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

Optimal. Leaf size=276 \[ \frac {5 \cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{64 b^2 c^2}+\frac {27 \cos \left (\frac {3 a}{b}\right ) \text {Ci}\left (\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2}+\frac {25 \cos \left (\frac {5 a}{b}\right ) \text {Ci}\left (\frac {5 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2}+\frac {7 \cos \left (\frac {7 a}{b}\right ) \text {Ci}\left (\frac {7 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2}+\frac {5 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{64 b^2 c^2}+\frac {27 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2}+\frac {25 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2}+\frac {7 \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2}-\frac {x \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )} \]

[Out]

-x*(-c^2*x^2+1)^3/b/c/(a+b*arcsin(c*x))+5/64*Ci((a+b*arcsin(c*x))/b)*cos(a/b)/b^2/c^2+27/64*Ci(3*(a+b*arcsin(c
*x))/b)*cos(3*a/b)/b^2/c^2+25/64*Ci(5*(a+b*arcsin(c*x))/b)*cos(5*a/b)/b^2/c^2+7/64*Ci(7*(a+b*arcsin(c*x))/b)*c
os(7*a/b)/b^2/c^2+5/64*Si((a+b*arcsin(c*x))/b)*sin(a/b)/b^2/c^2+27/64*Si(3*(a+b*arcsin(c*x))/b)*sin(3*a/b)/b^2
/c^2+25/64*Si(5*(a+b*arcsin(c*x))/b)*sin(5*a/b)/b^2/c^2+7/64*Si(7*(a+b*arcsin(c*x))/b)*sin(7*a/b)/b^2/c^2

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Rubi [A]  time = 0.87, antiderivative size = 272, normalized size of antiderivative = 0.99, number of steps used = 28, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {4721, 4661, 3312, 3303, 3299, 3302, 4723, 4406} \[ \frac {5 \cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{64 b^2 c^2}+\frac {27 \cos \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{64 b^2 c^2}+\frac {25 \cos \left (\frac {5 a}{b}\right ) \text {CosIntegral}\left (\frac {5 a}{b}+5 \sin ^{-1}(c x)\right )}{64 b^2 c^2}+\frac {7 \cos \left (\frac {7 a}{b}\right ) \text {CosIntegral}\left (\frac {7 a}{b}+7 \sin ^{-1}(c x)\right )}{64 b^2 c^2}+\frac {5 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{64 b^2 c^2}+\frac {27 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{64 b^2 c^2}+\frac {25 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 a}{b}+5 \sin ^{-1}(c x)\right )}{64 b^2 c^2}+\frac {7 \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 a}{b}+7 \sin ^{-1}(c x)\right )}{64 b^2 c^2}-\frac {x \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((x*(1 - c^2*x^2)^3)/(b*c*(a + b*ArcSin[c*x]))) + (5*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]])/(64*b^2*c^2) +
(27*Cos[(3*a)/b]*CosIntegral[(3*a)/b + 3*ArcSin[c*x]])/(64*b^2*c^2) + (25*Cos[(5*a)/b]*CosIntegral[(5*a)/b + 5
*ArcSin[c*x]])/(64*b^2*c^2) + (7*Cos[(7*a)/b]*CosIntegral[(7*a)/b + 7*ArcSin[c*x]])/(64*b^2*c^2) + (5*Sin[a/b]
*SinIntegral[a/b + ArcSin[c*x]])/(64*b^2*c^2) + (27*Sin[(3*a)/b]*SinIntegral[(3*a)/b + 3*ArcSin[c*x]])/(64*b^2
*c^2) + (25*Sin[(5*a)/b]*SinIntegral[(5*a)/b + 5*ArcSin[c*x]])/(64*b^2*c^2) + (7*Sin[(7*a)/b]*SinIntegral[(7*a
)/b + 7*ArcSin[c*x]])/(64*b^2*c^2)

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 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 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 4661

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^p/c, Subst[Int[(
a + b*x)^n*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] && I
GtQ[2*p, 0] && (IntegerQ[p] || GtQ[d, 0])

Rule 4721

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*d^IntPar
t[p]*(d + e*x^2)^FracPart[p])/(b*c*(n + 1)*(1 - c^2*x^2)^FracPart[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)*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(b*f*(n +
1)*(1 - c^2*x^2)^FracPart[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[m, -3] && IGtQ[2*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 \left (1-c^2 x^2\right )^{5/2}}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=-\frac {x \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {\int \frac {\left (1-c^2 x^2\right )^2}{a+b \sin ^{-1}(c x)} \, dx}{b c}-\frac {(7 c) \int \frac {x^2 \left (1-c^2 x^2\right )^2}{a+b \sin ^{-1}(c x)} \, dx}{b}\\ &=-\frac {x \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {\cos ^5(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c^2}-\frac {7 \operatorname {Subst}\left (\int \frac {\cos ^5(x) \sin ^2(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {x \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {\operatorname {Subst}\left (\int \left (\frac {5 \cos (x)}{8 (a+b x)}+\frac {5 \cos (3 x)}{16 (a+b x)}+\frac {\cos (5 x)}{16 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^2}-\frac {7 \operatorname {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^2}\\ &=-\frac {x \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {\cos (5 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^2}+\frac {7 \operatorname {Subst}\left (\int \frac {\cos (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^2}+\frac {7 \operatorname {Subst}\left (\int \frac {\cos (7 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^2}+\frac {5 \operatorname {Subst}\left (\int \frac {\cos (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^2}+\frac {21 \operatorname {Subst}\left (\int \frac {\cos (5 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^2}-\frac {35 \operatorname {Subst}\left (\int \frac {\cos (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^2}+\frac {5 \operatorname {Subst}\left (\int \frac {\cos (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^2}\\ &=-\frac {x \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {\left (35 \cos \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 )}{64 b c^2}+\frac {\left (5 \cos \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 b c^2}+\frac {\left (7 \cos \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 )}{64 b c^2}+\frac {\left (5 \cos \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 b c^2}+\frac {\cos \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 b c^2}+\frac {\left (21 \cos \left (\frac {5 a}{b}\right )\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 )}{64 b c^2}+\frac {\left (7 \cos \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 )}{64 b c^2}-\frac {\left (35 \sin \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 )}{64 b c^2}+\frac {\left (5 \sin \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 b c^2}+\frac {\left (7 \sin \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 )}{64 b c^2}+\frac {\left (5 \sin \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 b c^2}+\frac {\sin \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 b c^2}+\frac {\left (21 \sin \left (\frac {5 a}{b}\right )\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 )}{64 b c^2}+\frac {\left (7 \sin \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 )}{64 b c^2}\\ &=-\frac {x \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {5 \cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{64 b^2 c^2}+\frac {27 \cos \left (\frac {3 a}{b}\right ) \text {Ci}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{64 b^2 c^2}+\frac {25 \cos \left (\frac {5 a}{b}\right ) \text {Ci}\left (\frac {5 a}{b}+5 \sin ^{-1}(c x)\right )}{64 b^2 c^2}+\frac {7 \cos \left (\frac {7 a}{b}\right ) \text {Ci}\left (\frac {7 a}{b}+7 \sin ^{-1}(c x)\right )}{64 b^2 c^2}+\frac {5 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{64 b^2 c^2}+\frac {27 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{64 b^2 c^2}+\frac {25 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 a}{b}+5 \sin ^{-1}(c x)\right )}{64 b^2 c^2}+\frac {7 \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 a}{b}+7 \sin ^{-1}(c x)\right )}{64 b^2 c^2}\\ \end {align*}

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Mathematica [A]  time = 1.09, size = 404, normalized size = 1.46 \[ \frac {5 \cos \left (\frac {a}{b}\right ) \left (a+b \sin ^{-1}(c x)\right ) \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )+27 \cos \left (\frac {3 a}{b}\right ) \left (a+b \sin ^{-1}(c x)\right ) \text {Ci}\left (3 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+25 a \cos \left (\frac {5 a}{b}\right ) \text {Ci}\left (5 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+25 b \cos \left (\frac {5 a}{b}\right ) \sin ^{-1}(c x) \text {Ci}\left (5 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+7 a \cos \left (\frac {7 a}{b}\right ) \text {Ci}\left (7 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+7 b \cos \left (\frac {7 a}{b}\right ) \sin ^{-1}(c x) \text {Ci}\left (7 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+5 a \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )+5 b \sin \left (\frac {a}{b}\right ) \sin ^{-1}(c x) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )+27 a \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+27 b \sin \left (\frac {3 a}{b}\right ) \sin ^{-1}(c x) \text {Si}\left (3 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+25 a \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+25 b \sin \left (\frac {5 a}{b}\right ) \sin ^{-1}(c x) \text {Si}\left (5 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+7 a \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (7 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+7 b \sin \left (\frac {7 a}{b}\right ) \sin ^{-1}(c x) \text {Si}\left (7 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+64 b c^7 x^7-192 b c^5 x^5+192 b c^3 x^3-64 b c x}{64 b^2 c^2 \left (a+b \sin ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-64*b*c*x + 192*b*c^3*x^3 - 192*b*c^5*x^5 + 64*b*c^7*x^7 + 5*(a + b*ArcSin[c*x])*Cos[a/b]*CosIntegral[a/b + A
rcSin[c*x]] + 27*(a + b*ArcSin[c*x])*Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c*x])] + 25*a*Cos[(5*a)/b]*CosIn
tegral[5*(a/b + ArcSin[c*x])] + 25*b*ArcSin[c*x]*Cos[(5*a)/b]*CosIntegral[5*(a/b + ArcSin[c*x])] + 7*a*Cos[(7*
a)/b]*CosIntegral[7*(a/b + ArcSin[c*x])] + 7*b*ArcSin[c*x]*Cos[(7*a)/b]*CosIntegral[7*(a/b + ArcSin[c*x])] + 5
*a*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 5*b*ArcSin[c*x]*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 27*a*Si
n[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] + 27*b*ArcSin[c*x]*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x]
)] + 25*a*Sin[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c*x])] + 25*b*ArcSin[c*x]*Sin[(5*a)/b]*SinIntegral[5*(a/b +
ArcSin[c*x])] + 7*a*Sin[(7*a)/b]*SinIntegral[7*(a/b + ArcSin[c*x])] + 7*b*ArcSin[c*x]*Sin[(7*a)/b]*SinIntegra
l[7*(a/b + ArcSin[c*x])])/(64*b^2*c^2*(a + b*ArcSin[c*x]))

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fricas [F]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{4} x^{5} - 2 \, c^{2} x^{3} + x\right )} \sqrt {-c^{2} x^{2} + 1}}{b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((c^4*x^5 - 2*c^2*x^3 + x)*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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giac [B]  time = 1.27, size = 2026, normalized size = 7.34 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7*b*arcsin(c*x)*cos(a/b)^7*cos_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 7*b*arcsin(
c*x)*cos(a/b)^6*sin(a/b)*sin_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 7*a*cos(a/b)^
7*cos_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 7*a*cos(a/b)^6*sin(a/b)*sin_integral
(7*a/b + 7*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) - 49/4*b*arcsin(c*x)*cos(a/b)^5*cos_integral(7*a/b +
7*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 25/4*b*arcsin(c*x)*cos(a/b)^5*cos_integral(5*a/b + 5*arcsi
n(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) - 35/4*b*arcsin(c*x)*cos(a/b)^4*sin(a/b)*sin_integral(7*a/b + 7*arcs
in(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 25/4*b*arcsin(c*x)*cos(a/b)^4*sin(a/b)*sin_integral(5*a/b + 5*arc
sin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) - 49/4*a*cos(a/b)^5*cos_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^2*a
rcsin(c*x) + a*b^2*c^2) + 25/4*a*cos(a/b)^5*cos_integral(5*a/b + 5*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c
^2) - 35/4*a*cos(a/b)^4*sin(a/b)*sin_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 25/4*
a*cos(a/b)^4*sin(a/b)*sin_integral(5*a/b + 5*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + (c^2*x^2 - 1)^3*
b*c*x/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 49/8*b*arcsin(c*x)*cos(a/b)^3*cos_integral(7*a/b + 7*arcsin(c*x))/(b
^3*c^2*arcsin(c*x) + a*b^2*c^2) - 125/16*b*arcsin(c*x)*cos(a/b)^3*cos_integral(5*a/b + 5*arcsin(c*x))/(b^3*c^2
*arcsin(c*x) + a*b^2*c^2) + 27/16*b*arcsin(c*x)*cos(a/b)^3*cos_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^2*arcsin
(c*x) + a*b^2*c^2) + 21/8*b*arcsin(c*x)*cos(a/b)^2*sin(a/b)*sin_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^2*arcsi
n(c*x) + a*b^2*c^2) - 75/16*b*arcsin(c*x)*cos(a/b)^2*sin(a/b)*sin_integral(5*a/b + 5*arcsin(c*x))/(b^3*c^2*arc
sin(c*x) + a*b^2*c^2) + 27/16*b*arcsin(c*x)*cos(a/b)^2*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^2*a
rcsin(c*x) + a*b^2*c^2) + 49/8*a*cos(a/b)^3*cos_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c
^2) - 125/16*a*cos(a/b)^3*cos_integral(5*a/b + 5*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 27/16*a*cos(
a/b)^3*cos_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 21/8*a*cos(a/b)^2*sin(a/b)*sin_
integral(7*a/b + 7*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) - 75/16*a*cos(a/b)^2*sin(a/b)*sin_integral(5
*a/b + 5*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 27/16*a*cos(a/b)^2*sin(a/b)*sin_integral(3*a/b + 3*a
rcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) - 49/64*b*arcsin(c*x)*cos(a/b)*cos_integral(7*a/b + 7*arcsin(c*x
))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 125/64*b*arcsin(c*x)*cos(a/b)*cos_integral(5*a/b + 5*arcsin(c*x))/(b^3*
c^2*arcsin(c*x) + a*b^2*c^2) - 81/64*b*arcsin(c*x)*cos(a/b)*cos_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^2*arcsi
n(c*x) + a*b^2*c^2) + 5/64*b*arcsin(c*x)*cos(a/b)*cos_integral(a/b + arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2
*c^2) - 7/64*b*arcsin(c*x)*sin(a/b)*sin_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 25
/64*b*arcsin(c*x)*sin(a/b)*sin_integral(5*a/b + 5*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) - 27/64*b*arc
sin(c*x)*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 5/64*b*arcsin(c*x)*s
in(a/b)*sin_integral(a/b + arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) - 49/64*a*cos(a/b)*cos_integral(7*a/
b + 7*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 125/64*a*cos(a/b)*cos_integral(5*a/b + 5*arcsin(c*x))/(
b^3*c^2*arcsin(c*x) + a*b^2*c^2) - 81/64*a*cos(a/b)*cos_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^2*arcsin(c*x) +
a*b^2*c^2) + 5/64*a*cos(a/b)*cos_integral(a/b + arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) - 7/64*a*sin(a
/b)*sin_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 25/64*a*sin(a/b)*sin_integral(5*a/
b + 5*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) - 27/64*a*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b
^3*c^2*arcsin(c*x) + a*b^2*c^2) + 5/64*a*sin(a/b)*sin_integral(a/b + arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2
*c^2)

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maple [A]  time = 0.11, size = 455, normalized size = 1.65 \[ \frac {27 \arcsin \left (c x \right ) \Si \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b +27 \arcsin \left (c x \right ) \Ci \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b +5 \arcsin \left (c x \right ) \Si \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +5 \arcsin \left (c x \right ) \Ci \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +7 \arcsin \left (c x \right ) \Si \left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \sin \left (\frac {7 a}{b}\right ) b +7 \arcsin \left (c x \right ) \Ci \left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \cos \left (\frac {7 a}{b}\right ) b +25 \arcsin \left (c x \right ) \Si \left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right ) b +25 \arcsin \left (c x \right ) \Ci \left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right ) b +27 \Si \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a +27 \Ci \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a +5 \Si \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +5 \Ci \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a +7 \Si \left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \sin \left (\frac {7 a}{b}\right ) a +7 \Ci \left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \cos \left (\frac {7 a}{b}\right ) a +25 \Si \left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right ) a +25 \Ci \left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right ) a -5 x b c -9 \sin \left (3 \arcsin \left (c x \right )\right ) b -\sin \left (7 \arcsin \left (c x \right )\right ) b -5 \sin \left (5 \arcsin \left (c x \right )\right ) b}{64 c^{2} \left (a +b \arcsin \left (c x \right )\right ) b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64/c^2*(27*arcsin(c*x)*Si(3*arcsin(c*x)+3*a/b)*sin(3*a/b)*b+27*arcsin(c*x)*Ci(3*arcsin(c*x)+3*a/b)*cos(3*a/b
)*b+5*arcsin(c*x)*Si(arcsin(c*x)+a/b)*sin(a/b)*b+5*arcsin(c*x)*Ci(arcsin(c*x)+a/b)*cos(a/b)*b+7*arcsin(c*x)*Si
(7*arcsin(c*x)+7*a/b)*sin(7*a/b)*b+7*arcsin(c*x)*Ci(7*arcsin(c*x)+7*a/b)*cos(7*a/b)*b+25*arcsin(c*x)*Si(5*arcs
in(c*x)+5*a/b)*sin(5*a/b)*b+25*arcsin(c*x)*Ci(5*arcsin(c*x)+5*a/b)*cos(5*a/b)*b+27*Si(3*arcsin(c*x)+3*a/b)*sin
(3*a/b)*a+27*Ci(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*a+5*Si(arcsin(c*x)+a/b)*sin(a/b)*a+5*Ci(arcsin(c*x)+a/b)*cos(a
/b)*a+7*Si(7*arcsin(c*x)+7*a/b)*sin(7*a/b)*a+7*Ci(7*arcsin(c*x)+7*a/b)*cos(7*a/b)*a+25*Si(5*arcsin(c*x)+5*a/b)
*sin(5*a/b)*a+25*Ci(5*arcsin(c*x)+5*a/b)*cos(5*a/b)*a-5*x*b*c-9*sin(3*arcsin(c*x))*b-sin(7*arcsin(c*x))*b-5*si
n(5*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 \[ \frac {c^{6} x^{7} - 3 \, c^{4} x^{5} + 3 \, c^{2} x^{3} - x - \frac {{\left (7 \, c^{6} \int \frac {x^{6}}{b \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a}\,{d x} - 15 \, c^{4} \int \frac {x^{4}}{b \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a}\,{d x} + 9 \, c^{2} \int \frac {x^{2}}{b \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a}\,{d x} - \int \frac {1}{b \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a}\,{d x}\right )} {\left (b^{2} c \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a b c\right )}}{b c}}{b^{2} c \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^6*x^7 - 3*c^4*x^5 + 3*c^2*x^3 - (b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c)*integrate((7*c^6
*x^6 - 15*c^4*x^4 + 9*c^2*x^2 - 1)/(b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c), x) - x)/(b^2*c*
arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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