\(\int \frac {x^3 (1-c^2 x^2)^{5/2}}{a+b \arcsin (c x)} \, dx\) [333]

Optimal result

Integrand size = 28, antiderivative size = 245 \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \arcsin (c x)} \, dx=-\frac {3 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{128 b c^4}-\frac {\operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{32 b c^4}+\frac {3 \operatorname {CosIntegral}\left (\frac {7 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {7 a}{b}\right )}{256 b c^4}+\frac {\operatorname {CosIntegral}\left (\frac {9 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {9 a}{b}\right )}{256 b c^4}+\frac {3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{128 b c^4}+\frac {\cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{32 b c^4}-\frac {3 \cos \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )}{256 b c^4}-\frac {\cos \left (\frac {9 a}{b}\right ) \text {Si}\left (\frac {9 (a+b \arcsin (c x))}{b}\right )}{256 b c^4} \]

[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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {4809, 4491, 3384, 3380, 3383} \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \arcsin (c x)} \, dx=-\frac {3 \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{128 b c^4}-\frac {\sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{32 b c^4}+\frac {3 \sin \left (\frac {7 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )}{256 b c^4}+\frac {\sin \left (\frac {9 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {9 (a+b \arcsin (c x))}{b}\right )}{256 b c^4}+\frac {3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{128 b c^4}+\frac {\cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{32 b c^4}-\frac {3 \cos \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )}{256 b c^4}-\frac {\cos \left (\frac {9 a}{b}\right ) \text {Si}\left (\frac {9 (a+b \arcsin (c x))}{b}\right )}{256 b c^4} \]

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

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 4809

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\cos ^6\left (\frac {a}{b}-\frac {x}{b}\right ) \sin ^3\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{b c^4} \\ & = -\frac {\text {Subst}\left (\int \left (-\frac {\sin \left (\frac {9 a}{b}-\frac {9 x}{b}\right )}{256 x}-\frac {3 \sin \left (\frac {7 a}{b}-\frac {7 x}{b}\right )}{256 x}+\frac {\sin \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{32 x}+\frac {3 \sin \left (\frac {a}{b}-\frac {x}{b}\right )}{128 x}\right ) \, dx,x,a+b \arcsin (c x)\right )}{b c^4} \\ & = \frac {\text {Subst}\left (\int \frac {\sin \left (\frac {9 a}{b}-\frac {9 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{256 b c^4}+\frac {3 \text {Subst}\left (\int \frac {\sin \left (\frac {7 a}{b}-\frac {7 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{256 b c^4}-\frac {3 \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{128 b c^4}-\frac {\text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{32 b c^4} \\ & = \frac {\left (3 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{128 b c^4}+\frac {\cos \left (\frac {3 a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{32 b c^4}-\frac {\left (3 \cos \left (\frac {7 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {7 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{256 b c^4}-\frac {\cos \left (\frac {9 a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {9 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{256 b c^4}-\frac {\left (3 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{128 b c^4}-\frac {\sin \left (\frac {3 a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{32 b c^4}+\frac {\left (3 \sin \left (\frac {7 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {7 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{256 b c^4}+\frac {\sin \left (\frac {9 a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {9 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{256 b c^4} \\ & = -\frac {3 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{128 b c^4}-\frac {\operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{32 b c^4}+\frac {3 \operatorname {CosIntegral}\left (\frac {7 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {7 a}{b}\right )}{256 b c^4}+\frac {\operatorname {CosIntegral}\left (\frac {9 (a+b \arcsin (c x))}{b}\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 \arcsin (c x)}{b}\right )}{128 b c^4}+\frac {\cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{32 b c^4}-\frac {3 \cos \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )}{256 b c^4}-\frac {\cos \left (\frac {9 a}{b}\right ) \text {Si}\left (\frac {9 (a+b \arcsin (c x))}{b}\right )}{256 b c^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.97 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.73 \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \arcsin (c x)} \, dx=\frac {-6 \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right ) \sin \left (\frac {a}{b}\right )-8 \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {3 a}{b}\right )+3 \operatorname {CosIntegral}\left (7 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {7 a}{b}\right )+\operatorname {CosIntegral}\left (9 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {9 a}{b}\right )+6 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )+8 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-3 \cos \left (\frac {7 a}{b}\right ) \text {Si}\left (7 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-\cos \left (\frac {9 a}{b}\right ) \text {Si}\left (9 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{256 b c^4} \]

[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)

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.76

method	result	size
default	\(\frac {6 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )-6 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )+8 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right )-8 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right )-3 \,\operatorname {Si}\left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \cos \left (\frac {7 a}{b}\right )-\operatorname {Si}\left (9 \arcsin \left (c x \right )+\frac {9 a}{b}\right ) \cos \left (\frac {9 a}{b}\right )+\operatorname {Ci}\left (9 \arcsin \left (c x \right )+\frac {9 a}{b}\right ) \sin \left (\frac {9 a}{b}\right )+3 \,\operatorname {Ci}\left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \sin \left (\frac {7 a}{b}\right )}{256 c^{4} b}\)	\(185\)

[In]

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

[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)-
8*Ci(3*arcsin(c*x)+3*a/b)*sin(3*a/b)-3*Si(7*arcsin(c*x)+7*a/b)*cos(7*a/b)-Si(9*arcsin(c*x)+9*a/b)*cos(9*a/b)+C
i(9*arcsin(c*x)+9*a/b)*sin(9*a/b)+3*Ci(7*arcsin(c*x)+7*a/b)*sin(7*a/b))/b

Fricas [F]

\[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \arcsin (c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{3}}{b \arcsin \left (c x\right ) + a} \,d x } \]

[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)

Sympy [F]

\[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \arcsin (c x)} \, dx=\int \frac {x^{3} \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}{a + b \operatorname {asin}{\left (c x \right )}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \arcsin (c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{3}}{b \arcsin \left (c x\right ) + a} \,d x } \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 746 vs. \(2 (229) = 458\).

Time = 0.35 (sec) , antiderivative size = 746, normalized size of antiderivative = 3.04 \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \arcsin (c x)} \, dx=\text {Too large to display} \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \arcsin (c x)} \, dx=\int \frac {x^3\,{\left (1-c^2\,x^2\right )}^{5/2}}{a+b\,\mathrm {asin}\left (c\,x\right )} \,d x \]

[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)