3.333 \(\int \frac {x^3 (1-c^2 x^2)^{5/2}}{a+b \sin ^{-1}(c x)} \, dx\)

Optimal. Leaf size=245 \[ -\frac {3 \sin \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{128 b c^4}-\frac {\sin \left (\frac {3 a}{b}\right ) \text {Ci}\left (\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 b c^4}+\frac {3 \sin \left (\frac {7 a}{b}\right ) \text {Ci}\left (\frac {7 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{256 b c^4}+\frac {\sin \left (\frac {9 a}{b}\right ) \text {Ci}\left (\frac {9 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{256 b c^4}+\frac {3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{128 b c^4}+\frac {\cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 b c^4}-\frac {3 \cos \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{256 b c^4}-\frac {\cos \left (\frac {9 a}{b}\right ) \text {Si}\left (\frac {9 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{256 b c^4} \]

[Out]

3/128*cos(a/b)*Si((a+b*arcsin(c*x))/b)/b/c^4+1/32*cos(3*a/b)*Si(3*(a+b*arcsin(c*x))/b)/b/c^4-3/256*cos(7*a/b)*
Si(7*(a+b*arcsin(c*x))/b)/b/c^4-1/256*cos(9*a/b)*Si(9*(a+b*arcsin(c*x))/b)/b/c^4-3/128*Ci((a+b*arcsin(c*x))/b)
*sin(a/b)/b/c^4-1/32*Ci(3*(a+b*arcsin(c*x))/b)*sin(3*a/b)/b/c^4+3/256*Ci(7*(a+b*arcsin(c*x))/b)*sin(7*a/b)/b/c
^4+1/256*Ci(9*(a+b*arcsin(c*x))/b)*sin(9*a/b)/b/c^4

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Rubi [A]  time = 0.51, antiderivative size = 241, normalized size of antiderivative = 0.98, number of steps used = 15, 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} \[ -\frac {3 \sin \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{128 b c^4}-\frac {\sin \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{32 b c^4}+\frac {3 \sin \left (\frac {7 a}{b}\right ) \text {CosIntegral}\left (\frac {7 a}{b}+7 \sin ^{-1}(c x)\right )}{256 b c^4}+\frac {\sin \left (\frac {9 a}{b}\right ) \text {CosIntegral}\left (\frac {9 a}{b}+9 \sin ^{-1}(c x)\right )}{256 b c^4}+\frac {3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{128 b c^4}+\frac {\cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{32 b c^4}-\frac {3 \cos \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 a}{b}+7 \sin ^{-1}(c x)\right )}{256 b c^4}-\frac {\cos \left (\frac {9 a}{b}\right ) \text {Si}\left (\frac {9 a}{b}+9 \sin ^{-1}(c x)\right )}{256 b c^4} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x]),x]

[Out]

(-3*CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b])/(128*b*c^4) - (CosIntegral[(3*a)/b + 3*ArcSin[c*x]]*Sin[(3*a)/b])
/(32*b*c^4) + (3*CosIntegral[(7*a)/b + 7*ArcSin[c*x]]*Sin[(7*a)/b])/(256*b*c^4) + (CosIntegral[(9*a)/b + 9*Arc
Sin[c*x]]*Sin[(9*a)/b])/(256*b*c^4) + (3*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]])/(128*b*c^4) + (Cos[(3*a)/b]*
SinIntegral[(3*a)/b + 3*ArcSin[c*x]])/(32*b*c^4) - (3*Cos[(7*a)/b]*SinIntegral[(7*a)/b + 7*ArcSin[c*x]])/(256*
b*c^4) - (Cos[(9*a)/b]*SinIntegral[(9*a)/b + 9*ArcSin[c*x]])/(256*b*c^4)

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^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \sin ^{-1}(c x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cos ^6(x) \sin ^3(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{c^4}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {3 \sin (x)}{128 (a+b x)}+\frac {\sin (3 x)}{32 (a+b x)}-\frac {3 \sin (7 x)}{256 (a+b x)}-\frac {\sin (9 x)}{256 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^4}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\sin (9 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{256 c^4}-\frac {3 \operatorname {Subst}\left (\int \frac {\sin (7 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{256 c^4}+\frac {3 \operatorname {Subst}\left (\int \frac {\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{128 c^4}+\frac {\operatorname {Subst}\left (\int \frac {\sin (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{32 c^4}\\ &=\frac {\left (3 \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 )}{128 c^4}+\frac {\cos \left (\frac {3 a}{b}\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 )}{32 c^4}-\frac {\left (3 \cos \left (\frac {7 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{256 c^4}-\frac {\cos \left (\frac {9 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {9 a}{b}+9 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{256 c^4}-\frac {\left (3 \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 )}{128 c^4}-\frac {\sin \left (\frac {3 a}{b}\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 )}{32 c^4}+\frac {\left (3 \sin \left (\frac {7 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{256 c^4}+\frac {\sin \left (\frac {9 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {9 a}{b}+9 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{256 c^4}\\ &=-\frac {3 \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c x)\right ) \sin \left (\frac {a}{b}\right )}{128 b c^4}-\frac {\text {Ci}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right ) \sin \left (\frac {3 a}{b}\right )}{32 b c^4}+\frac {3 \text {Ci}\left (\frac {7 a}{b}+7 \sin ^{-1}(c x)\right ) \sin \left (\frac {7 a}{b}\right )}{256 b c^4}+\frac {\text {Ci}\left (\frac {9 a}{b}+9 \sin ^{-1}(c x)\right ) \sin \left (\frac {9 a}{b}\right )}{256 b c^4}+\frac {3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{128 b c^4}+\frac {\cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{32 b c^4}-\frac {3 \cos \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 a}{b}+7 \sin ^{-1}(c x)\right )}{256 b c^4}-\frac {\cos \left (\frac {9 a}{b}\right ) \text {Si}\left (\frac {9 a}{b}+9 \sin ^{-1}(c x)\right )}{256 b c^4}\\ \end {align*}

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Mathematica [A]  time = 1.27, size = 180, normalized size = 0.73 \[ \frac {-6 \sin \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )-8 \sin \left (\frac {3 a}{b}\right ) \text {Ci}\left (3 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+3 \sin \left (\frac {7 a}{b}\right ) \text {Ci}\left (7 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+\sin \left (\frac {9 a}{b}\right ) \text {Ci}\left (9 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+6 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )+8 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )-3 \cos \left (\frac {7 a}{b}\right ) \text {Si}\left (7 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )-\cos \left (\frac {9 a}{b}\right ) \text {Si}\left (9 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )}{256 b c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x]),x]

[Out]

(-6*CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b] - 8*CosIntegral[3*(a/b + ArcSin[c*x])]*Sin[(3*a)/b] + 3*CosIntegra
l[7*(a/b + ArcSin[c*x])]*Sin[(7*a)/b] + CosIntegral[9*(a/b + ArcSin[c*x])]*Sin[(9*a)/b] + 6*Cos[a/b]*SinIntegr
al[a/b + ArcSin[c*x]] + 8*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] - 3*Cos[(7*a)/b]*SinIntegral[7*(a/b
+ ArcSin[c*x])] - Cos[(9*a)/b]*SinIntegral[9*(a/b + ArcSin[c*x])])/(256*b*c^4)

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fricas [F]  time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{4} x^{7} - 2 \, c^{2} x^{5} + x^{3}\right )} \sqrt {-c^{2} x^{2} + 1}}{b \arcsin \left (c x\right ) + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x)),x, algorithm="fricas")

[Out]

integral((c^4*x^7 - 2*c^2*x^5 + x^3)*sqrt(-c^2*x^2 + 1)/(b*arcsin(c*x) + a), x)

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giac [B]  time = 0.51, size = 746, normalized size = 3.04 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x)),x, algorithm="giac")

[Out]

cos(a/b)^8*cos_integral(9*a/b + 9*arcsin(c*x))*sin(a/b)/(b*c^4) - cos(a/b)^9*sin_integral(9*a/b + 9*arcsin(c*x
))/(b*c^4) - 7/4*cos(a/b)^6*cos_integral(9*a/b + 9*arcsin(c*x))*sin(a/b)/(b*c^4) + 3/4*cos(a/b)^6*cos_integral
(7*a/b + 7*arcsin(c*x))*sin(a/b)/(b*c^4) + 9/4*cos(a/b)^7*sin_integral(9*a/b + 9*arcsin(c*x))/(b*c^4) - 3/4*co
s(a/b)^7*sin_integral(7*a/b + 7*arcsin(c*x))/(b*c^4) + 15/16*cos(a/b)^4*cos_integral(9*a/b + 9*arcsin(c*x))*si
n(a/b)/(b*c^4) - 15/16*cos(a/b)^4*cos_integral(7*a/b + 7*arcsin(c*x))*sin(a/b)/(b*c^4) - 27/16*cos(a/b)^5*sin_
integral(9*a/b + 9*arcsin(c*x))/(b*c^4) + 21/16*cos(a/b)^5*sin_integral(7*a/b + 7*arcsin(c*x))/(b*c^4) - 5/32*
cos(a/b)^2*cos_integral(9*a/b + 9*arcsin(c*x))*sin(a/b)/(b*c^4) + 9/32*cos(a/b)^2*cos_integral(7*a/b + 7*arcsi
n(c*x))*sin(a/b)/(b*c^4) - 1/8*cos(a/b)^2*cos_integral(3*a/b + 3*arcsin(c*x))*sin(a/b)/(b*c^4) + 15/32*cos(a/b
)^3*sin_integral(9*a/b + 9*arcsin(c*x))/(b*c^4) - 21/32*cos(a/b)^3*sin_integral(7*a/b + 7*arcsin(c*x))/(b*c^4)
+ 1/8*cos(a/b)^3*sin_integral(3*a/b + 3*arcsin(c*x))/(b*c^4) + 1/256*cos_integral(9*a/b + 9*arcsin(c*x))*sin(
a/b)/(b*c^4) - 3/256*cos_integral(7*a/b + 7*arcsin(c*x))*sin(a/b)/(b*c^4) + 1/32*cos_integral(3*a/b + 3*arcsin
(c*x))*sin(a/b)/(b*c^4) - 3/128*cos_integral(a/b + arcsin(c*x))*sin(a/b)/(b*c^4) - 9/256*cos(a/b)*sin_integral
(9*a/b + 9*arcsin(c*x))/(b*c^4) + 21/256*cos(a/b)*sin_integral(7*a/b + 7*arcsin(c*x))/(b*c^4) - 3/32*cos(a/b)*
sin_integral(3*a/b + 3*arcsin(c*x))/(b*c^4) + 3/128*cos(a/b)*sin_integral(a/b + arcsin(c*x))/(b*c^4)

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maple [A]  time = 0.08, size = 185, normalized size = 0.76 \[ \frac {6 \Si \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )-6 \Ci \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )+8 \Si \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right )-\Si \left (9 \arcsin \left (c x \right )+\frac {9 a}{b}\right ) \cos \left (\frac {9 a}{b}\right )+\Ci \left (9 \arcsin \left (c x \right )+\frac {9 a}{b}\right ) \sin \left (\frac {9 a}{b}\right )-8 \Ci \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right )-3 \Si \left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \cos \left (\frac {7 a}{b}\right )+3 \Ci \left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \sin \left (\frac {7 a}{b}\right )}{256 c^{4} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x)),x)

[Out]

1/256/c^4*(6*Si(arcsin(c*x)+a/b)*cos(a/b)-6*Ci(arcsin(c*x)+a/b)*sin(a/b)+8*Si(3*arcsin(c*x)+3*a/b)*cos(3*a/b)-
Si(9*arcsin(c*x)+9*a/b)*cos(9*a/b)+Ci(9*arcsin(c*x)+9*a/b)*sin(9*a/b)-8*Ci(3*arcsin(c*x)+3*a/b)*sin(3*a/b)-3*S
i(7*arcsin(c*x)+7*a/b)*cos(7*a/b)+3*Ci(7*arcsin(c*x)+7*a/b)*sin(7*a/b))/b

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{3}}{b \arcsin \left (c x\right ) + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x)),x, algorithm="maxima")

[Out]

integrate((-c^2*x^2 + 1)^(5/2)*x^3/(b*arcsin(c*x) + a), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,{\left (1-c^2\,x^2\right )}^{5/2}}{a+b\,\mathrm {asin}\left (c\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(1 - c^2*x^2)^(5/2))/(a + b*asin(c*x)),x)

[Out]

int((x^3*(1 - c^2*x^2)^(5/2))/(a + b*asin(c*x)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}{a + b \operatorname {asin}{\left (c x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(-c**2*x**2+1)**(5/2)/(a+b*asin(c*x)),x)

[Out]

Integral(x**3*(-(c*x - 1)*(c*x + 1))**(5/2)/(a + b*asin(c*x)), x)

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