3.3.15 \(\int \frac {(c e+d e x)^4}{a+b \text {ArcSin}(c+d x)} \, dx\) [215]

Optimal. Leaf size=213 \[ \frac {e^4 \cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a+b \text {ArcSin}(c+d x)}{b}\right )}{8 b d}-\frac {3 e^4 \cos \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 (a+b \text {ArcSin}(c+d x))}{b}\right )}{16 b d}+\frac {e^4 \cos \left (\frac {5 a}{b}\right ) \text {CosIntegral}\left (\frac {5 (a+b \text {ArcSin}(c+d x))}{b}\right )}{16 b d}+\frac {e^4 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \text {ArcSin}(c+d x)}{b}\right )}{8 b d}-\frac {3 e^4 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \text {ArcSin}(c+d x))}{b}\right )}{16 b d}+\frac {e^4 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \text {ArcSin}(c+d x))}{b}\right )}{16 b d} \]

[Out]

1/8*e^4*Ci((a+b*arcsin(d*x+c))/b)*cos(a/b)/b/d-3/16*e^4*Ci(3*(a+b*arcsin(d*x+c))/b)*cos(3*a/b)/b/d+1/16*e^4*Ci
(5*(a+b*arcsin(d*x+c))/b)*cos(5*a/b)/b/d+1/8*e^4*Si((a+b*arcsin(d*x+c))/b)*sin(a/b)/b/d-3/16*e^4*Si(3*(a+b*arc
sin(d*x+c))/b)*sin(3*a/b)/b/d+1/16*e^4*Si(5*(a+b*arcsin(d*x+c))/b)*sin(5*a/b)/b/d

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Rubi [A]
time = 0.29, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {4889, 12, 4731, 4491, 3384, 3380, 3383} \begin {gather*} \frac {e^4 \cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a+b \text {ArcSin}(c+d x)}{b}\right )}{8 b d}-\frac {3 e^4 \cos \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 (a+b \text {ArcSin}(c+d x))}{b}\right )}{16 b d}+\frac {e^4 \cos \left (\frac {5 a}{b}\right ) \text {CosIntegral}\left (\frac {5 (a+b \text {ArcSin}(c+d x))}{b}\right )}{16 b d}+\frac {e^4 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \text {ArcSin}(c+d x)}{b}\right )}{8 b d}-\frac {3 e^4 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \text {ArcSin}(c+d x))}{b}\right )}{16 b d}+\frac {e^4 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \text {ArcSin}(c+d x))}{b}\right )}{16 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e^4*Cos[a/b]*CosIntegral[(a + b*ArcSin[c + d*x])/b])/(8*b*d) - (3*e^4*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcS
in[c + d*x]))/b])/(16*b*d) + (e^4*Cos[(5*a)/b]*CosIntegral[(5*(a + b*ArcSin[c + d*x]))/b])/(16*b*d) + (e^4*Sin
[a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/b])/(8*b*d) - (3*e^4*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c + d
*x]))/b])/(16*b*d) + (e^4*Sin[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c + d*x]))/b])/(16*b*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 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 4731

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/(b*c^(m + 1)), Subst[Int[x^n*Sin[-
a/b + x/b]^m*Cos[-a/b + x/b], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 4889

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]

Rubi steps

\begin {align*} \int \frac {(c e+d e x)^4}{a+b \sin ^{-1}(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {e^4 x^4}{a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 \text {Subst}\left (\int \frac {x^4}{a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 \text {Subst}\left (\int \frac {\cos (x) \sin ^4(x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{d}\\ &=\frac {e^4 \text {Subst}\left (\int \left (\frac {\cos (x)}{8 (a+b x)}-\frac {3 \cos (3 x)}{16 (a+b x)}+\frac {\cos (5 x)}{16 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{d}\\ &=\frac {e^4 \text {Subst}\left (\int \frac {\cos (5 x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{16 d}+\frac {e^4 \text {Subst}\left (\int \frac {\cos (x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 d}-\frac {\left (3 e^4\right ) \text {Subst}\left (\int \frac {\cos (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{16 d}\\ &=\frac {\left (e^4 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 d}-\frac {\left (3 e^4 \cos \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 )}{16 d}+\frac {\left (e^4 \cos \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 )}{16 d}+\frac {\left (e^4 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 d}-\frac {\left (3 e^4 \sin \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 )}{16 d}+\frac {\left (e^4 \sin \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 )}{16 d}\\ &=\frac {e^4 \cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )}{8 b d}-\frac {3 e^4 \cos \left (\frac {3 a}{b}\right ) \text {Ci}\left (\frac {3 a}{b}+3 \sin ^{-1}(c+d x)\right )}{16 b d}+\frac {e^4 \cos \left (\frac {5 a}{b}\right ) \text {Ci}\left (\frac {5 a}{b}+5 \sin ^{-1}(c+d x)\right )}{16 b d}+\frac {e^4 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )}{8 b d}-\frac {3 e^4 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c+d x)\right )}{16 b d}+\frac {e^4 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 a}{b}+5 \sin ^{-1}(c+d x)\right )}{16 b d}\\ \end {align*}

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Mathematica [A]
time = 0.24, size = 150, normalized size = 0.70 \begin {gather*} \frac {e^4 \left (2 \cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )-3 \cos \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (3 \left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )\right )+\cos \left (\frac {5 a}{b}\right ) \text {CosIntegral}\left (5 \left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )\right )+2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )-3 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )\right )+\sin \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )\right )\right )}{16 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e^4*(2*Cos[a/b]*CosIntegral[a/b + ArcSin[c + d*x]] - 3*Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c + d*x])] +
Cos[(5*a)/b]*CosIntegral[5*(a/b + ArcSin[c + d*x])] + 2*Sin[a/b]*SinIntegral[a/b + ArcSin[c + d*x]] - 3*Sin[(3
*a)/b]*SinIntegral[3*(a/b + ArcSin[c + d*x])] + Sin[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c + d*x])]))/(16*b*d)

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Maple [A]
time = 0.12, size = 155, normalized size = 0.73

method	result	size
derivativedivides	\(-\frac {e^{4} \left (3 \sinIntegral \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right )+3 \cosineIntegral \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right )-\sinIntegral \left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right )-\cosineIntegral \left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right )-2 \sinIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )-2 \cosineIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )\right )}{16 d b}\)	\(155\)
default	\(-\frac {e^{4} \left (3 \sinIntegral \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right )+3 \cosineIntegral \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right )-\sinIntegral \left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right )-\cosineIntegral \left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right )-2 \sinIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )-2 \cosineIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )\right )}{16 d b}\)	\(155\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16/d*e^4*(3*Si(3*arcsin(d*x+c)+3*a/b)*sin(3*a/b)+3*Ci(3*arcsin(d*x+c)+3*a/b)*cos(3*a/b)-Si(5*arcsin(d*x+c)+
5*a/b)*sin(5*a/b)-Ci(5*arcsin(d*x+c)+5*a/b)*cos(5*a/b)-2*Si(arcsin(d*x+c)+a/b)*sin(a/b)-2*Ci(arcsin(d*x+c)+a/b
)*cos(a/b))/b

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to 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)),x, algorithm="maxima")

[Out]

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

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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)),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*arcsin(d*x + c) + a), 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 + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int \frac {d^{4} x^{4}}{a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int \frac {4 c d^{3} x^{3}}{a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int \frac {6 c^{2} d^{2} x^{2}}{a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int \frac {4 c^{3} d x}{a + b \operatorname {asin}{\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)),x)

[Out]

e**4*(Integral(c**4/(a + b*asin(c + d*x)), x) + Integral(d**4*x**4/(a + b*asin(c + d*x)), x) + Integral(4*c*d*
*3*x**3/(a + b*asin(c + d*x)), x) + Integral(6*c**2*d**2*x**2/(a + b*asin(c + d*x)), x) + Integral(4*c**3*d*x/
(a + b*asin(c + d*x)), x))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 419 vs. \(2 (201) = 402\).
time = 0.44, size = 419, normalized size = 1.97 \begin {gather*} \frac {e^{4} \cos \left (\frac {a}{b}\right )^{5} \operatorname {Ci}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (d x + c\right )\right )}{b d} + \frac {e^{4} \cos \left (\frac {a}{b}\right )^{4} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (d x + c\right )\right )}{b d} - \frac {5 \, e^{4} \cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (d x + c\right )\right )}{4 \, b d} - \frac {3 \, e^{4} \cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (d x + c\right )\right )}{4 \, b d} - \frac {3 \, e^{4} \cos \left (\frac {a}{b}\right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (d x + c\right )\right )}{4 \, b d} - \frac {3 \, e^{4} \cos \left (\frac {a}{b}\right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (d x + c\right )\right )}{4 \, b d} + \frac {5 \, e^{4} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (d x + c\right )\right )}{16 \, b d} + \frac {9 \, e^{4} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (d x + c\right )\right )}{16 \, b d} + \frac {e^{4} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (d x + c\right )\right )}{8 \, b d} + \frac {e^{4} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (d x + c\right )\right )}{16 \, b d} + \frac {3 \, e^{4} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (d x + c\right )\right )}{16 \, b d} + \frac {e^{4} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (d x + c\right )\right )}{8 \, b d} \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)),x, algorithm="giac")

[Out]

e^4*cos(a/b)^5*cos_integral(5*a/b + 5*arcsin(d*x + c))/(b*d) + e^4*cos(a/b)^4*sin(a/b)*sin_integral(5*a/b + 5*
arcsin(d*x + c))/(b*d) - 5/4*e^4*cos(a/b)^3*cos_integral(5*a/b + 5*arcsin(d*x + c))/(b*d) - 3/4*e^4*cos(a/b)^3
*cos_integral(3*a/b + 3*arcsin(d*x + c))/(b*d) - 3/4*e^4*cos(a/b)^2*sin(a/b)*sin_integral(5*a/b + 5*arcsin(d*x
+ c))/(b*d) - 3/4*e^4*cos(a/b)^2*sin(a/b)*sin_integral(3*a/b + 3*arcsin(d*x + c))/(b*d) + 5/16*e^4*cos(a/b)*c
os_integral(5*a/b + 5*arcsin(d*x + c))/(b*d) + 9/16*e^4*cos(a/b)*cos_integral(3*a/b + 3*arcsin(d*x + c))/(b*d)
+ 1/8*e^4*cos(a/b)*cos_integral(a/b + arcsin(d*x + c))/(b*d) + 1/16*e^4*sin(a/b)*sin_integral(5*a/b + 5*arcsi
n(d*x + c))/(b*d) + 3/16*e^4*sin(a/b)*sin_integral(3*a/b + 3*arcsin(d*x + c))/(b*d) + 1/8*e^4*sin(a/b)*sin_int
egral(a/b + arcsin(d*x + c))/(b*d)

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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}{a+b\,\mathrm {asin}\left (c+d\,x\right )} \,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)),x)

[Out]

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

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