3.3.33 \(\int \frac {(c e+d e x)^4}{(a+b \text {ArcSin}(c+d x))^4} \, dx\) [233]

Optimal. Leaf size=416 \[ -\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{3 b d (a+b \text {ArcSin}(c+d x))^3}-\frac {2 e^4 (c+d x)^3}{3 b^2 d (a+b \text {ArcSin}(c+d x))^2}+\frac {5 e^4 (c+d x)^5}{6 b^2 d (a+b \text {ArcSin}(c+d x))^2}-\frac {2 e^4 (c+d x)^2 \sqrt {1-(c+d x)^2}}{b^3 d (a+b \text {ArcSin}(c+d x))}+\frac {25 e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{6 b^3 d (a+b \text {ArcSin}(c+d x))}-\frac {e^4 \text {CosIntegral}\left (\frac {a+b \text {ArcSin}(c+d x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{48 b^4 d}+\frac {27 e^4 \text {CosIntegral}\left (\frac {3 (a+b \text {ArcSin}(c+d x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{32 b^4 d}-\frac {125 e^4 \text {CosIntegral}\left (\frac {5 (a+b \text {ArcSin}(c+d x))}{b}\right ) \sin \left (\frac {5 a}{b}\right )}{96 b^4 d}+\frac {e^4 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \text {ArcSin}(c+d x)}{b}\right )}{48 b^4 d}-\frac {27 e^4 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \text {ArcSin}(c+d x))}{b}\right )}{32 b^4 d}+\frac {125 e^4 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \text {ArcSin}(c+d x))}{b}\right )}{96 b^4 d} \]

[Out]

-2/3*e^4*(d*x+c)^3/b^2/d/(a+b*arcsin(d*x+c))^2+5/6*e^4*(d*x+c)^5/b^2/d/(a+b*arcsin(d*x+c))^2+1/48*e^4*cos(a/b)
*Si((a+b*arcsin(d*x+c))/b)/b^4/d-27/32*e^4*cos(3*a/b)*Si(3*(a+b*arcsin(d*x+c))/b)/b^4/d+125/96*e^4*cos(5*a/b)*
Si(5*(a+b*arcsin(d*x+c))/b)/b^4/d-1/48*e^4*Ci((a+b*arcsin(d*x+c))/b)*sin(a/b)/b^4/d+27/32*e^4*Ci(3*(a+b*arcsin
(d*x+c))/b)*sin(3*a/b)/b^4/d-125/96*e^4*Ci(5*(a+b*arcsin(d*x+c))/b)*sin(5*a/b)/b^4/d-1/3*e^4*(d*x+c)^4*(1-(d*x
+c)^2)^(1/2)/b/d/(a+b*arcsin(d*x+c))^3-2*e^4*(d*x+c)^2*(1-(d*x+c)^2)^(1/2)/b^3/d/(a+b*arcsin(d*x+c))+25/6*e^4*
(d*x+c)^4*(1-(d*x+c)^2)^(1/2)/b^3/d/(a+b*arcsin(d*x+c))

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Rubi [A]
time = 0.56, antiderivative size = 416, normalized size of antiderivative = 1.00, number of steps used = 24, 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^4 \sin \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a+b \text {ArcSin}(c+d x)}{b}\right )}{48 b^4 d}+\frac {27 e^4 \sin \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 (a+b \text {ArcSin}(c+d x))}{b}\right )}{32 b^4 d}-\frac {125 e^4 \sin \left (\frac {5 a}{b}\right ) \text {CosIntegral}\left (\frac {5 (a+b \text {ArcSin}(c+d x))}{b}\right )}{96 b^4 d}+\frac {e^4 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \text {ArcSin}(c+d x)}{b}\right )}{48 b^4 d}-\frac {27 e^4 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \text {ArcSin}(c+d x))}{b}\right )}{32 b^4 d}+\frac {125 e^4 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \text {ArcSin}(c+d x))}{b}\right )}{96 b^4 d}+\frac {25 e^4 \sqrt {1-(c+d x)^2} (c+d x)^4}{6 b^3 d (a+b \text {ArcSin}(c+d x))}-\frac {2 e^4 \sqrt {1-(c+d x)^2} (c+d x)^2}{b^3 d (a+b \text {ArcSin}(c+d x))}+\frac {5 e^4 (c+d x)^5}{6 b^2 d (a+b \text {ArcSin}(c+d x))^2}-\frac {2 e^4 (c+d x)^3}{3 b^2 d (a+b \text {ArcSin}(c+d x))^2}-\frac {e^4 \sqrt {1-(c+d x)^2} (c+d x)^4}{3 b d (a+b \text {ArcSin}(c+d x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(e^4*(c + d*x)^4*Sqrt[1 - (c + d*x)^2])/(b*d*(a + b*ArcSin[c + d*x])^3) - (2*e^4*(c + d*x)^3)/(3*b^2*d*(a
+ b*ArcSin[c + d*x])^2) + (5*e^4*(c + d*x)^5)/(6*b^2*d*(a + b*ArcSin[c + d*x])^2) - (2*e^4*(c + d*x)^2*Sqrt[1
- (c + d*x)^2])/(b^3*d*(a + b*ArcSin[c + d*x])) + (25*e^4*(c + d*x)^4*Sqrt[1 - (c + d*x)^2])/(6*b^3*d*(a + b*
ArcSin[c + d*x])) - (e^4*CosIntegral[(a + b*ArcSin[c + d*x])/b]*Sin[a/b])/(48*b^4*d) + (27*e^4*CosIntegral[(3*
(a + b*ArcSin[c + d*x]))/b]*Sin[(3*a)/b])/(32*b^4*d) - (125*e^4*CosIntegral[(5*(a + b*ArcSin[c + d*x]))/b]*Sin
[(5*a)/b])/(96*b^4*d) + (e^4*Cos[a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/b])/(48*b^4*d) - (27*e^4*Cos[(3*a)/b
]*SinIntegral[(3*(a + b*ArcSin[c + d*x]))/b])/(32*b^4*d) + (125*e^4*Cos[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[
c + d*x]))/b])/(96*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)^4}{\left (a+b \sin ^{-1}(c+d x)\right )^4} \, dx &=\frac {\text {Subst}\left (\int \frac {e^4 x^4}{\left (a+b \sin ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 \text {Subst}\left (\int \frac {x^4}{\left (a+b \sin ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac {\left (4 e^4\right ) \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 )}{3 b d}-\frac {\left (5 e^4\right ) \text {Subst}\left (\int \frac {x^5}{\sqrt {1-x^2} \left (a+b \sin ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{3 b d}\\ &=-\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}-\frac {2 e^4 (c+d x)^3}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {5 e^4 (c+d x)^5}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {\left (2 e^4\right ) \text {Subst}\left (\int \frac {x^2}{\left (a+b \sin ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{b^2 d}-\frac {\left (25 e^4\right ) \text {Subst}\left (\int \frac {x^4}{\left (a+b \sin ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{6 b^2 d}\\ &=-\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}-\frac {2 e^4 (c+d x)^3}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {5 e^4 (c+d x)^5}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {2 e^4 (c+d x)^2 \sqrt {1-(c+d x)^2}}{b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {25 e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{6 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {\left (2 e^4\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 )}{b^3 d}-\frac {\left (25 e^4\right ) \text {Subst}\left (\int \left (-\frac {\sin (x)}{8 (a+b x)}+\frac {9 \sin (3 x)}{16 (a+b x)}-\frac {5 \sin (5 x)}{16 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{6 b^3 d}\\ &=-\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}-\frac {2 e^4 (c+d x)^3}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {5 e^4 (c+d x)^5}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {2 e^4 (c+d x)^2 \sqrt {1-(c+d x)^2}}{b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {25 e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{6 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac {e^4 \text {Subst}\left (\int \frac {\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{2 b^3 d}+\frac {\left (25 e^4\right ) \text {Subst}\left (\int \frac {\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{48 b^3 d}+\frac {\left (125 e^4\right ) \text {Subst}\left (\int \frac {\sin (5 x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{96 b^3 d}+\frac {\left (3 e^4\right ) \text {Subst}\left (\int \frac {\sin (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{2 b^3 d}-\frac {\left (75 e^4\right ) \text {Subst}\left (\int \frac {\sin (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{32 b^3 d}\\ &=-\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}-\frac {2 e^4 (c+d x)^3}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {5 e^4 (c+d x)^5}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {2 e^4 (c+d x)^2 \sqrt {1-(c+d x)^2}}{b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {25 e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{6 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac {\left (e^4 \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 )}{2 b^3 d}+\frac {\left (25 e^4 \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 )}{48 b^3 d}+\frac {\left (3 e^4 \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 )}{2 b^3 d}-\frac {\left (75 e^4 \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 )}{32 b^3 d}+\frac {\left (125 e^4 \cos \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{96 b^3 d}+\frac {\left (e^4 \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 )}{2 b^3 d}-\frac {\left (25 e^4 \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 )}{48 b^3 d}-\frac {\left (3 e^4 \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 )}{2 b^3 d}+\frac {\left (75 e^4 \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 )}{32 b^3 d}-\frac {\left (125 e^4 \sin \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{96 b^3 d}\\ &=-\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}-\frac {2 e^4 (c+d x)^3}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {5 e^4 (c+d x)^5}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {2 e^4 (c+d x)^2 \sqrt {1-(c+d x)^2}}{b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {25 e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{6 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac {e^4 \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right ) \sin \left (\frac {a}{b}\right )}{48 b^4 d}+\frac {27 e^4 \text {Ci}\left (\frac {3 a}{b}+3 \sin ^{-1}(c+d x)\right ) \sin \left (\frac {3 a}{b}\right )}{32 b^4 d}-\frac {125 e^4 \text {Ci}\left (\frac {5 a}{b}+5 \sin ^{-1}(c+d x)\right ) \sin \left (\frac {5 a}{b}\right )}{96 b^4 d}+\frac {e^4 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )}{48 b^4 d}-\frac {27 e^4 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c+d x)\right )}{32 b^4 d}+\frac {125 e^4 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 a}{b}+5 \sin ^{-1}(c+d x)\right )}{96 b^4 d}\\ \end {align*}

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Mathematica [A]
time = 1.10, size = 414, normalized size = 1.00 \begin {gather*} \frac {e^4 \left (-\frac {32 b^3 (c+d x)^4 \sqrt {1-(c+d x)^2}}{(a+b \text {ArcSin}(c+d x))^3}+\frac {16 b^2 \left (-4 (c+d x)^3+5 (c+d x)^5\right )}{(a+b \text {ArcSin}(c+d x))^2}+\frac {16 b \sqrt {1-(c+d x)^2} \left (-12 (c+d x)^2+25 (c+d x)^4\right )}{a+b \text {ArcSin}(c+d x)}+384 \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 )+544 \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 )-125 \left (10 \text {CosIntegral}\left (\frac {a}{b}+\text {ArcSin}(c+d x)\right ) \sin \left (\frac {a}{b}\right )-5 \text {CosIntegral}\left (3 \left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )\right ) \sin \left (\frac {3 a}{b}\right )+\text {CosIntegral}\left (5 \left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )\right ) \sin \left (\frac {5 a}{b}\right )-10 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )+5 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )\right )-\cos \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )\right )\right )\right )}{96 b^4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e^4*((-32*b^3*(c + d*x)^4*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])^3 + (16*b^2*(-4*(c + d*x)^3 + 5*(c +
d*x)^5))/(a + b*ArcSin[c + d*x])^2 + (16*b*Sqrt[1 - (c + d*x)^2]*(-12*(c + d*x)^2 + 25*(c + d*x)^4))/(a + b*A
rcSin[c + d*x]) + 384*(-(CosIntegral[a/b + ArcSin[c + d*x]]*Sin[a/b]) + Cos[a/b]*SinIntegral[a/b + ArcSin[c +
d*x]]) + 544*(3*CosIntegral[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]*SinIntegral[a/b + ArcSin[c + d*x]] + Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c + d*x])]) - 1
25*(10*CosIntegral[a/b + ArcSin[c + d*x]]*Sin[a/b] - 5*CosIntegral[3*(a/b + ArcSin[c + d*x])]*Sin[(3*a)/b] + C
osIntegral[5*(a/b + ArcSin[c + d*x])]*Sin[(5*a)/b] - 10*Cos[a/b]*SinIntegral[a/b + ArcSin[c + d*x]] + 5*Cos[(3
*a)/b]*SinIntegral[3*(a/b + ArcSin[c + d*x])] - Cos[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c + d*x])])))/(96*b^4
*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1137\) vs. \(2(390)=780\).
time = 0.36, size = 1138, normalized size = 2.74

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1138\)
default	\(\text {Expression too large to display}\)	\(1138\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96/d*e^4*(-50*arcsin(d*x+c)*cos(5*arcsin(d*x+c))*a*b^2-2*arcsin(d*x+c)^3*Si(arcsin(d*x+c)+a/b)*cos(a/b)*b^3
+2*arcsin(d*x+c)^3*Ci(arcsin(d*x+c)+a/b)*sin(a/b)*b^3-4*(1-(d*x+c)^2)^(1/2)*arcsin(d*x+c)*a*b^2+81*arcsin(d*x+
c)^3*Si(3*arcsin(d*x+c)+3*a/b)*cos(3*a/b)*b^3-81*arcsin(d*x+c)^3*Ci(3*arcsin(d*x+c)+3*a/b)*sin(3*a/b)*b^3+54*c
os(3*arcsin(d*x+c))*arcsin(d*x+c)*a*b^2-125*arcsin(d*x+c)^3*Si(5*arcsin(d*x+c)+5*a/b)*cos(5*a/b)*b^3+125*arcsi
n(d*x+c)^3*Ci(5*arcsin(d*x+c)+5*a/b)*sin(5*a/b)*b^3+6*arcsin(d*x+c)*Ci(arcsin(d*x+c)+a/b)*sin(a/b)*a^2*b+243*a
rcsin(d*x+c)^2*Si(3*arcsin(d*x+c)+3*a/b)*cos(3*a/b)*a*b^2-243*arcsin(d*x+c)^2*Ci(3*arcsin(d*x+c)+3*a/b)*sin(3*
a/b)*a*b^2+243*arcsin(d*x+c)*Si(3*arcsin(d*x+c)+3*a/b)*cos(3*a/b)*a^2*b-243*arcsin(d*x+c)*Ci(3*arcsin(d*x+c)+3
*a/b)*sin(3*a/b)*a^2*b-375*arcsin(d*x+c)^2*Si(5*arcsin(d*x+c)+5*a/b)*cos(5*a/b)*a*b^2+375*arcsin(d*x+c)^2*Ci(5
*arcsin(d*x+c)+5*a/b)*sin(5*a/b)*a*b^2-375*arcsin(d*x+c)*Si(5*arcsin(d*x+c)+5*a/b)*cos(5*a/b)*a^2*b+375*arcsin
(d*x+c)*Ci(5*arcsin(d*x+c)+5*a/b)*sin(5*a/b)*a^2*b-6*arcsin(d*x+c)^2*Si(arcsin(d*x+c)+a/b)*cos(a/b)*a*b^2+6*ar
csin(d*x+c)^2*Ci(arcsin(d*x+c)+a/b)*sin(a/b)*a*b^2-6*arcsin(d*x+c)*Si(arcsin(d*x+c)+a/b)*cos(a/b)*a^2*b-2*a*b^
2*(d*x+c)+4*(1-(d*x+c)^2)^(1/2)*b^3-6*cos(3*arcsin(d*x+c))*b^3+2*cos(5*arcsin(d*x+c))*b^3-2*(1-(d*x+c)^2)^(1/2
)*a^2*b+9*sin(3*arcsin(d*x+c))*arcsin(d*x+c)*b^3+81*Si(3*arcsin(d*x+c)+3*a/b)*cos(3*a/b)*a^3-81*Ci(3*arcsin(d*
x+c)+3*a/b)*sin(3*a/b)*a^3+27*cos(3*arcsin(d*x+c))*a^2*b+9*sin(3*arcsin(d*x+c))*a*b^2-25*arcsin(d*x+c)^2*cos(5
*arcsin(d*x+c))*b^3-5*arcsin(d*x+c)*sin(5*arcsin(d*x+c))*b^3-125*Si(5*arcsin(d*x+c)+5*a/b)*cos(5*a/b)*a^3+125*
Ci(5*arcsin(d*x+c)+5*a/b)*sin(5*a/b)*a^3-25*cos(5*arcsin(d*x+c))*a^2*b-5*sin(5*arcsin(d*x+c))*a*b^2-2*arcsin(d
*x+c)*b^3*(d*x+c)-2*Si(arcsin(d*x+c)+a/b)*cos(a/b)*a^3+2*Ci(arcsin(d*x+c)+a/b)*sin(a/b)*a^3-2*(1-(d*x+c)^2)^(1
/2)*arcsin(d*x+c)^2*b^3+27*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)^4/(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)^4/(a+b*arcsin(d*x+c))^4,x, algorithm="fricas")

[Out]

integral((d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 + 4*c^3*d*x + c^4)*e^4/(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^{4} \left (\int \frac {c^{4}}{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^{4} x^{4}}{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 {4 c 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 {6 c^{2} 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 {4 c^{3} 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)**4/(a+b*asin(d*x+c))**4,x)

[Out]

e**4*(Integral(c**4/(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**4*x**4/(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(4*c*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(6*c**2
*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*as
in(c + d*x)**4), x) + Integral(4*c**3*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. 5870 vs. \(2 (390) = 780\).
time = 0.82, size = 5870, normalized size = 14.11 \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)^4/(a+b*arcsin(d*x+c))^4,x, algorithm="giac")

[Out]

-125/6*b^3*e^4*arcsin(d*x + c)^3*cos(a/b)^4*cos_integral(5*a/b + 5*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) + 125/6*b^3*e^4*arcsin(d*x +
c)^3*cos(a/b)^5*sin_integral(5*a/b + 5*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) - 125/2*a*b^2*e^4*arcsin(d*x + c)^2*cos(a/b)^4*cos_integral(5*a/b
+ 5*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) + 125/2*a*b^2*e^4*arcsin(d*x + c)^2*cos(a/b)^5*sin_integral(5*a/b + 5*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) + 125/8*b^3*e^4*a
rcsin(d*x + c)^3*cos(a/b)^2*cos_integral(5*a/b + 5*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) - 125/2*a^2*b*e^4*arcsin(d*x + c)*cos(a/b)^4*
cos_integral(5*a/b + 5*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^4*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) - 625/24*b^3*e^4*arcsin(d*x + c)^3*cos(a/b)^3*sin_integral(5*a/b + 5*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) + 125/2*a^2*b*e^4*arcsin(d*x
+ c)*cos(a/b)^5*sin_integral(5*a/b + 5*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*b^3*e^4*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) + 375/8*a*b^2*e^4*arcsin(d*x + c)^2*cos(a/b)^2*cos_integral(5*a/b + 5*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) - 125/6*a^3*e^4*cos
(a/b)^4*cos_integral(5*a/b + 5*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^4*arcsin(d*x + c)^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) - 625/8*a*b^2*e^4*arcsin(d*x + c)^2*cos(a/b)^3*sin_integral(5*a/b + 5*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) + 125/6*a^3*e^4*c
os(a/b)^5*sin_integral(5*a/b + 5*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) - 81/8*a*b^2*e^4*arcsin(d*x + c)^2*cos(a/b)^3*sin_integral(3*a/b + 3*arc
sin(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
) - 125/96*b^3*e^4*arcsin(d*x + c)^3*cos_integral(5*a/b + 5*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) + 375/8*a^2*b*e^4*arcsin(d*x + c)*co
s(a/b)^2*cos_integral(5*a/b + 5*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) - 27/32*b^3*e^4*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) + 81/8*a^2*b*e^4*arcsin(d*x + c)*cos(a/b)^2*cos_integral(3*a/b + 3*arcsin(d*x + c))*sin(a/b)/(b^7*d*arc
sin(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) - 1/48*b^3*e^4*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) + 625/96*b^3*e^4*arcsin(d*x + c)^3*cos(a/b)*sin_integral(5*a/b + 5
*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) - 625/8*a^2*b*e^4*arcsin(d*x + c)*cos(a/b)^3*sin_integral(5*a/b + 5*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) + 81/32*b^3*e^4*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) - 81/8*a^2*b*e^4*arcsin(d*x + c)*cos(a/b)^3*sin_integral(3*a/b + 3*arc
sin(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
) + 1/48*b^3*e^4*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) + 25/6*((d*x + c)^2 - 1)^2*sqrt(-(d*x + c)
^2 + 1)*b^3*e^4*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) + 5/6*((d*x + c)^2 - 1)^2...

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

[Out]

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

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