3.4.99 \(\int \frac {x^3 (1-c^2 x^2)^{5/2}}{(a+b \text {ArcSin}(c x))^2} \, dx\) [399]

Optimal. Leaf size=278 \[ -\frac {x^3 \left (1-c^2 x^2\right )^3}{b c (a+b \text {ArcSin}(c x))}+\frac {3 \cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right )}{128 b^2 c^4}+\frac {3 \cos \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 (a+b \text {ArcSin}(c x))}{b}\right )}{32 b^2 c^4}-\frac {21 \cos \left (\frac {7 a}{b}\right ) \text {CosIntegral}\left (\frac {7 (a+b \text {ArcSin}(c x))}{b}\right )}{256 b^2 c^4}-\frac {9 \cos \left (\frac {9 a}{b}\right ) \text {CosIntegral}\left (\frac {9 (a+b \text {ArcSin}(c x))}{b}\right )}{256 b^2 c^4}+\frac {3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right )}{128 b^2 c^4}+\frac {3 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \text {ArcSin}(c x))}{b}\right )}{32 b^2 c^4}-\frac {21 \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \text {ArcSin}(c x))}{b}\right )}{256 b^2 c^4}-\frac {9 \sin \left (\frac {9 a}{b}\right ) \text {Si}\left (\frac {9 (a+b \text {ArcSin}(c x))}{b}\right )}{256 b^2 c^4} \]

[Out]

-x^3*(-c^2*x^2+1)^3/b/c/(a+b*arcsin(c*x))+3/128*Ci((a+b*arcsin(c*x))/b)*cos(a/b)/b^2/c^4+3/32*Ci(3*(a+b*arcsin
(c*x))/b)*cos(3*a/b)/b^2/c^4-21/256*Ci(7*(a+b*arcsin(c*x))/b)*cos(7*a/b)/b^2/c^4-9/256*Ci(9*(a+b*arcsin(c*x))/
b)*cos(9*a/b)/b^2/c^4+3/128*Si((a+b*arcsin(c*x))/b)*sin(a/b)/b^2/c^4+3/32*Si(3*(a+b*arcsin(c*x))/b)*sin(3*a/b)
/b^2/c^4-21/256*Si(7*(a+b*arcsin(c*x))/b)*sin(7*a/b)/b^2/c^4-9/256*Si(9*(a+b*arcsin(c*x))/b)*sin(9*a/b)/b^2/c^
4

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Rubi [A]
time = 0.74, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4799, 4809, 4491, 3384, 3380, 3383} \begin {gather*} \frac {3 \cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right )}{128 b^2 c^4}+\frac {3 \cos \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 (a+b \text {ArcSin}(c x))}{b}\right )}{32 b^2 c^4}-\frac {21 \cos \left (\frac {7 a}{b}\right ) \text {CosIntegral}\left (\frac {7 (a+b \text {ArcSin}(c x))}{b}\right )}{256 b^2 c^4}-\frac {9 \cos \left (\frac {9 a}{b}\right ) \text {CosIntegral}\left (\frac {9 (a+b \text {ArcSin}(c x))}{b}\right )}{256 b^2 c^4}+\frac {3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right )}{128 b^2 c^4}+\frac {3 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \text {ArcSin}(c x))}{b}\right )}{32 b^2 c^4}-\frac {21 \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \text {ArcSin}(c x))}{b}\right )}{256 b^2 c^4}-\frac {9 \sin \left (\frac {9 a}{b}\right ) \text {Si}\left (\frac {9 (a+b \text {ArcSin}(c x))}{b}\right )}{256 b^2 c^4}-\frac {x^3 \left (1-c^2 x^2\right )^3}{b c (a+b \text {ArcSin}(c x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((x^3*(1 - c^2*x^2)^3)/(b*c*(a + b*ArcSin[c*x]))) + (3*Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(128*b^2*
c^4) + (3*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c*x]))/b])/(32*b^2*c^4) - (21*Cos[(7*a)/b]*CosIntegral[(7*
(a + b*ArcSin[c*x]))/b])/(256*b^2*c^4) - (9*Cos[(9*a)/b]*CosIntegral[(9*(a + b*ArcSin[c*x]))/b])/(256*b^2*c^4)
+ (3*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(128*b^2*c^4) + (3*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSi
n[c*x]))/b])/(32*b^2*c^4) - (21*Sin[(7*a)/b]*SinIntegral[(7*(a + b*ArcSin[c*x]))/b])/(256*b^2*c^4) - (9*Sin[(9
*a)/b]*SinIntegral[(9*(a + b*ArcSin[c*x]))/b])/(256*b^2*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 4799

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[
(f*x)^m*Sqrt[1 - c^2*x^2]*(d + e*x^2)^p*((a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1))), x] + (-Dist[f*(m/(b*c*(n
+ 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n +
1), x], x] + Dist[c*((m + 2*p + 1)/(b*f*(n + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m + 1)*(1 -
c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0]
&& LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 2*p + 1, 0] && IGtQ[m, -3]

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*} \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=-\frac {x^3 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {3 \int \frac {x^2 \left (1-c^2 x^2\right )^2}{a+b \sin ^{-1}(c x)} \, dx}{b c}-\frac {(9 c) \int \frac {x^4 \left (1-c^2 x^2\right )^2}{a+b \sin ^{-1}(c x)} \, dx}{b}\\ &=-\frac {x^3 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {3 \text {Subst}\left (\int \frac {\cos ^5(x) \sin ^2(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c^4}-\frac {9 \text {Subst}\left (\int \frac {\cos ^5(x) \sin ^4(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c^4}\\ &=-\frac {x^3 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {3 \text {Subst}\left (\int \left (\frac {5 \cos (x)}{64 (a+b x)}-\frac {\cos (3 x)}{64 (a+b x)}-\frac {3 \cos (5 x)}{64 (a+b x)}-\frac {\cos (7 x)}{64 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^4}-\frac {9 \text {Subst}\left (\int \left (\frac {3 \cos (x)}{128 (a+b x)}-\frac {\cos (3 x)}{64 (a+b x)}-\frac {\cos (5 x)}{64 (a+b x)}+\frac {\cos (7 x)}{256 (a+b x)}+\frac {\cos (9 x)}{256 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^4}\\ &=-\frac {x^3 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {9 \text {Subst}\left (\int \frac {\cos (7 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{256 b c^4}-\frac {9 \text {Subst}\left (\int \frac {\cos (9 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{256 b c^4}-\frac {3 \text {Subst}\left (\int \frac {\cos (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}-\frac {3 \text {Subst}\left (\int \frac {\cos (7 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}+\frac {9 \text {Subst}\left (\int \frac {\cos (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}-\frac {27 \text {Subst}\left (\int \frac {\cos (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{128 b c^4}+\frac {15 \text {Subst}\left (\int \frac {\cos (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}\\ &=-\frac {x^3 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {\left (27 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{128 b c^4}+\frac {\left (15 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}-\frac {\left (3 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}+\frac {\left (9 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}-\frac {\left (9 \cos \left (\frac {7 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{256 b c^4}-\frac {\left (3 \cos \left (\frac {7 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}-\frac {\left (9 \cos \left (\frac {9 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {9 a}{b}+9 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{256 b c^4}-\frac {\left (27 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{128 b c^4}+\frac {\left (15 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}-\frac {\left (3 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}+\frac {\left (9 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}-\frac {\left (9 \sin \left (\frac {7 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{256 b c^4}-\frac {\left (3 \sin \left (\frac {7 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}-\frac {\left (9 \sin \left (\frac {9 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {9 a}{b}+9 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{256 b c^4}\\ &=-\frac {x^3 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {3 \cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{128 b^2 c^4}+\frac {3 \cos \left (\frac {3 a}{b}\right ) \text {Ci}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{32 b^2 c^4}-\frac {21 \cos \left (\frac {7 a}{b}\right ) \text {Ci}\left (\frac {7 a}{b}+7 \sin ^{-1}(c x)\right )}{256 b^2 c^4}-\frac {9 \cos \left (\frac {9 a}{b}\right ) \text {Ci}\left (\frac {9 a}{b}+9 \sin ^{-1}(c x)\right )}{256 b^2 c^4}+\frac {3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{128 b^2 c^4}+\frac {3 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{32 b^2 c^4}-\frac {21 \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 a}{b}+7 \sin ^{-1}(c x)\right )}{256 b^2 c^4}-\frac {9 \sin \left (\frac {9 a}{b}\right ) \text {Si}\left (\frac {9 a}{b}+9 \sin ^{-1}(c x)\right )}{256 b^2 c^4}\\ \end {align*}

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Mathematica [A]
time = 1.07, size = 408, normalized size = 1.47 \begin {gather*} -\frac {256 b c^3 x^3-768 b c^5 x^5+768 b c^7 x^7-256 b c^9 x^9-6 (a+b \text {ArcSin}(c x)) \cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\text {ArcSin}(c x)\right )-24 (a+b \text {ArcSin}(c x)) \cos \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (3 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+21 a \cos \left (\frac {7 a}{b}\right ) \text {CosIntegral}\left (7 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+21 b \text {ArcSin}(c x) \cos \left (\frac {7 a}{b}\right ) \text {CosIntegral}\left (7 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+9 a \cos \left (\frac {9 a}{b}\right ) \text {CosIntegral}\left (9 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+9 b \text {ArcSin}(c x) \cos \left (\frac {9 a}{b}\right ) \text {CosIntegral}\left (9 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )-6 a \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\text {ArcSin}(c x)\right )-6 b \text {ArcSin}(c x) \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\text {ArcSin}(c x)\right )-24 a \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )-24 b \text {ArcSin}(c x) \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+21 a \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (7 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+21 b \text {ArcSin}(c x) \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (7 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+9 a \sin \left (\frac {9 a}{b}\right ) \text {Si}\left (9 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+9 b \text {ArcSin}(c x) \sin \left (\frac {9 a}{b}\right ) \text {Si}\left (9 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )}{256 b^2 c^4 (a+b \text {ArcSin}(c x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/256*(256*b*c^3*x^3 - 768*b*c^5*x^5 + 768*b*c^7*x^7 - 256*b*c^9*x^9 - 6*(a + b*ArcSin[c*x])*Cos[a/b]*CosInte
gral[a/b + ArcSin[c*x]] - 24*(a + b*ArcSin[c*x])*Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c*x])] + 21*a*Cos[(7
*a)/b]*CosIntegral[7*(a/b + ArcSin[c*x])] + 21*b*ArcSin[c*x]*Cos[(7*a)/b]*CosIntegral[7*(a/b + ArcSin[c*x])] +
9*a*Cos[(9*a)/b]*CosIntegral[9*(a/b + ArcSin[c*x])] + 9*b*ArcSin[c*x]*Cos[(9*a)/b]*CosIntegral[9*(a/b + ArcSi
n[c*x])] - 6*a*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] - 6*b*ArcSin[c*x]*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x
]] - 24*a*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] - 24*b*ArcSin[c*x]*Sin[(3*a)/b]*SinIntegral[3*(a/b +
ArcSin[c*x])] + 21*a*Sin[(7*a)/b]*SinIntegral[7*(a/b + ArcSin[c*x])] + 21*b*ArcSin[c*x]*Sin[(7*a)/b]*SinInteg
ral[7*(a/b + ArcSin[c*x])] + 9*a*Sin[(9*a)/b]*SinIntegral[9*(a/b + ArcSin[c*x])] + 9*b*ArcSin[c*x]*Sin[(9*a)/b
]*SinIntegral[9*(a/b + ArcSin[c*x])])/(b^2*c^4*(a + b*ArcSin[c*x]))

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Maple [A]
time = 0.13, size = 454, normalized size = 1.63

method	result	size
default	\(\frac {6 \arcsin \left (c x \right ) \sinIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +6 \arcsin \left (c x \right ) \cosineIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b -9 \arcsin \left (c x \right ) \sinIntegral \left (9 \arcsin \left (c x \right )+\frac {9 a}{b}\right ) \sin \left (\frac {9 a}{b}\right ) b -9 \arcsin \left (c x \right ) \cosineIntegral \left (9 \arcsin \left (c x \right )+\frac {9 a}{b}\right ) \cos \left (\frac {9 a}{b}\right ) b +24 \arcsin \left (c x \right ) \sinIntegral \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b +24 \arcsin \left (c x \right ) \cosineIntegral \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b -21 \arcsin \left (c x \right ) \sinIntegral \left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \sin \left (\frac {7 a}{b}\right ) b -21 \arcsin \left (c x \right ) \cosineIntegral \left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \cos \left (\frac {7 a}{b}\right ) b +6 \sinIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +6 \cosineIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -9 \sinIntegral \left (9 \arcsin \left (c x \right )+\frac {9 a}{b}\right ) \sin \left (\frac {9 a}{b}\right ) a -9 \cosineIntegral \left (9 \arcsin \left (c x \right )+\frac {9 a}{b}\right ) \cos \left (\frac {9 a}{b}\right ) a +24 \sinIntegral \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a +24 \cosineIntegral \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a -21 \sinIntegral \left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \sin \left (\frac {7 a}{b}\right ) a -21 \cosineIntegral \left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \cos \left (\frac {7 a}{b}\right ) a -6 x b c +\sin \left (9 \arcsin \left (c x \right )\right ) b -8 \sin \left (3 \arcsin \left (c x \right )\right ) b +3 \sin \left (7 \arcsin \left (c x \right )\right ) b}{256 c^{4} \left (a +b \arcsin \left (c x \right )\right ) b^{2}}\)	\(454\)

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))^2,x,method=_RETURNVERBOSE)

[Out]

1/256/c^4*(6*arcsin(c*x)*Si(arcsin(c*x)+a/b)*sin(a/b)*b+6*arcsin(c*x)*Ci(arcsin(c*x)+a/b)*cos(a/b)*b-9*arcsin(
c*x)*Si(9*arcsin(c*x)+9*a/b)*sin(9*a/b)*b-9*arcsin(c*x)*Ci(9*arcsin(c*x)+9*a/b)*cos(9*a/b)*b+24*arcsin(c*x)*Si
(3*arcsin(c*x)+3*a/b)*sin(3*a/b)*b+24*arcsin(c*x)*Ci(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*b-21*arcsin(c*x)*Si(7*arc
sin(c*x)+7*a/b)*sin(7*a/b)*b-21*arcsin(c*x)*Ci(7*arcsin(c*x)+7*a/b)*cos(7*a/b)*b+6*Si(arcsin(c*x)+a/b)*sin(a/b
)*a+6*Ci(arcsin(c*x)+a/b)*cos(a/b)*a-9*Si(9*arcsin(c*x)+9*a/b)*sin(9*a/b)*a-9*Ci(9*arcsin(c*x)+9*a/b)*cos(9*a/
b)*a+24*Si(3*arcsin(c*x)+3*a/b)*sin(3*a/b)*a+24*Ci(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*a-21*Si(7*arcsin(c*x)+7*a/b
)*sin(7*a/b)*a-21*Ci(7*arcsin(c*x)+7*a/b)*cos(7*a/b)*a-6*x*b*c+sin(9*arcsin(c*x))*b-8*sin(3*arcsin(c*x))*b+3*s
in(7*arcsin(c*x))*b)/(a+b*arcsin(c*x))/b^2

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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))^2,x, algorithm="maxima")

[Out]

(c^6*x^9 - 3*c^4*x^7 + 3*c^2*x^5 - x^3 - (b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c)*integrate(
3*(3*c^6*x^8 - 7*c^4*x^6 + 5*c^2*x^4 - x^2)/(b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c), x))/(b
^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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))^2,x, algorithm="fricas")

[Out]

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

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

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))**2,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2479 vs. \(2 (260) = 520\).
time = 0.54, size = 2479, normalized size = 8.92 \begin {gather*} \text {Too large to display} \end {gather*}

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))^2,x, algorithm="giac")

[Out]

-9*b*arcsin(c*x)*cos(a/b)^9*cos_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 9*b*arcsin
(c*x)*cos(a/b)^8*sin(a/b)*sin_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 9*a*cos(a/b)
^9*cos_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 9*a*cos(a/b)^8*sin(a/b)*sin_integra
l(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 81/4*b*arcsin(c*x)*cos(a/b)^7*cos_integral(9*a/b
+ 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 21/4*b*arcsin(c*x)*cos(a/b)^7*cos_integral(7*a/b + 7*arcs
in(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 63/4*b*arcsin(c*x)*cos(a/b)^6*sin(a/b)*sin_integral(9*a/b + 9*arc
sin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 21/4*b*arcsin(c*x)*cos(a/b)^6*sin(a/b)*sin_integral(7*a/b + 7*ar
csin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 81/4*a*cos(a/b)^7*cos_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*
arcsin(c*x) + a*b^2*c^4) - 21/4*a*cos(a/b)^7*cos_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*
c^4) + 63/4*a*cos(a/b)^6*sin(a/b)*sin_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 21/4
*a*cos(a/b)^6*sin(a/b)*sin_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 243/16*b*arcsin
(c*x)*cos(a/b)^5*cos_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 147/16*b*arcsin(c*x)*
cos(a/b)^5*cos_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 135/16*b*arcsin(c*x)*cos(a/
b)^4*sin(a/b)*sin_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 105/16*b*arcsin(c*x)*cos
(a/b)^4*sin(a/b)*sin_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + (c^2*x^2 - 1)^4*b*c*x
/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 243/16*a*cos(a/b)^5*cos_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c
*x) + a*b^2*c^4) + 147/16*a*cos(a/b)^5*cos_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) -
135/16*a*cos(a/b)^4*sin(a/b)*sin_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 105/16*a
*cos(a/b)^4*sin(a/b)*sin_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + (c^2*x^2 - 1)^3*b
*c*x/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 135/32*b*arcsin(c*x)*cos(a/b)^3*cos_integral(9*a/b + 9*arcsin(c*x))/(
b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 147/32*b*arcsin(c*x)*cos(a/b)^3*cos_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^
4*arcsin(c*x) + a*b^2*c^4) + 3/8*b*arcsin(c*x)*cos(a/b)^3*cos_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^4*arcsin(
c*x) + a*b^2*c^4) + 45/32*b*arcsin(c*x)*cos(a/b)^2*sin(a/b)*sin_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsi
n(c*x) + a*b^2*c^4) - 63/32*b*arcsin(c*x)*cos(a/b)^2*sin(a/b)*sin_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^4*arc
sin(c*x) + a*b^2*c^4) + 3/8*b*arcsin(c*x)*cos(a/b)^2*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^4*arc
sin(c*x) + a*b^2*c^4) + 135/32*a*cos(a/b)^3*cos_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c
^4) - 147/32*a*cos(a/b)^3*cos_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 3/8*a*cos(a/
b)^3*cos_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 45/32*a*cos(a/b)^2*sin(a/b)*sin_i
ntegral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 63/32*a*cos(a/b)^2*sin(a/b)*sin_integral(7*
a/b + 7*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 3/8*a*cos(a/b)^2*sin(a/b)*sin_integral(3*a/b + 3*arcs
in(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 81/256*b*arcsin(c*x)*cos(a/b)*cos_integral(9*a/b + 9*arcsin(c*x))
/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 147/256*b*arcsin(c*x)*cos(a/b)*cos_integral(7*a/b + 7*arcsin(c*x))/(b^3*c
^4*arcsin(c*x) + a*b^2*c^4) - 9/32*b*arcsin(c*x)*cos(a/b)*cos_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^4*arcsin(
c*x) + a*b^2*c^4) + 3/128*b*arcsin(c*x)*cos(a/b)*cos_integral(a/b + arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*
c^4) - 9/256*b*arcsin(c*x)*sin(a/b)*sin_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 21
/256*b*arcsin(c*x)*sin(a/b)*sin_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 3/32*b*arc
sin(c*x)*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 3/128*b*arcsin(c*x)*
sin(a/b)*sin_integral(a/b + arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 81/256*a*cos(a/b)*cos_integral(9*
a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 147/256*a*cos(a/b)*cos_integral(7*a/b + 7*arcsin(c*x)
)/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 9/32*a*cos(a/b)*cos_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^4*arcsin(c*x)
+ a*b^2*c^4) + 3/128*a*cos(a/b)*cos_integral(a/b + arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 9/256*a*s
in(a/b)*sin_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 21/256*a*sin(a/b)*sin_integral
(7*a/b + 7*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 3/32*a*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x)
)/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 3/128*a*sin(a/b)*sin_integral(a/b + arcsin(c*x))/(b^3*c^4*arcsin(c*x) +
a*b^2*c^4)

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

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))^2,x)

[Out]

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

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