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

Optimal. Leaf size=245 \[ -\frac{5 \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{64 b c^2}-\frac{9 \sin \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b c^2}-\frac{5 \sin \left (\frac{5 a}{b}\right ) \text{CosIntegral}\left (\frac{5 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b c^2}-\frac{\sin \left (\frac{7 a}{b}\right ) \text{CosIntegral}\left (\frac{7 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b c^2}+\frac{5 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{64 b c^2}+\frac{9 \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b c^2}+\frac{5 \cos \left (\frac{5 a}{b}\right ) \text{Si}\left (\frac{5 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b c^2}+\frac{\cos \left (\frac{7 a}{b}\right ) \text{Si}\left (\frac{7 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b c^2} \]

[Out]

(-5*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(64*b*c^2) - (9*CosIntegral[(3*(a + b*ArcSin[c*x]))/b]*Sin[(3
*a)/b])/(64*b*c^2) - (5*CosIntegral[(5*(a + b*ArcSin[c*x]))/b]*Sin[(5*a)/b])/(64*b*c^2) - (CosIntegral[(7*(a +
b*ArcSin[c*x]))/b]*Sin[(7*a)/b])/(64*b*c^2) + (5*Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(64*b*c^2) + (9
*Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(64*b*c^2) + (5*Cos[(5*a)/b]*SinIntegral[(5*(a + b*ArcSi
n[c*x]))/b])/(64*b*c^2) + (Cos[(7*a)/b]*SinIntegral[(7*(a + b*ArcSin[c*x]))/b])/(64*b*c^2)

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Rubi [A]  time = 0.447002, antiderivative size = 241, normalized size of antiderivative = 0.98, number of steps used = 15, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {4723, 4406, 3303, 3299, 3302} \[ -\frac{5 \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{64 b c^2}-\frac{9 \sin \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{64 b c^2}-\frac{5 \sin \left (\frac{5 a}{b}\right ) \text{CosIntegral}\left (\frac{5 a}{b}+5 \sin ^{-1}(c x)\right )}{64 b c^2}-\frac{\sin \left (\frac{7 a}{b}\right ) \text{CosIntegral}\left (\frac{7 a}{b}+7 \sin ^{-1}(c x)\right )}{64 b c^2}+\frac{5 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{64 b c^2}+\frac{9 \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{64 b c^2}+\frac{5 \cos \left (\frac{5 a}{b}\right ) \text{Si}\left (\frac{5 a}{b}+5 \sin ^{-1}(c x)\right )}{64 b c^2}+\frac{\cos \left (\frac{7 a}{b}\right ) \text{Si}\left (\frac{7 a}{b}+7 \sin ^{-1}(c x)\right )}{64 b c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b])/(64*b*c^2) - (9*CosIntegral[(3*a)/b + 3*ArcSin[c*x]]*Sin[(3*a)/b]
)/(64*b*c^2) - (5*CosIntegral[(5*a)/b + 5*ArcSin[c*x]]*Sin[(5*a)/b])/(64*b*c^2) - (CosIntegral[(7*a)/b + 7*Arc
Sin[c*x]]*Sin[(7*a)/b])/(64*b*c^2) + (5*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]])/(64*b*c^2) + (9*Cos[(3*a)/b]*
SinIntegral[(3*a)/b + 3*ArcSin[c*x]])/(64*b*c^2) + (5*Cos[(5*a)/b]*SinIntegral[(5*a)/b + 5*ArcSin[c*x]])/(64*b
*c^2) + (Cos[(7*a)/b]*SinIntegral[(7*a)/b + 7*ArcSin[c*x]])/(64*b*c^2)

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])

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 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 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]

Rubi steps

\begin{align*} \int \frac{x \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 (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{c^2}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{5 \sin (x)}{64 (a+b x)}+\frac{9 \sin (3 x)}{64 (a+b x)}+\frac{5 \sin (5 x)}{64 (a+b x)}+\frac{\sin (7 x)}{64 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^2}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sin (7 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 c^2}+\frac{5 \operatorname{Subst}\left (\int \frac{\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 c^2}+\frac{5 \operatorname{Subst}\left (\int \frac{\sin (5 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 c^2}+\frac{9 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 c^2}\\ &=\frac{\left (5 \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 )}{64 c^2}+\frac{\left (9 \cos \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 c^2}+\frac{\left (5 \cos \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 c^2}+\frac{\cos \left (\frac{7 a}{b}\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 c^2}-\frac{\left (5 \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 )}{64 c^2}-\frac{\left (9 \sin \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 c^2}-\frac{\left (5 \sin \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 c^2}-\frac{\sin \left (\frac{7 a}{b}\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 c^2}\\ &=-\frac{5 \text{Ci}\left (\frac{a}{b}+\sin ^{-1}(c x)\right ) \sin \left (\frac{a}{b}\right )}{64 b c^2}-\frac{9 \text{Ci}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right ) \sin \left (\frac{3 a}{b}\right )}{64 b c^2}-\frac{5 \text{Ci}\left (\frac{5 a}{b}+5 \sin ^{-1}(c x)\right ) \sin \left (\frac{5 a}{b}\right )}{64 b c^2}-\frac{\text{Ci}\left (\frac{7 a}{b}+7 \sin ^{-1}(c x)\right ) \sin \left (\frac{7 a}{b}\right )}{64 b c^2}+\frac{5 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{64 b c^2}+\frac{9 \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{64 b c^2}+\frac{5 \cos \left (\frac{5 a}{b}\right ) \text{Si}\left (\frac{5 a}{b}+5 \sin ^{-1}(c x)\right )}{64 b c^2}+\frac{\cos \left (\frac{7 a}{b}\right ) \text{Si}\left (\frac{7 a}{b}+7 \sin ^{-1}(c x)\right )}{64 b c^2}\\ \end{align*}

Mathematica [A]  time = 0.914393, size = 180, normalized size = 0.73 \[ \frac{-5 \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )-9 \sin \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )-5 \sin \left (\frac{5 a}{b}\right ) \text{CosIntegral}\left (5 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )-\sin \left (\frac{7 a}{b}\right ) \text{CosIntegral}\left (7 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+5 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )+9 \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+5 \cos \left (\frac{5 a}{b}\right ) \text{Si}\left (5 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+\cos \left (\frac{7 a}{b}\right ) \text{Si}\left (7 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )}{64 b c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b] - 9*CosIntegral[3*(a/b + ArcSin[c*x])]*Sin[(3*a)/b] - 5*CosIntegra
l[5*(a/b + ArcSin[c*x])]*Sin[(5*a)/b] - CosIntegral[7*(a/b + ArcSin[c*x])]*Sin[(7*a)/b] + 5*Cos[a/b]*SinIntegr
al[a/b + ArcSin[c*x]] + 9*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] + 5*Cos[(5*a)/b]*SinIntegral[5*(a/b
+ ArcSin[c*x])] + Cos[(7*a)/b]*SinIntegral[7*(a/b + ArcSin[c*x])])/(64*b*c^2)

________________________________________________________________________________________

Maple [A]  time = 0.048, size = 185, normalized size = 0.8 \begin{align*}{\frac{1}{64\,{c}^{2}b} \left ( 9\,{\it Si} \left ( 3\,\arcsin \left ( cx \right ) +3\,{\frac{a}{b}} \right ) \cos \left ( 3\,{\frac{a}{b}} \right ) -9\,{\it Ci} \left ( 3\,\arcsin \left ( cx \right ) +3\,{\frac{a}{b}} \right ) \sin \left ( 3\,{\frac{a}{b}} \right ) +5\,{\it Si} \left ( \arcsin \left ( cx \right ) +{\frac{a}{b}} \right ) \cos \left ({\frac{a}{b}} \right ) -5\,{\it Ci} \left ( \arcsin \left ( cx \right ) +{\frac{a}{b}} \right ) \sin \left ({\frac{a}{b}} \right ) +{\it Si} \left ( 7\,\arcsin \left ( cx \right ) +7\,{\frac{a}{b}} \right ) \cos \left ( 7\,{\frac{a}{b}} \right ) -{\it Ci} \left ( 7\,\arcsin \left ( cx \right ) +7\,{\frac{a}{b}} \right ) \sin \left ( 7\,{\frac{a}{b}} \right ) +5\,{\it Si} \left ( 5\,\arcsin \left ( cx \right ) +5\,{\frac{a}{b}} \right ) \cos \left ( 5\,{\frac{a}{b}} \right ) -5\,{\it Ci} \left ( 5\,\arcsin \left ( cx \right ) +5\,{\frac{a}{b}} \right ) \sin \left ( 5\,{\frac{a}{b}} \right ) \right ) } \end{align*}

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)),x)

[Out]

1/64/c^2*(9*Si(3*arcsin(c*x)+3*a/b)*cos(3*a/b)-9*Ci(3*arcsin(c*x)+3*a/b)*sin(3*a/b)+5*Si(arcsin(c*x)+a/b)*cos(
a/b)-5*Ci(arcsin(c*x)+a/b)*sin(a/b)+Si(7*arcsin(c*x)+7*a/b)*cos(7*a/b)-Ci(7*arcsin(c*x)+7*a/b)*sin(7*a/b)+5*Si
(5*arcsin(c*x)+5*a/b)*cos(5*a/b)-5*Ci(5*arcsin(c*x)+5*a/b)*sin(5*a/b))/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{5}{2}} x}{b \arcsin \left (c x\right ) + a}\,{d x} \end{align*}

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)),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c^{4} x^{5} - 2 \, c^{2} x^{3} + x\right )} \sqrt{-c^{2} x^{2} + 1}}{b \arcsin \left (c x\right ) + a}, x\right ) \end{align*}

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)),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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)),x)

[Out]

Timed out

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Giac [B]  time = 1.4237, size = 829, normalized size = 3.38 \begin{align*} \text{result too large to display} \end{align*}

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)),x, algorithm="giac")

[Out]

-cos(a/b)^6*cos_integral(7*a/b + 7*arcsin(c*x))*sin(a/b)/(b*c^2) + cos(a/b)^7*sin_integral(7*a/b + 7*arcsin(c*
x))/(b*c^2) + 5/4*cos(a/b)^4*cos_integral(7*a/b + 7*arcsin(c*x))*sin(a/b)/(b*c^2) - 5/4*cos(a/b)^4*cos_integra
l(5*a/b + 5*arcsin(c*x))*sin(a/b)/(b*c^2) - 7/4*cos(a/b)^5*sin_integral(7*a/b + 7*arcsin(c*x))/(b*c^2) + 5/4*c
os(a/b)^5*sin_integral(5*a/b + 5*arcsin(c*x))/(b*c^2) - 3/8*cos(a/b)^2*cos_integral(7*a/b + 7*arcsin(c*x))*sin
(a/b)/(b*c^2) + 15/16*cos(a/b)^2*cos_integral(5*a/b + 5*arcsin(c*x))*sin(a/b)/(b*c^2) - 9/16*cos(a/b)^2*cos_in
tegral(3*a/b + 3*arcsin(c*x))*sin(a/b)/(b*c^2) + 7/8*cos(a/b)^3*sin_integral(7*a/b + 7*arcsin(c*x))/(b*c^2) -
25/16*cos(a/b)^3*sin_integral(5*a/b + 5*arcsin(c*x))/(b*c^2) + 9/16*cos(a/b)^3*sin_integral(3*a/b + 3*arcsin(c
*x))/(b*c^2) + 1/64*cos_integral(7*a/b + 7*arcsin(c*x))*sin(a/b)/(b*c^2) - 5/64*cos_integral(5*a/b + 5*arcsin(
c*x))*sin(a/b)/(b*c^2) + 9/64*cos_integral(3*a/b + 3*arcsin(c*x))*sin(a/b)/(b*c^2) - 5/64*cos_integral(a/b + a
rcsin(c*x))*sin(a/b)/(b*c^2) - 7/64*cos(a/b)*sin_integral(7*a/b + 7*arcsin(c*x))/(b*c^2) + 25/64*cos(a/b)*sin_
integral(5*a/b + 5*arcsin(c*x))/(b*c^2) - 27/64*cos(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b*c^2) + 5/64*co
s(a/b)*sin_integral(a/b + arcsin(c*x))/(b*c^2)