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

Optimal. Leaf size=145 \[ -\frac{e^3 \sin \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{4 b d}+\frac{e^3 \sin \left (\frac{4 a}{b}\right ) \text{CosIntegral}\left (\frac{4 \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{8 b d}+\frac{e^3 \cos \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{4 b d}-\frac{e^3 \cos \left (\frac{4 a}{b}\right ) \text{Si}\left (\frac{4 \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{8 b d} \]

[Out]

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

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Rubi [A]  time = 0.314128, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {4805, 12, 4635, 4406, 3303, 3299, 3302} \[ -\frac{e^3 \sin \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (\frac{2 a}{b}+2 \sin ^{-1}(c+d x)\right )}{4 b d}+\frac{e^3 \sin \left (\frac{4 a}{b}\right ) \text{CosIntegral}\left (\frac{4 a}{b}+4 \sin ^{-1}(c+d x)\right )}{8 b d}+\frac{e^3 \cos \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 a}{b}+2 \sin ^{-1}(c+d x)\right )}{4 b d}-\frac{e^3 \cos \left (\frac{4 a}{b}\right ) \text{Si}\left (\frac{4 a}{b}+4 \sin ^{-1}(c+d x)\right )}{8 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e^3*CosIntegral[(2*a)/b + 2*ArcSin[c + d*x]]*Sin[(2*a)/b])/(4*b*d) + (e^3*CosIntegral[(4*a)/b + 4*ArcSin[c +
d*x]]*Sin[(4*a)/b])/(8*b*d) + (e^3*Cos[(2*a)/b]*SinIntegral[(2*a)/b + 2*ArcSin[c + d*x]])/(4*b*d) - (e^3*Cos[
(4*a)/b]*SinIntegral[(4*a)/b + 4*ArcSin[c + d*x]])/(8*b*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 4635

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

Rule 4406

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

Mathematica [A]  time = 0.236173, size = 109, normalized size = 0.75 \[ \frac{e^3 \left (-2 \sin \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (2 \left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )\right )+\sin \left (\frac{4 a}{b}\right ) \text{CosIntegral}\left (4 \left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )\right )+2 \cos \left (\frac{2 a}{b}\right ) \text{Si}\left (2 \left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )\right )-\cos \left (\frac{4 a}{b}\right ) \text{Si}\left (4 \left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )\right )\right )}{8 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.034, size = 112, normalized size = 0.8 \begin{align*} -{\frac{{e}^{3}}{8\,bd} \left ({\it Si} \left ( 4\,\arcsin \left ( dx+c \right ) +4\,{\frac{a}{b}} \right ) \cos \left ( 4\,{\frac{a}{b}} \right ) -{\it Ci} \left ( 4\,\arcsin \left ( dx+c \right ) +4\,{\frac{a}{b}} \right ) \sin \left ( 4\,{\frac{a}{b}} \right ) -2\,{\it Si} \left ( 2\,\arcsin \left ( dx+c \right ) +2\,{\frac{a}{b}} \right ) \cos \left ( 2\,{\frac{a}{b}} \right ) +2\,{\it Ci} \left ( 2\,\arcsin \left ( dx+c \right ) +2\,{\frac{a}{b}} \right ) \sin \left ( 2\,{\frac{a}{b}} \right ) \right ) } \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)),x)

[Out]

-1/8/d*e^3*(Si(4*arcsin(d*x+c)+4*a/b)*cos(4*a/b)-Ci(4*arcsin(d*x+c)+4*a/b)*sin(4*a/b)-2*Si(2*arcsin(d*x+c)+2*a
/b)*cos(2*a/b)+2*Ci(2*arcsin(d*x+c)+2*a/b)*sin(2*a/b))/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d e x + c e\right )}^{3}}{b \arcsin \left (d x + c\right ) + a}\,{d x} \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)),x, algorithm="maxima")

[Out]

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

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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 \arcsin \left (d x + c\right ) + a}, 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)),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*arcsin(d*x + c) + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} e^{3} \left (\int \frac{c^{3}}{a + b \operatorname{asin}{\left (c + d x \right )}}\, dx + \int \frac{d^{3} x^{3}}{a + b \operatorname{asin}{\left (c + d x \right )}}\, dx + \int \frac{3 c d^{2} x^{2}}{a + b \operatorname{asin}{\left (c + d x \right )}}\, dx + \int \frac{3 c^{2} d x}{a + b \operatorname{asin}{\left (c + d x \right )}}\, dx\right ) \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)),x)

[Out]

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

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Giac [A]  time = 1.21758, size = 363, normalized size = 2.5 \begin{align*} \frac{\cos \left (\frac{a}{b}\right )^{3} \operatorname{Ci}\left (\frac{4 \, a}{b} + 4 \, \arcsin \left (d x + c\right )\right ) e^{3} \sin \left (\frac{a}{b}\right )}{b d} - \frac{\cos \left (\frac{a}{b}\right )^{4} e^{3} \operatorname{Si}\left (\frac{4 \, a}{b} + 4 \, \arcsin \left (d x + c\right )\right )}{b d} - \frac{\cos \left (\frac{a}{b}\right ) \operatorname{Ci}\left (\frac{4 \, a}{b} + 4 \, \arcsin \left (d x + c\right )\right ) e^{3} \sin \left (\frac{a}{b}\right )}{2 \, b d} - \frac{\cos \left (\frac{a}{b}\right ) \operatorname{Ci}\left (\frac{2 \, a}{b} + 2 \, \arcsin \left (d x + c\right )\right ) e^{3} \sin \left (\frac{a}{b}\right )}{2 \, b d} + \frac{\cos \left (\frac{a}{b}\right )^{2} e^{3} \operatorname{Si}\left (\frac{4 \, a}{b} + 4 \, \arcsin \left (d x + c\right )\right )}{b d} + \frac{\cos \left (\frac{a}{b}\right )^{2} e^{3} \operatorname{Si}\left (\frac{2 \, a}{b} + 2 \, \arcsin \left (d x + c\right )\right )}{2 \, b d} - \frac{e^{3} \operatorname{Si}\left (\frac{4 \, a}{b} + 4 \, \arcsin \left (d x + c\right )\right )}{8 \, b d} - \frac{e^{3} \operatorname{Si}\left (\frac{2 \, a}{b} + 2 \, \arcsin \left (d x + c\right )\right )}{4 \, b d} \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)),x, algorithm="giac")

[Out]

cos(a/b)^3*cos_integral(4*a/b + 4*arcsin(d*x + c))*e^3*sin(a/b)/(b*d) - cos(a/b)^4*e^3*sin_integral(4*a/b + 4*
arcsin(d*x + c))/(b*d) - 1/2*cos(a/b)*cos_integral(4*a/b + 4*arcsin(d*x + c))*e^3*sin(a/b)/(b*d) - 1/2*cos(a/b
)*cos_integral(2*a/b + 2*arcsin(d*x + c))*e^3*sin(a/b)/(b*d) + cos(a/b)^2*e^3*sin_integral(4*a/b + 4*arcsin(d*
x + c))/(b*d) + 1/2*cos(a/b)^2*e^3*sin_integral(2*a/b + 2*arcsin(d*x + c))/(b*d) - 1/8*e^3*sin_integral(4*a/b
+ 4*arcsin(d*x + c))/(b*d) - 1/4*e^3*sin_integral(2*a/b + 2*arcsin(d*x + c))/(b*d)