x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(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]*Cos
Integral[(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)

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 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 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)^3}{\left (a+b \sin ^{-1}(c+d x)\right )^4} \, dx &=\frac {\text {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 \text {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 \text {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 ) \text {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 \text {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 ) \text {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 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.70, size = 320, normalized size = 0.92 \begin {gather*} \frac {e^3 \left (-\frac {2 b^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{(a+b \text {ArcSin}(c+d x))^3}+\frac {b^2 \left (-3 (c+d x)^2+4 (c+d x)^4\right )}{(a+b \text {ArcSin}(c+d x))^2}+\frac {2 b \sqrt {1-(c+d x)^2} \left (-3 (c+d x)+8 (c+d x)^3\right )}{a+b \text {ArcSin}(c+d x)}+6 \log (a+b \text {ArcSin}(c+d x))+30 \left (\cos \left (\frac {2 a}{b}\right ) \text {CosIntegral}\left (2 \left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )\right )-\log (a+b \text {ArcSin}(c+d x))+\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )\right )\right )+8 \left (-4 \cos \left (\frac {2 a}{b}\right ) \text {CosIntegral}\left (2 \left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )\right )+\cos \left (\frac {4 a}{b}\right ) \text {CosIntegral}\left (4 \left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )\right )+3 \log (a+b \text {ArcSin}(c+d x))-4 \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )\right )+\sin \left (\frac {4 a}{b}\right ) \text {Si}\left (4 \left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )\right )\right )\right )}{6 b^4 d} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(782\) vs. \(2(324)=648\).
time = 0.11, size = 783, normalized size = 2.26

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(783\)
default	\(\text {Expression too large to display}\)	\(783\)

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,method=_RETURNVERBOSE)

[Out]

-1/24/d*e^3*(-4*arcsin(d*x+c)^2*sin(2*arcsin(d*x+c))*b^3-96*arcsin(d*x+c)*Si(4*arcsin(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)^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)^2*Si(4*arcsi
n(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
)*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+2*sin
(2*arcsin(d*x+c))*b^3-sin(4*arcsin(d*x+c))*b^3-32*Ci(4*arcsin(d*x+c)+4*a/b)*cos(4*a/b)*a^3-4*sin(2*arcsin(d*x+
c))*a^2*b+2*cos(2*arcsin(d*x+c))*a*b^2+8*sin(4*arcsin(d*x+c))*a^2*b-2*cos(4*arcsin(d*x+c))*a*b^2+8*arcsin(d*x+
c)^2*sin(4*arcsin(d*x+c))*b^3+2*arcsin(d*x+c)*cos(2*arcsin(d*x+c))*b^3-2*arcsin(d*x+c)*cos(4*arcsin(d*x+c))*b^
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^3-32*Si(4*arcsin(d*x+c)+
4*a/b)*sin(4*a/b)*a^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^3-32*arcsin(d*x+c)^3*Si(4*arcsin(d*x+c)+4*a/b)*sin(4*a/b)*b^3-32*arcsin(d*x+c)^3*C
i(4*arcsin(d*x+c)+4*a/b)*cos(4*a/b)*b^3-8*arcsin(d*x+c)*sin(2*arcsin(d*x+c))*a*b^2+16*arcsin(d*x+c)*sin(4*arcs
in(d*x+c))*a*b^2)/(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)^3/(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)^3/(a+b*arcsin(d*x+c))^4,x, algorithm="fricas")

[Out]

integral((d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{3} \left (\int \frac {c^{3}}{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^{3} x^{3}}{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 {3 c 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 {3 c^{2} 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)**3/(a+b*asin(d*x+c))**4,x)

[Out]

e**3*(Integral(c**3/(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**3*x**3/(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(3*c*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(3*c**2
*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. 4040 vs. \(2 (324) = 648\).
time = 0.85, size = 4040, normalized size = 11.68 \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)^3/(a+b*arcsin(d*x+c))^4,x, algorithm="giac")

[Out]

32/3*b^3*e^3*arcsin(d*x + c)^3*cos(a/b)^4*cos_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*e^3*arcsin(d*x + c)^3*cos(a/b
)^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*e^3*arcsin(d*x + c)^2*cos(a/b)^4*cos_integral(4*a/b + 4*ar
csin(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*e^3*arcsin(d*x + c)^2*cos(a/b)^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*e^3*arcsin(d*x
+ c)^3*cos(a/b)^2*cos_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*e^3*arcsin(d*x + c)*cos(a/b)^4*cos_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*e^3*arcsin(d*x + c)^3*cos(a/b)^2*cos_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*b^3*e^3*arcsin(d*x + c)^3*
cos(a/b)*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*e^3*arcsin(d*x + c)*cos(a/b)^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*e^3*arcsin(d*x + c)^3*cos(a/b)*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*e^3*ar
csin(d*x + c)^2*cos(a/b)^2*cos_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*e^3*cos(a/b)^4*cos_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*e^3*arcsin(d*x + c)^2*cos(a/b)^2*cos_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*a*b^2*e^3*arcsin(d*x + c)^2*cos(a/b
)*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*e^3*cos(a/b)^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*e^3*arcsin(d*x + c)^2*cos(a/b)*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*e^3*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) + 4/3*b^3*e^3*arcsin(d*x + c)^3*cos_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*e^3*arcsin(d*x + c)*cos(
a/b)^2*cos_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) + 1/3*b^3*e^3*arcsin(d*x + c)^3*cos_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) - 2*a^2*b*e^3*a
rcsin(d*x + c)*cos(a/b)^2*cos_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*a^2*b*e^3*arcsin(d*x + c)*cos(a/b)*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*e^3*arcsin(d*x + c)*cos(a/b)*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*e^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) + 5/3*sqrt(-(d*x + c)^2 + 1)*(d*x + c)*b^3*e^3*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) + 2/3*((d*x +
c)^2 - 1)^2*b^3*e^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*arcsi
n(d*x + c) + a^3*b^4*d) + 4*a*b^2*e^3*arcsin(d*x + c)^2*cos_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*e^3*cos(a/b)^2*
cos_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*a
rcsin(d*x + c) + a^3*b^4*d) + a*b^2*e^3*arcsin(...

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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 )}^3}{{\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)^3/(a + b*asin(c + d*x))^4,x)

[Out]

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

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