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

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

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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 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 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 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])

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*}

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Mathematica [A]  time = 0.15, size = 43, normalized size = 0.64 \[ \frac {c^3 \left (35 \text {Ci}\left (\sin ^{-1}(a x)\right )+21 \text {Ci}\left (3 \sin ^{-1}(a x)\right )+7 \text {Ci}\left (5 \sin ^{-1}(a x)\right )+\text {Ci}\left (7 \sin ^{-1}(a x)\right )\right )}{64 a} \]

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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fricas [F]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}}{\arcsin \left (a x\right )}, x\right ) \]

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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giac [A]  time = 0.44, size = 59, normalized size = 0.88 \[ \frac {c^{3} \operatorname {Ci}\left (7 \, \arcsin \left (a x\right )\right )}{64 \, a} + \frac {7 \, c^{3} \operatorname {Ci}\left (5 \, \arcsin \left (a x\right )\right )}{64 \, a} + \frac {21 \, c^{3} \operatorname {Ci}\left (3 \, \arcsin \left (a x\right )\right )}{64 \, a} + \frac {35 \, c^{3} \operatorname {Ci}\left (\arcsin \left (a x\right )\right )}{64 \, a} \]

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

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maple [A]  time = 0.12, size = 42, normalized size = 0.63 \[ \frac {c^{3} \left (35 \Ci \left (\arcsin \left (a x \right )\right )+21 \Ci \left (3 \arcsin \left (a x \right )\right )+7 \Ci \left (5 \arcsin \left (a x \right )\right )+\Ci \left (7 \arcsin \left (a x \right )\right )\right )}{64 a} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (a^{2} c x^{2} - c\right )}^{3}}{\arcsin \left (a x\right )}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c-a^2\,c\,x^2\right )}^3}{\mathrm {asin}\left (a\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c - a^2*c*x^2)^3/asin(a*x),x)

[Out]

int((c - a^2*c*x^2)^3/asin(a*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - c^{3} \left (\int \frac {3 a^{2} x^{2}}{\operatorname {asin}{\left (a x \right )}}\, dx + \int \left (- \frac {3 a^{4} x^{4}}{\operatorname {asin}{\left (a x \right )}}\right )\, dx + \int \frac {a^{6} x^{6}}{\operatorname {asin}{\left (a x \right )}}\, dx + \int \left (- \frac {1}{\operatorname {asin}{\left (a x \right )}}\right )\, dx\right ) \]

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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