3.3.36 \(\int \frac {c e+d e x}{(a+b \text {ArcSin}(c+d x))^4} \, dx\) [236]

Optimal. Leaf size=208 \[ -\frac {e (c+d x) \sqrt {1-(c+d x)^2}}{3 b d (a+b \text {ArcSin}(c+d x))^3}-\frac {e}{6 b^2 d (a+b \text {ArcSin}(c+d x))^2}+\frac {e (c+d x)^2}{3 b^2 d (a+b \text {ArcSin}(c+d x))^2}+\frac {2 e (c+d x) \sqrt {1-(c+d x)^2}}{3 b^3 d (a+b \text {ArcSin}(c+d x))}-\frac {2 e \cos \left (\frac {2 a}{b}\right ) \text {CosIntegral}\left (\frac {2 (a+b \text {ArcSin}(c+d x))}{b}\right )}{3 b^4 d}-\frac {2 e \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \text {ArcSin}(c+d x))}{b}\right )}{3 b^4 d} \]

[Out]

-1/6*e/b^2/d/(a+b*arcsin(d*x+c))^2+1/3*e*(d*x+c)^2/b^2/d/(a+b*arcsin(d*x+c))^2-2/3*e*Ci(2*(a+b*arcsin(d*x+c))/
b)*cos(2*a/b)/b^4/d-2/3*e*Si(2*(a+b*arcsin(d*x+c))/b)*sin(2*a/b)/b^4/d-1/3*e*(d*x+c)*(1-(d*x+c)^2)^(1/2)/b/d/(
a+b*arcsin(d*x+c))^3+2/3*e*(d*x+c)*(1-(d*x+c)^2)^(1/2)/b^3/d/(a+b*arcsin(d*x+c))

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Rubi [A]
time = 0.24, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4889, 12, 4729, 4807, 4727, 3384, 3380, 3383, 4737} \begin {gather*} -\frac {2 e \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 {2 e \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 {2 e \sqrt {1-(c+d x)^2} (c+d x)}{3 b^3 d (a+b \text {ArcSin}(c+d x))}+\frac {e (c+d x)^2}{3 b^2 d (a+b \text {ArcSin}(c+d x))^2}-\frac {e}{6 b^2 d (a+b \text {ArcSin}(c+d x))^2}-\frac {e \sqrt {1-(c+d x)^2} (c+d x)}{3 b d (a+b \text {ArcSin}(c+d x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(e*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(b*d*(a + b*ArcSin[c + d*x])^3) - e/(6*b^2*d*(a + b*ArcSin[c + d*x])^
2) + (e*(c + d*x)^2)/(3*b^2*d*(a + b*ArcSin[c + d*x])^2) + (2*e*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(3*b^3*d*(a +
b*ArcSin[c + d*x])) - (2*e*Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcSin[c + d*x]))/b])/(3*b^4*d) - (2*e*Sin[(2*a
)/b]*SinIntegral[(2*(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 4737

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

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}{\left (a+b \sin ^{-1}(c+d x)\right )^4} \, dx &=\frac {\text {Subst}\left (\int \frac {e x}{\left (a+b \sin ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=\frac {e \text {Subst}\left (\int \frac {x}{\left (a+b \sin ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e (c+d x) \sqrt {1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac {e \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \left (a+b \sin ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{3 b d}-\frac {(2 e) \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 )}{3 b d}\\ &=-\frac {e (c+d x) \sqrt {1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}-\frac {e}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {e (c+d x)^2}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {(2 e) \text {Subst}\left (\int \frac {x}{\left (a+b \sin ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{3 b^2 d}\\ &=-\frac {e (c+d x) \sqrt {1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}-\frac {e}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {e (c+d x)^2}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {2 e (c+d x) \sqrt {1-(c+d x)^2}}{3 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac {(2 e) \text {Subst}\left (\int \frac {\cos (2 x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{3 b^3 d}\\ &=-\frac {e (c+d x) \sqrt {1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}-\frac {e}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {e (c+d x)^2}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {2 e (c+d x) \sqrt {1-(c+d x)^2}}{3 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac {\left (2 e \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 (2 e \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 {e (c+d x) \sqrt {1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}-\frac {e}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {e (c+d x)^2}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {2 e (c+d x) \sqrt {1-(c+d x)^2}}{3 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac {2 e \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 {2 e \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}\\ \end {align*}

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Mathematica [A]
time = 0.52, size = 186, normalized size = 0.89 \begin {gather*} \frac {e \left (-\frac {2 b^3 (c+d x) \sqrt {1-(c+d x)^2}}{(a+b \text {ArcSin}(c+d x))^3}+\frac {b^2 \left (-1+2 (c+d x)^2\right )}{(a+b \text {ArcSin}(c+d x))^2}+\frac {4 b (c+d x) \sqrt {1-(c+d x)^2}}{a+b \text {ArcSin}(c+d x)}-4 \log (a+b \text {ArcSin}(c+d x))-4 \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 )\right )}{6 b^4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*((-2*b^3*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])^3 + (b^2*(-1 + 2*(c + d*x)^2))/(a + b*Arc
Sin[c + d*x])^2 + (4*b*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x]) - 4*Log[a + b*ArcSin[c + d*x]]
- 4*(Cos[(2*a)/b]*CosIntegral[2*(a/b + ArcSin[c + d*x])] - Log[a + b*ArcSin[c + d*x]] + Sin[(2*a)/b]*SinInteg
ral[2*(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. \(398\) vs. \(2(192)=384\).
time = 0.04, size = 399, normalized size = 1.92

method	result	size
derivativedivides	\(-\frac {e \left (4 \arcsin \left (d x +c \right )^{3} \sinIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) b^{3}+4 \arcsin \left (d x +c \right )^{3} \cosineIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b^{3}+12 \arcsin \left (d x +c \right )^{2} \sinIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a \,b^{2}+12 \arcsin \left (d x +c \right )^{2} \cosineIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a \,b^{2}-2 \arcsin \left (d x +c \right )^{2} \sin \left (2 \arcsin \left (d x +c \right )\right ) b^{3}+12 \arcsin \left (d x +c \right ) \sinIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a^{2} b +12 \arcsin \left (d x +c \right ) \cosineIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a^{2} b -4 \arcsin \left (d x +c \right ) \sin \left (2 \arcsin \left (d x +c \right )\right ) a \,b^{2}+\arcsin \left (d x +c \right ) \cos \left (2 \arcsin \left (d x +c \right )\right ) b^{3}+4 \sinIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a^{3}+4 \cosineIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a^{3}-2 \sin \left (2 \arcsin \left (d x +c \right )\right ) a^{2} b +\sin \left (2 \arcsin \left (d x +c \right )\right ) b^{3}+\cos \left (2 \arcsin \left (d x +c \right )\right ) a \,b^{2}\right )}{6 d \left (a +b \arcsin \left (d x +c \right )\right )^{3} b^{4}}\)	\(399\)
default	\(-\frac {e \left (4 \arcsin \left (d x +c \right )^{3} \sinIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) b^{3}+4 \arcsin \left (d x +c \right )^{3} \cosineIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b^{3}+12 \arcsin \left (d x +c \right )^{2} \sinIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a \,b^{2}+12 \arcsin \left (d x +c \right )^{2} \cosineIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a \,b^{2}-2 \arcsin \left (d x +c \right )^{2} \sin \left (2 \arcsin \left (d x +c \right )\right ) b^{3}+12 \arcsin \left (d x +c \right ) \sinIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a^{2} b +12 \arcsin \left (d x +c \right ) \cosineIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a^{2} b -4 \arcsin \left (d x +c \right ) \sin \left (2 \arcsin \left (d x +c \right )\right ) a \,b^{2}+\arcsin \left (d x +c \right ) \cos \left (2 \arcsin \left (d x +c \right )\right ) b^{3}+4 \sinIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a^{3}+4 \cosineIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a^{3}-2 \sin \left (2 \arcsin \left (d x +c \right )\right ) a^{2} b +\sin \left (2 \arcsin \left (d x +c \right )\right ) b^{3}+\cos \left (2 \arcsin \left (d x +c \right )\right ) a \,b^{2}\right )}{6 d \left (a +b \arcsin \left (d x +c \right )\right )^{3} b^{4}}\)	\(399\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6/d*e*(4*arcsin(d*x+c)^3*Si(2*arcsin(d*x+c)+2*a/b)*sin(2*a/b)*b^3+4*arcsin(d*x+c)^3*Ci(2*arcsin(d*x+c)+2*a/
b)*cos(2*a/b)*b^3+12*arcsin(d*x+c)^2*Si(2*arcsin(d*x+c)+2*a/b)*sin(2*a/b)*a*b^2+12*arcsin(d*x+c)^2*Ci(2*arcsin
(d*x+c)+2*a/b)*cos(2*a/b)*a*b^2-2*arcsin(d*x+c)^2*sin(2*arcsin(d*x+c))*b^3+12*arcsin(d*x+c)*Si(2*arcsin(d*x+c)
+2*a/b)*sin(2*a/b)*a^2*b+12*arcsin(d*x+c)*Ci(2*arcsin(d*x+c)+2*a/b)*cos(2*a/b)*a^2*b-4*arcsin(d*x+c)*sin(2*arc
sin(d*x+c))*a*b^2+arcsin(d*x+c)*cos(2*arcsin(d*x+c))*b^3+4*Si(2*arcsin(d*x+c)+2*a/b)*sin(2*a/b)*a^3+4*Ci(2*arc
sin(d*x+c)+2*a/b)*cos(2*a/b)*a^3-2*sin(2*arcsin(d*x+c))*a^2*b+sin(2*arcsin(d*x+c))*b^3+cos(2*arcsin(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)/(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)/(a+b*arcsin(d*x+c))^4,x, algorithm="fricas")

[Out]

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

[Out]

e*(Integral(c/(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*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. 1665 vs. \(2 (192) = 384\).
time = 0.74, size = 1665, normalized size = 8.00 \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)/(a+b*arcsin(d*x+c))^4,x, algorithm="giac")

[Out]

-4/3*b^3*e*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) - 4/3*b^3*e*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) - 4*a*b^2*e*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) - 4*a*b
^2*e*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) + 2/3*b^3*e*arcsin(d*x + c)^3*cos_integra
l(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) - 4*a^2*b*e*arcsin(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) - 4*a^2*b*e*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) + 2/3*sqrt(-(d*x + c)^2 + 1)*(d*x + c)*b^3*e*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*a*b^2*e
*arcsin(d*x + c)^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) - 4/3*a^3*e*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) - 4/3*a^3*e*
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) + 4/3*sqrt(-(d*x + c)^2 + 1)*(d*x + c)*a*b^2*e*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*((d*x +
c)^2 - 1)*b^3*e*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*arcsin(d*x + c)*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) + 2/3*sqrt(-(d*x + c)^2 + 1)*(d*x
+ c)*a^2*b*e/(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/(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)*a*b^2*e/(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*e*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*a^3*e*cos_integral(2*a/b + 2*arcs
in(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)
+ 1/6*a*b^2*e/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*
d)

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

[Out]

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

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