3.316 \(\int \frac {x^4 \sqrt {1-c^2 x^2}}{a+b \sin ^{-1}(c x)} \, dx\)

Optimal. Leaf size=206 \[ -\frac {\cos \left (\frac {2 a}{b}\right ) \text {Ci}\left (\frac {2 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 b c^5}-\frac {\cos \left (\frac {4 a}{b}\right ) \text {Ci}\left (\frac {4 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b c^5}+\frac {\cos \left (\frac {6 a}{b}\right ) \text {Ci}\left (\frac {6 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 b c^5}-\frac {\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 b c^5}-\frac {\sin \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b c^5}+\frac {\sin \left (\frac {6 a}{b}\right ) \text {Si}\left (\frac {6 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 b c^5}+\frac {\log \left (a+b \sin ^{-1}(c x)\right )}{16 b c^5} \]

[Out]

-1/32*Ci(2*(a+b*arcsin(c*x))/b)*cos(2*a/b)/b/c^5-1/16*Ci(4*(a+b*arcsin(c*x))/b)*cos(4*a/b)/b/c^5+1/32*Ci(6*(a+
b*arcsin(c*x))/b)*cos(6*a/b)/b/c^5+1/16*ln(a+b*arcsin(c*x))/b/c^5-1/32*Si(2*(a+b*arcsin(c*x))/b)*sin(2*a/b)/b/
c^5-1/16*Si(4*(a+b*arcsin(c*x))/b)*sin(4*a/b)/b/c^5+1/32*Si(6*(a+b*arcsin(c*x))/b)*sin(6*a/b)/b/c^5

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Rubi [A]  time = 0.46, antiderivative size = 206, normalized size of antiderivative = 1.00, 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, 4406, 3303, 3299, 3302} \[ -\frac {\cos \left (\frac {2 a}{b}\right ) \text {CosIntegral}\left (\frac {2 a}{b}+2 \sin ^{-1}(c x)\right )}{32 b c^5}-\frac {\cos \left (\frac {4 a}{b}\right ) \text {CosIntegral}\left (\frac {4 a}{b}+4 \sin ^{-1}(c x)\right )}{16 b c^5}+\frac {\cos \left (\frac {6 a}{b}\right ) \text {CosIntegral}\left (\frac {6 a}{b}+6 \sin ^{-1}(c x)\right )}{32 b c^5}-\frac {\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \sin ^{-1}(c x)\right )}{32 b c^5}-\frac {\sin \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 a}{b}+4 \sin ^{-1}(c x)\right )}{16 b c^5}+\frac {\sin \left (\frac {6 a}{b}\right ) \text {Si}\left (\frac {6 a}{b}+6 \sin ^{-1}(c x)\right )}{32 b c^5}+\frac {\log \left (a+b \sin ^{-1}(c x)\right )}{16 b c^5} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]),x]

[Out]

-(Cos[(2*a)/b]*CosIntegral[(2*a)/b + 2*ArcSin[c*x]])/(32*b*c^5) - (Cos[(4*a)/b]*CosIntegral[(4*a)/b + 4*ArcSin
[c*x]])/(16*b*c^5) + (Cos[(6*a)/b]*CosIntegral[(6*a)/b + 6*ArcSin[c*x]])/(32*b*c^5) + Log[a + b*ArcSin[c*x]]/(
16*b*c^5) - (Sin[(2*a)/b]*SinIntegral[(2*a)/b + 2*ArcSin[c*x]])/(32*b*c^5) - (Sin[(4*a)/b]*SinIntegral[(4*a)/b
+ 4*ArcSin[c*x]])/(16*b*c^5) + (Sin[(6*a)/b]*SinIntegral[(6*a)/b + 6*ArcSin[c*x]])/(32*b*c^5)

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]

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 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 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])

Rubi steps

\begin {align*} \int \frac {x^4 \sqrt {1-c^2 x^2}}{a+b \sin ^{-1}(c x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cos ^2(x) \sin ^4(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{c^5}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{16 (a+b x)}-\frac {\cos (2 x)}{32 (a+b x)}-\frac {\cos (4 x)}{16 (a+b x)}+\frac {\cos (6 x)}{32 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^5}\\ &=\frac {\log \left (a+b \sin ^{-1}(c x)\right )}{16 b c^5}-\frac {\operatorname {Subst}\left (\int \frac {\cos (2 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{32 c^5}+\frac {\operatorname {Subst}\left (\int \frac {\cos (6 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{32 c^5}-\frac {\operatorname {Subst}\left (\int \frac {\cos (4 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 c^5}\\ &=\frac {\log \left (a+b \sin ^{-1}(c x)\right )}{16 b c^5}-\frac {\cos \left (\frac {2 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{32 c^5}-\frac {\cos \left (\frac {4 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 c^5}+\frac {\cos \left (\frac {6 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {6 a}{b}+6 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{32 c^5}-\frac {\sin \left (\frac {2 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{32 c^5}-\frac {\sin \left (\frac {4 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 c^5}+\frac {\sin \left (\frac {6 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {6 a}{b}+6 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{32 c^5}\\ &=-\frac {\cos \left (\frac {2 a}{b}\right ) \text {Ci}\left (\frac {2 a}{b}+2 \sin ^{-1}(c x)\right )}{32 b c^5}-\frac {\cos \left (\frac {4 a}{b}\right ) \text {Ci}\left (\frac {4 a}{b}+4 \sin ^{-1}(c x)\right )}{16 b c^5}+\frac {\cos \left (\frac {6 a}{b}\right ) \text {Ci}\left (\frac {6 a}{b}+6 \sin ^{-1}(c x)\right )}{32 b c^5}+\frac {\log \left (a+b \sin ^{-1}(c x)\right )}{16 b c^5}-\frac {\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \sin ^{-1}(c x)\right )}{32 b c^5}-\frac {\sin \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 a}{b}+4 \sin ^{-1}(c x)\right )}{16 b c^5}+\frac {\sin \left (\frac {6 a}{b}\right ) \text {Si}\left (\frac {6 a}{b}+6 \sin ^{-1}(c x)\right )}{32 b c^5}\\ \end {align*}

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Mathematica [A]  time = 0.49, size = 152, normalized size = 0.74 \[ -\frac {\cos \left (\frac {2 a}{b}\right ) \text {Ci}\left (2 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+2 \cos \left (\frac {4 a}{b}\right ) \text {Ci}\left (4 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )-\cos \left (\frac {6 a}{b}\right ) \text {Ci}\left (6 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+2 \sin \left (\frac {4 a}{b}\right ) \text {Si}\left (4 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )-\sin \left (\frac {6 a}{b}\right ) \text {Si}\left (6 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )-2 \log \left (a+b \sin ^{-1}(c x)\right )}{32 b c^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]),x]

[Out]

-1/32*(Cos[(2*a)/b]*CosIntegral[2*(a/b + ArcSin[c*x])] + 2*Cos[(4*a)/b]*CosIntegral[4*(a/b + ArcSin[c*x])] - C
os[(6*a)/b]*CosIntegral[6*(a/b + ArcSin[c*x])] - 2*Log[a + b*ArcSin[c*x]] + Sin[(2*a)/b]*SinIntegral[2*(a/b +
ArcSin[c*x])] + 2*Sin[(4*a)/b]*SinIntegral[4*(a/b + ArcSin[c*x])] - Sin[(6*a)/b]*SinIntegral[6*(a/b + ArcSin[c
*x])])/(b*c^5)

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fricas [F]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{4}}{b \arcsin \left (c x\right ) + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*x^2+1)^(1/2)/(a+b*arcsin(c*x)),x, algorithm="fricas")

[Out]

integral(sqrt(-c^2*x^2 + 1)*x^4/(b*arcsin(c*x) + a), x)

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giac [B]  time = 0.42, size = 472, normalized size = 2.29 \[ \frac {\cos \left (\frac {a}{b}\right )^{6} \operatorname {Ci}\left (\frac {6 \, a}{b} + 6 \, \arcsin \left (c x\right )\right )}{b c^{5}} + \frac {\cos \left (\frac {a}{b}\right )^{5} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {6 \, a}{b} + 6 \, \arcsin \left (c x\right )\right )}{b c^{5}} - \frac {3 \, \cos \left (\frac {a}{b}\right )^{4} \operatorname {Ci}\left (\frac {6 \, a}{b} + 6 \, \arcsin \left (c x\right )\right )}{2 \, b c^{5}} - \frac {\cos \left (\frac {a}{b}\right )^{4} \operatorname {Ci}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{2 \, b c^{5}} - \frac {\cos \left (\frac {a}{b}\right )^{3} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {6 \, a}{b} + 6 \, \arcsin \left (c x\right )\right )}{b c^{5}} - \frac {\cos \left (\frac {a}{b}\right )^{3} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{2 \, b c^{5}} + \frac {9 \, \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {6 \, a}{b} + 6 \, \arcsin \left (c x\right )\right )}{16 \, b c^{5}} + \frac {\cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{2 \, b c^{5}} - \frac {\cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{16 \, b c^{5}} + \frac {3 \, \cos \left (\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {6 \, a}{b} + 6 \, \arcsin \left (c x\right )\right )}{16 \, b c^{5}} + \frac {\cos \left (\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{4 \, b c^{5}} - \frac {\cos \left (\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{16 \, b c^{5}} - \frac {\operatorname {Ci}\left (\frac {6 \, a}{b} + 6 \, \arcsin \left (c x\right )\right )}{32 \, b c^{5}} - \frac {\operatorname {Ci}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{16 \, b c^{5}} + \frac {\operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{32 \, b c^{5}} + \frac {\log \left (b \arcsin \left (c x\right ) + a\right )}{16 \, b c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*x^2+1)^(1/2)/(a+b*arcsin(c*x)),x, algorithm="giac")

[Out]

cos(a/b)^6*cos_integral(6*a/b + 6*arcsin(c*x))/(b*c^5) + cos(a/b)^5*sin(a/b)*sin_integral(6*a/b + 6*arcsin(c*x
))/(b*c^5) - 3/2*cos(a/b)^4*cos_integral(6*a/b + 6*arcsin(c*x))/(b*c^5) - 1/2*cos(a/b)^4*cos_integral(4*a/b +
4*arcsin(c*x))/(b*c^5) - cos(a/b)^3*sin(a/b)*sin_integral(6*a/b + 6*arcsin(c*x))/(b*c^5) - 1/2*cos(a/b)^3*sin(
a/b)*sin_integral(4*a/b + 4*arcsin(c*x))/(b*c^5) + 9/16*cos(a/b)^2*cos_integral(6*a/b + 6*arcsin(c*x))/(b*c^5)
+ 1/2*cos(a/b)^2*cos_integral(4*a/b + 4*arcsin(c*x))/(b*c^5) - 1/16*cos(a/b)^2*cos_integral(2*a/b + 2*arcsin(
c*x))/(b*c^5) + 3/16*cos(a/b)*sin(a/b)*sin_integral(6*a/b + 6*arcsin(c*x))/(b*c^5) + 1/4*cos(a/b)*sin(a/b)*sin
_integral(4*a/b + 4*arcsin(c*x))/(b*c^5) - 1/16*cos(a/b)*sin(a/b)*sin_integral(2*a/b + 2*arcsin(c*x))/(b*c^5)
- 1/32*cos_integral(6*a/b + 6*arcsin(c*x))/(b*c^5) - 1/16*cos_integral(4*a/b + 4*arcsin(c*x))/(b*c^5) + 1/32*c
os_integral(2*a/b + 2*arcsin(c*x))/(b*c^5) + 1/16*log(b*arcsin(c*x) + a)/(b*c^5)

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maple [A]  time = 0.09, size = 193, normalized size = 0.94 \[ -\frac {\Si \left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) \sin \left (\frac {4 a}{b}\right )}{16 c^{5} b}-\frac {\Ci \left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) \cos \left (\frac {4 a}{b}\right )}{16 c^{5} b}-\frac {\Si \left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right )}{32 c^{5} b}-\frac {\Ci \left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right )}{32 c^{5} b}+\frac {\Si \left (6 \arcsin \left (c x \right )+\frac {6 a}{b}\right ) \sin \left (\frac {6 a}{b}\right )}{32 c^{5} b}+\frac {\Ci \left (6 \arcsin \left (c x \right )+\frac {6 a}{b}\right ) \cos \left (\frac {6 a}{b}\right )}{32 c^{5} b}+\frac {\ln \left (a +b \arcsin \left (c x \right )\right )}{16 b \,c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(-c^2*x^2+1)^(1/2)/(a+b*arcsin(c*x)),x)

[Out]

-1/16/c^5/b*Si(4*arcsin(c*x)+4*a/b)*sin(4*a/b)-1/16/c^5/b*Ci(4*arcsin(c*x)+4*a/b)*cos(4*a/b)-1/32/c^5/b*Si(2*a
rcsin(c*x)+2*a/b)*sin(2*a/b)-1/32/c^5/b*Ci(2*arcsin(c*x)+2*a/b)*cos(2*a/b)+1/32/c^5/b*Si(6*arcsin(c*x)+6*a/b)*
sin(6*a/b)+1/32/c^5/b*Ci(6*arcsin(c*x)+6*a/b)*cos(6*a/b)+1/16*ln(a+b*arcsin(c*x))/b/c^5

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-c^{2} x^{2} + 1} x^{4}}{b \arcsin \left (c x\right ) + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*x^2+1)^(1/2)/(a+b*arcsin(c*x)),x, algorithm="maxima")

[Out]

integrate(sqrt(-c^2*x^2 + 1)*x^4/(b*arcsin(c*x) + a), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^4\,\sqrt {1-c^2\,x^2}}{a+b\,\mathrm {asin}\left (c\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(1 - c^2*x^2)^(1/2))/(a + b*asin(c*x)),x)

[Out]

int((x^4*(1 - c^2*x^2)^(1/2))/(a + b*asin(c*x)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{a + b \operatorname {asin}{\left (c x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(-c**2*x**2+1)**(1/2)/(a+b*asin(c*x)),x)

[Out]

Integral(x**4*sqrt(-(c*x - 1)*(c*x + 1))/(a + b*asin(c*x)), x)

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