3.42 \(\int \frac{x^6}{\sin ^{-1}(a x)} \, dx\)

Optimal. Leaf size=55 \[ \frac{5 \text{CosIntegral}\left (\sin ^{-1}(a x)\right )}{64 a^7}-\frac{9 \text{CosIntegral}\left (3 \sin ^{-1}(a x)\right )}{64 a^7}+\frac{5 \text{CosIntegral}\left (5 \sin ^{-1}(a x)\right )}{64 a^7}-\frac{\text{CosIntegral}\left (7 \sin ^{-1}(a x)\right )}{64 a^7} \]

[Out]

(5*CosIntegral[ArcSin[a*x]])/(64*a^7) - (9*CosIntegral[3*ArcSin[a*x]])/(64*a^7) + (5*CosIntegral[5*ArcSin[a*x]
])/(64*a^7) - CosIntegral[7*ArcSin[a*x]]/(64*a^7)

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Rubi [A]  time = 0.0965148, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4635, 4406, 3302} \[ \frac{5 \text{CosIntegral}\left (\sin ^{-1}(a x)\right )}{64 a^7}-\frac{9 \text{CosIntegral}\left (3 \sin ^{-1}(a x)\right )}{64 a^7}+\frac{5 \text{CosIntegral}\left (5 \sin ^{-1}(a x)\right )}{64 a^7}-\frac{\text{CosIntegral}\left (7 \sin ^{-1}(a x)\right )}{64 a^7} \]

Antiderivative was successfully verified.

[In]

Int[x^6/ArcSin[a*x],x]

[Out]

(5*CosIntegral[ArcSin[a*x]])/(64*a^7) - (9*CosIntegral[3*ArcSin[a*x]])/(64*a^7) + (5*CosIntegral[5*ArcSin[a*x]
])/(64*a^7) - CosIntegral[7*ArcSin[a*x]]/(64*a^7)

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^6}{\sin ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cos (x) \sin ^6(x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{a^7}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{5 \cos (x)}{64 x}-\frac{9 \cos (3 x)}{64 x}+\frac{5 \cos (5 x)}{64 x}-\frac{\cos (7 x)}{64 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^7}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\cos (7 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a^7}+\frac{5 \operatorname{Subst}\left (\int \frac{\cos (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a^7}+\frac{5 \operatorname{Subst}\left (\int \frac{\cos (5 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a^7}-\frac{9 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a^7}\\ &=\frac{5 \text{Ci}\left (\sin ^{-1}(a x)\right )}{64 a^7}-\frac{9 \text{Ci}\left (3 \sin ^{-1}(a x)\right )}{64 a^7}+\frac{5 \text{Ci}\left (5 \sin ^{-1}(a x)\right )}{64 a^7}-\frac{\text{Ci}\left (7 \sin ^{-1}(a x)\right )}{64 a^7}\\ \end{align*}

Mathematica [A]  time = 0.0164933, size = 40, normalized size = 0.73 \[ -\frac{-5 \text{CosIntegral}\left (\sin ^{-1}(a x)\right )+9 \text{CosIntegral}\left (3 \sin ^{-1}(a x)\right )-5 \text{CosIntegral}\left (5 \sin ^{-1}(a x)\right )+\text{CosIntegral}\left (7 \sin ^{-1}(a x)\right )}{64 a^7} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/ArcSin[a*x],x]

[Out]

-(-5*CosIntegral[ArcSin[a*x]] + 9*CosIntegral[3*ArcSin[a*x]] - 5*CosIntegral[5*ArcSin[a*x]] + CosIntegral[7*Ar
cSin[a*x]])/(64*a^7)

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Maple [A]  time = 0.039, size = 40, normalized size = 0.7 \begin{align*}{\frac{1}{{a}^{7}} \left ({\frac{5\,{\it Ci} \left ( \arcsin \left ( ax \right ) \right ) }{64}}-{\frac{9\,{\it Ci} \left ( 3\,\arcsin \left ( ax \right ) \right ) }{64}}+{\frac{5\,{\it Ci} \left ( 5\,\arcsin \left ( ax \right ) \right ) }{64}}-{\frac{{\it Ci} \left ( 7\,\arcsin \left ( ax \right ) \right ) }{64}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/arcsin(a*x),x)

[Out]

1/a^7*(5/64*Ci(arcsin(a*x))-9/64*Ci(3*arcsin(a*x))+5/64*Ci(5*arcsin(a*x))-1/64*Ci(7*arcsin(a*x)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{\arcsin \left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/arcsin(a*x),x, algorithm="maxima")

[Out]

integrate(x^6/arcsin(a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/arcsin(a*x),x, algorithm="fricas")

[Out]

integral(x^6/arcsin(a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/asin(a*x),x)

[Out]

Integral(x**6/asin(a*x), x)

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Giac [A]  time = 1.31765, size = 63, normalized size = 1.15 \begin{align*} -\frac{\operatorname{Ci}\left (7 \, \arcsin \left (a x\right )\right )}{64 \, a^{7}} + \frac{5 \, \operatorname{Ci}\left (5 \, \arcsin \left (a x\right )\right )}{64 \, a^{7}} - \frac{9 \, \operatorname{Ci}\left (3 \, \arcsin \left (a x\right )\right )}{64 \, a^{7}} + \frac{5 \, \operatorname{Ci}\left (\arcsin \left (a x\right )\right )}{64 \, a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/arcsin(a*x),x, algorithm="giac")

[Out]

-1/64*cos_integral(7*arcsin(a*x))/a^7 + 5/64*cos_integral(5*arcsin(a*x))/a^7 - 9/64*cos_integral(3*arcsin(a*x)
)/a^7 + 5/64*cos_integral(arcsin(a*x))/a^7