3.349 \(\int \frac{x^5}{\sqrt{1-c^2 x^2} (a+b \sin ^{-1}(c x))} \, dx\)

Optimal. Leaf size=183 \[ -\frac{5 \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{8 b c^6}+\frac{5 \sin \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b c^6}-\frac{\sin \left (\frac{5 a}{b}\right ) \text{CosIntegral}\left (\frac{5 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b c^6}+\frac{5 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{8 b c^6}-\frac{5 \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b c^6}+\frac{\cos \left (\frac{5 a}{b}\right ) \text{Si}\left (\frac{5 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b c^6} \]

[Out]

(-5*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(8*b*c^6) + (5*CosIntegral[(3*(a + b*ArcSin[c*x]))/b]*Sin[(3*
a)/b])/(16*b*c^6) - (CosIntegral[(5*(a + b*ArcSin[c*x]))/b]*Sin[(5*a)/b])/(16*b*c^6) + (5*Cos[a/b]*SinIntegral
[(a + b*ArcSin[c*x])/b])/(8*b*c^6) - (5*Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(16*b*c^6) + (Cos
[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c*x]))/b])/(16*b*c^6)

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Rubi [A]  time = 0.37279, antiderivative size = 179, normalized size of antiderivative = 0.98, number of steps used = 12, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {4723, 3312, 3303, 3299, 3302} \[ -\frac{5 \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{8 b c^6}+\frac{5 \sin \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{16 b c^6}-\frac{\sin \left (\frac{5 a}{b}\right ) \text{CosIntegral}\left (\frac{5 a}{b}+5 \sin ^{-1}(c x)\right )}{16 b c^6}+\frac{5 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{8 b c^6}-\frac{5 \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{16 b c^6}+\frac{\cos \left (\frac{5 a}{b}\right ) \text{Si}\left (\frac{5 a}{b}+5 \sin ^{-1}(c x)\right )}{16 b c^6} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])),x]

[Out]

(-5*CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b])/(8*b*c^6) + (5*CosIntegral[(3*a)/b + 3*ArcSin[c*x]]*Sin[(3*a)/b])
/(16*b*c^6) - (CosIntegral[(5*a)/b + 5*ArcSin[c*x]]*Sin[(5*a)/b])/(16*b*c^6) + (5*Cos[a/b]*SinIntegral[a/b + A
rcSin[c*x]])/(8*b*c^6) - (5*Cos[(3*a)/b]*SinIntegral[(3*a)/b + 3*ArcSin[c*x]])/(16*b*c^6) + (Cos[(5*a)/b]*SinI
ntegral[(5*a)/b + 5*ArcSin[c*x]])/(16*b*c^6)

Rule 4723

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^p/c^(
m + 1), Subst[Int[(a + b*x)^n*Sin[x]^m*Cos[x]^(2*p + 1), x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n},
x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

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{x^5}{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sin ^5(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{c^6}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{5 \sin (x)}{8 (a+b x)}-\frac{5 \sin (3 x)}{16 (a+b x)}+\frac{\sin (5 x)}{16 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^6}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sin (5 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 c^6}-\frac{5 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 c^6}+\frac{5 \operatorname{Subst}\left (\int \frac{\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 c^6}\\ &=\frac{\left (5 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 c^6}-\frac{\left (5 \cos \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 c^6}+\frac{\cos \left (\frac{5 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 c^6}-\frac{\left (5 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 c^6}+\frac{\left (5 \sin \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 c^6}-\frac{\sin \left (\frac{5 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 c^6}\\ &=-\frac{5 \text{Ci}\left (\frac{a}{b}+\sin ^{-1}(c x)\right ) \sin \left (\frac{a}{b}\right )}{8 b c^6}+\frac{5 \text{Ci}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right ) \sin \left (\frac{3 a}{b}\right )}{16 b c^6}-\frac{\text{Ci}\left (\frac{5 a}{b}+5 \sin ^{-1}(c x)\right ) \sin \left (\frac{5 a}{b}\right )}{16 b c^6}+\frac{5 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{8 b c^6}-\frac{5 \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{16 b c^6}+\frac{\cos \left (\frac{5 a}{b}\right ) \text{Si}\left (\frac{5 a}{b}+5 \sin ^{-1}(c x)\right )}{16 b c^6}\\ \end{align*}

Mathematica [A]  time = 0.319395, size = 136, normalized size = 0.74 \[ -\frac{10 \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )-5 \sin \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+\sin \left (\frac{5 a}{b}\right ) \text{CosIntegral}\left (5 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )-10 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )+5 \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )-\cos \left (\frac{5 a}{b}\right ) \text{Si}\left (5 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )}{16 b c^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])),x]

[Out]

-(10*CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b] - 5*CosIntegral[3*(a/b + ArcSin[c*x])]*Sin[(3*a)/b] + CosIntegral
[5*(a/b + ArcSin[c*x])]*Sin[(5*a)/b] - 10*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 5*Cos[(3*a)/b]*SinIntegral
[3*(a/b + ArcSin[c*x])] - Cos[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c*x])])/(16*b*c^6)

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Maple [A]  time = 0.048, size = 139, normalized size = 0.8 \begin{align*}{\frac{1}{16\,{c}^{6}b} \left ({\it Si} \left ( 5\,\arcsin \left ( cx \right ) +5\,{\frac{a}{b}} \right ) \cos \left ( 5\,{\frac{a}{b}} \right ) -{\it Ci} \left ( 5\,\arcsin \left ( cx \right ) +5\,{\frac{a}{b}} \right ) \sin \left ( 5\,{\frac{a}{b}} \right ) -5\,{\it Si} \left ( 3\,\arcsin \left ( cx \right ) +3\,{\frac{a}{b}} \right ) \cos \left ( 3\,{\frac{a}{b}} \right ) +5\,{\it Ci} \left ( 3\,\arcsin \left ( cx \right ) +3\,{\frac{a}{b}} \right ) \sin \left ( 3\,{\frac{a}{b}} \right ) +10\,{\it Si} \left ( \arcsin \left ( cx \right ) +{\frac{a}{b}} \right ) \cos \left ({\frac{a}{b}} \right ) -10\,{\it Ci} \left ( \arcsin \left ( cx \right ) +{\frac{a}{b}} \right ) \sin \left ({\frac{a}{b}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2),x)

[Out]

1/16/c^6*(Si(5*arcsin(c*x)+5*a/b)*cos(5*a/b)-Ci(5*arcsin(c*x)+5*a/b)*sin(5*a/b)-5*Si(3*arcsin(c*x)+3*a/b)*cos(
3*a/b)+5*Ci(3*arcsin(c*x)+3*a/b)*sin(3*a/b)+10*Si(arcsin(c*x)+a/b)*cos(a/b)-10*Ci(arcsin(c*x)+a/b)*sin(a/b))/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\sqrt{-c^{2} x^{2} + 1}{\left (b \arcsin \left (c x\right ) + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^5/(sqrt(-c^2*x^2 + 1)*(b*arcsin(c*x) + a)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} x^{2} + 1} x^{5}}{a c^{2} x^{2} +{\left (b c^{2} x^{2} - b\right )} \arcsin \left (c x\right ) - a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-c^2*x^2 + 1)*x^5/(a*c^2*x^2 + (b*c^2*x^2 - b)*arcsin(c*x) - a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\sqrt{- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname{asin}{\left (c x \right )}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(a+b*asin(c*x))/(-c**2*x**2+1)**(1/2),x)

[Out]

Integral(x**5/(sqrt(-(c*x - 1)*(c*x + 1))*(a + b*asin(c*x))), x)

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Giac [B]  time = 1.43218, size = 486, normalized size = 2.66 \begin{align*} -\frac{\cos \left (\frac{a}{b}\right )^{4} \operatorname{Ci}\left (\frac{5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right ) \sin \left (\frac{a}{b}\right )}{b c^{6}} + \frac{\cos \left (\frac{a}{b}\right )^{5} \operatorname{Si}\left (\frac{5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{b c^{6}} + \frac{3 \, \cos \left (\frac{a}{b}\right )^{2} \operatorname{Ci}\left (\frac{5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right ) \sin \left (\frac{a}{b}\right )}{4 \, b c^{6}} + \frac{5 \, \cos \left (\frac{a}{b}\right )^{2} \operatorname{Ci}\left (\frac{3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right ) \sin \left (\frac{a}{b}\right )}{4 \, b c^{6}} - \frac{5 \, \cos \left (\frac{a}{b}\right )^{3} \operatorname{Si}\left (\frac{5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{4 \, b c^{6}} - \frac{5 \, \cos \left (\frac{a}{b}\right )^{3} \operatorname{Si}\left (\frac{3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{4 \, b c^{6}} - \frac{\operatorname{Ci}\left (\frac{5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right ) \sin \left (\frac{a}{b}\right )}{16 \, b c^{6}} - \frac{5 \, \operatorname{Ci}\left (\frac{3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right ) \sin \left (\frac{a}{b}\right )}{16 \, b c^{6}} - \frac{5 \, \operatorname{Ci}\left (\frac{a}{b} + \arcsin \left (c x\right )\right ) \sin \left (\frac{a}{b}\right )}{8 \, b c^{6}} + \frac{5 \, \cos \left (\frac{a}{b}\right ) \operatorname{Si}\left (\frac{5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{16 \, b c^{6}} + \frac{15 \, \cos \left (\frac{a}{b}\right ) \operatorname{Si}\left (\frac{3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{16 \, b c^{6}} + \frac{5 \, \cos \left (\frac{a}{b}\right ) \operatorname{Si}\left (\frac{a}{b} + \arcsin \left (c x\right )\right )}{8 \, b c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

-cos(a/b)^4*cos_integral(5*a/b + 5*arcsin(c*x))*sin(a/b)/(b*c^6) + cos(a/b)^5*sin_integral(5*a/b + 5*arcsin(c*
x))/(b*c^6) + 3/4*cos(a/b)^2*cos_integral(5*a/b + 5*arcsin(c*x))*sin(a/b)/(b*c^6) + 5/4*cos(a/b)^2*cos_integra
l(3*a/b + 3*arcsin(c*x))*sin(a/b)/(b*c^6) - 5/4*cos(a/b)^3*sin_integral(5*a/b + 5*arcsin(c*x))/(b*c^6) - 5/4*c
os(a/b)^3*sin_integral(3*a/b + 3*arcsin(c*x))/(b*c^6) - 1/16*cos_integral(5*a/b + 5*arcsin(c*x))*sin(a/b)/(b*c
^6) - 5/16*cos_integral(3*a/b + 3*arcsin(c*x))*sin(a/b)/(b*c^6) - 5/8*cos_integral(a/b + arcsin(c*x))*sin(a/b)
/(b*c^6) + 5/16*cos(a/b)*sin_integral(5*a/b + 5*arcsin(c*x))/(b*c^6) + 15/16*cos(a/b)*sin_integral(3*a/b + 3*a
rcsin(c*x))/(b*c^6) + 5/8*cos(a/b)*sin_integral(a/b + arcsin(c*x))/(b*c^6)