3.44 \(\int \frac{x^4}{\sin ^{-1}(a x)} \, dx\)

Optimal. Leaf size=41 \[ \frac{\text{CosIntegral}\left (\sin ^{-1}(a x)\right )}{8 a^5}-\frac{3 \text{CosIntegral}\left (3 \sin ^{-1}(a x)\right )}{16 a^5}+\frac{\text{CosIntegral}\left (5 \sin ^{-1}(a x)\right )}{16 a^5} \]

[Out]

CosIntegral[ArcSin[a*x]]/(8*a^5) - (3*CosIntegral[3*ArcSin[a*x]])/(16*a^5) + CosIntegral[5*ArcSin[a*x]]/(16*a^
5)

________________________________________________________________________________________

Rubi [A]  time = 0.080892, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4635, 4406, 3302} \[ \frac{\text{CosIntegral}\left (\sin ^{-1}(a x)\right )}{8 a^5}-\frac{3 \text{CosIntegral}\left (3 \sin ^{-1}(a x)\right )}{16 a^5}+\frac{\text{CosIntegral}\left (5 \sin ^{-1}(a x)\right )}{16 a^5} \]

Antiderivative was successfully verified.

[In]

Int[x^4/ArcSin[a*x],x]

[Out]

CosIntegral[ArcSin[a*x]]/(8*a^5) - (3*CosIntegral[3*ArcSin[a*x]])/(16*a^5) + CosIntegral[5*ArcSin[a*x]]/(16*a^
5)

Rule 4635

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*S
in[x]^m*Cos[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 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 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^4}{\sin ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cos (x) \sin ^4(x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{a^5}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{\cos (x)}{8 x}-\frac{3 \cos (3 x)}{16 x}+\frac{\cos (5 x)}{16 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^5}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\cos (5 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{16 a^5}+\frac{\operatorname{Subst}\left (\int \frac{\cos (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^5}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{16 a^5}\\ &=\frac{\text{Ci}\left (\sin ^{-1}(a x)\right )}{8 a^5}-\frac{3 \text{Ci}\left (3 \sin ^{-1}(a x)\right )}{16 a^5}+\frac{\text{Ci}\left (5 \sin ^{-1}(a x)\right )}{16 a^5}\\ \end{align*}

Mathematica [A]  time = 0.0092898, size = 31, normalized size = 0.76 \[ \frac{2 \text{CosIntegral}\left (\sin ^{-1}(a x)\right )-3 \text{CosIntegral}\left (3 \sin ^{-1}(a x)\right )+\text{CosIntegral}\left (5 \sin ^{-1}(a x)\right )}{16 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/ArcSin[a*x],x]

[Out]

(2*CosIntegral[ArcSin[a*x]] - 3*CosIntegral[3*ArcSin[a*x]] + CosIntegral[5*ArcSin[a*x]])/(16*a^5)

________________________________________________________________________________________

Maple [A]  time = 0.023, size = 31, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{5}} \left ({\frac{{\it Ci} \left ( \arcsin \left ( ax \right ) \right ) }{8}}-{\frac{3\,{\it Ci} \left ( 3\,\arcsin \left ( ax \right ) \right ) }{16}}+{\frac{{\it Ci} \left ( 5\,\arcsin \left ( ax \right ) \right ) }{16}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/arcsin(a*x),x)

[Out]

1/a^5*(1/8*Ci(arcsin(a*x))-3/16*Ci(3*arcsin(a*x))+1/16*Ci(5*arcsin(a*x)))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\arcsin \left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arcsin(a*x),x, algorithm="maxima")

[Out]

integrate(x^4/arcsin(a*x), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arcsin(a*x),x, algorithm="fricas")

[Out]

integral(x^4/arcsin(a*x), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/asin(a*x),x)

[Out]

Integral(x**4/asin(a*x), x)

________________________________________________________________________________________

Giac [A]  time = 1.3525, size = 47, normalized size = 1.15 \begin{align*} \frac{\operatorname{Ci}\left (5 \, \arcsin \left (a x\right )\right )}{16 \, a^{5}} - \frac{3 \, \operatorname{Ci}\left (3 \, \arcsin \left (a x\right )\right )}{16 \, a^{5}} + \frac{\operatorname{Ci}\left (\arcsin \left (a x\right )\right )}{8 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arcsin(a*x),x, algorithm="giac")

[Out]

1/16*cos_integral(5*arcsin(a*x))/a^5 - 3/16*cos_integral(3*arcsin(a*x))/a^5 + 1/8*cos_integral(arcsin(a*x))/a^
5