3.311 \(\int \frac{(c-a^2 c x^2)^3}{\sin ^{-1}(a x)} \, dx\)

Optimal. Leaf size=67 \[ \frac{35 c^3 \text{CosIntegral}\left (\sin ^{-1}(a x)\right )}{64 a}+\frac{21 c^3 \text{CosIntegral}\left (3 \sin ^{-1}(a x)\right )}{64 a}+\frac{7 c^3 \text{CosIntegral}\left (5 \sin ^{-1}(a x)\right )}{64 a}+\frac{c^3 \text{CosIntegral}\left (7 \sin ^{-1}(a x)\right )}{64 a} \]

[Out]

(35*c^3*CosIntegral[ArcSin[a*x]])/(64*a) + (21*c^3*CosIntegral[3*ArcSin[a*x]])/(64*a) + (7*c^3*CosIntegral[5*A
rcSin[a*x]])/(64*a) + (c^3*CosIntegral[7*ArcSin[a*x]])/(64*a)

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Rubi [A]  time = 0.105305, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4661, 3312, 3302} \[ \frac{35 c^3 \text{CosIntegral}\left (\sin ^{-1}(a x)\right )}{64 a}+\frac{21 c^3 \text{CosIntegral}\left (3 \sin ^{-1}(a x)\right )}{64 a}+\frac{7 c^3 \text{CosIntegral}\left (5 \sin ^{-1}(a x)\right )}{64 a}+\frac{c^3 \text{CosIntegral}\left (7 \sin ^{-1}(a x)\right )}{64 a} \]

Antiderivative was successfully verified.

[In]

Int[(c - a^2*c*x^2)^3/ArcSin[a*x],x]

[Out]

(35*c^3*CosIntegral[ArcSin[a*x]])/(64*a) + (21*c^3*CosIntegral[3*ArcSin[a*x]])/(64*a) + (7*c^3*CosIntegral[5*A
rcSin[a*x]])/(64*a) + (c^3*CosIntegral[7*ArcSin[a*x]])/(64*a)

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 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 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{\left (c-a^2 c x^2\right )^3}{\sin ^{-1}(a x)} \, dx &=\frac{c^3 \operatorname{Subst}\left (\int \frac{\cos ^7(x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{c^3 \operatorname{Subst}\left (\int \left (\frac{35 \cos (x)}{64 x}+\frac{21 \cos (3 x)}{64 x}+\frac{7 \cos (5 x)}{64 x}+\frac{\cos (7 x)}{64 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{c^3 \operatorname{Subst}\left (\int \frac{\cos (7 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a}+\frac{\left (7 c^3\right ) \operatorname{Subst}\left (\int \frac{\cos (5 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a}+\frac{\left (21 c^3\right ) \operatorname{Subst}\left (\int \frac{\cos (3 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a}+\frac{\left (35 c^3\right ) \operatorname{Subst}\left (\int \frac{\cos (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a}\\ &=\frac{35 c^3 \text{Ci}\left (\sin ^{-1}(a x)\right )}{64 a}+\frac{21 c^3 \text{Ci}\left (3 \sin ^{-1}(a x)\right )}{64 a}+\frac{7 c^3 \text{Ci}\left (5 \sin ^{-1}(a x)\right )}{64 a}+\frac{c^3 \text{Ci}\left (7 \sin ^{-1}(a x)\right )}{64 a}\\ \end{align*}

Mathematica [A]  time = 0.116208, size = 43, normalized size = 0.64 \[ \frac{c^3 \left (35 \text{CosIntegral}\left (\sin ^{-1}(a x)\right )+21 \text{CosIntegral}\left (3 \sin ^{-1}(a x)\right )+7 \text{CosIntegral}\left (5 \sin ^{-1}(a x)\right )+\text{CosIntegral}\left (7 \sin ^{-1}(a x)\right )\right )}{64 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^3*(35*CosIntegral[ArcSin[a*x]] + 21*CosIntegral[3*ArcSin[a*x]] + 7*CosIntegral[5*ArcSin[a*x]] + CosIntegral
[7*ArcSin[a*x]]))/(64*a)

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Maple [A]  time = 0.044, size = 42, normalized size = 0.6 \begin{align*}{\frac{{c}^{3} \left ( 35\,{\it Ci} \left ( \arcsin \left ( ax \right ) \right ) +21\,{\it Ci} \left ( 3\,\arcsin \left ( ax \right ) \right ) +7\,{\it Ci} \left ( 5\,\arcsin \left ( ax \right ) \right ) +{\it Ci} \left ( 7\,\arcsin \left ( ax \right ) \right ) \right ) }{64\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64/a*c^3*(35*Ci(arcsin(a*x))+21*Ci(3*arcsin(a*x))+7*Ci(5*arcsin(a*x))+Ci(7*arcsin(a*x)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (a^{2} c x^{2} - c\right )}^{3}}{\arcsin \left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate((a^2*c*x^2 - c)^3/arcsin(a*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}}{\arcsin \left (a x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3)/arcsin(a*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - c^{3} \left (\int \frac{3 a^{2} x^{2}}{\operatorname{asin}{\left (a x \right )}}\, dx + \int - \frac{3 a^{4} x^{4}}{\operatorname{asin}{\left (a x \right )}}\, dx + \int \frac{a^{6} x^{6}}{\operatorname{asin}{\left (a x \right )}}\, dx + \int - \frac{1}{\operatorname{asin}{\left (a x \right )}}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c**3*(Integral(3*a**2*x**2/asin(a*x), x) + Integral(-3*a**4*x**4/asin(a*x), x) + Integral(a**6*x**6/asin(a*x)
, x) + Integral(-1/asin(a*x), x))

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Giac [A]  time = 1.40599, size = 80, normalized size = 1.19 \begin{align*} \frac{c^{3} \operatorname{Ci}\left (7 \, \arcsin \left (a x\right )\right )}{64 \, a} + \frac{7 \, c^{3} \operatorname{Ci}\left (5 \, \arcsin \left (a x\right )\right )}{64 \, a} + \frac{21 \, c^{3} \operatorname{Ci}\left (3 \, \arcsin \left (a x\right )\right )}{64 \, a} + \frac{35 \, c^{3} \operatorname{Ci}\left (\arcsin \left (a x\right )\right )}{64 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*c^3*cos_integral(7*arcsin(a*x))/a + 7/64*c^3*cos_integral(5*arcsin(a*x))/a + 21/64*c^3*cos_integral(3*arc
sin(a*x))/a + 35/64*c^3*cos_integral(arcsin(a*x))/a