3.38 \(\int \frac{1}{x^4 (a+b \csc ^{-1}(c x))} \, dx\)

Optimal. Leaf size=117 \[ -\frac{c^3 \cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\csc ^{-1}(c x)\right )}{4 b}+\frac{c^3 \cos \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 a}{b}+3 \csc ^{-1}(c x)\right )}{4 b}-\frac{c^3 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\csc ^{-1}(c x)\right )}{4 b}+\frac{c^3 \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \csc ^{-1}(c x)\right )}{4 b} \]

[Out]

-(c^3*Cos[a/b]*CosIntegral[a/b + ArcCsc[c*x]])/(4*b) + (c^3*Cos[(3*a)/b]*CosIntegral[(3*a)/b + 3*ArcCsc[c*x]])
/(4*b) - (c^3*Sin[a/b]*SinIntegral[a/b + ArcCsc[c*x]])/(4*b) + (c^3*Sin[(3*a)/b]*SinIntegral[(3*a)/b + 3*ArcCs
c[c*x]])/(4*b)

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Rubi [A]  time = 0.228755, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {5223, 4406, 3303, 3299, 3302} \[ -\frac{c^3 \cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\csc ^{-1}(c x)\right )}{4 b}+\frac{c^3 \cos \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 a}{b}+3 \csc ^{-1}(c x)\right )}{4 b}-\frac{c^3 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\csc ^{-1}(c x)\right )}{4 b}+\frac{c^3 \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \csc ^{-1}(c x)\right )}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*ArcCsc[c*x])),x]

[Out]

-(c^3*Cos[a/b]*CosIntegral[a/b + ArcCsc[c*x]])/(4*b) + (c^3*Cos[(3*a)/b]*CosIntegral[(3*a)/b + 3*ArcCsc[c*x]])
/(4*b) - (c^3*Sin[a/b]*SinIntegral[a/b + ArcCsc[c*x]])/(4*b) + (c^3*Sin[(3*a)/b]*SinIntegral[(3*a)/b + 3*ArcCs
c[c*x]])/(4*b)

Rule 5223

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b*
x)^n*Csc[x]^(m + 1)*Cot[x], x], x, ArcCsc[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (G
tQ[n, 0] || LtQ[m, -1])

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{1}{x^4 \left (a+b \csc ^{-1}(c x)\right )} \, dx &=-\left (c^3 \operatorname{Subst}\left (\int \frac{\cos (x) \sin ^2(x)}{a+b x} \, dx,x,\csc ^{-1}(c x)\right )\right )\\ &=-\left (c^3 \operatorname{Subst}\left (\int \left (\frac{\cos (x)}{4 (a+b x)}-\frac{\cos (3 x)}{4 (a+b x)}\right ) \, dx,x,\csc ^{-1}(c x)\right )\right )\\ &=-\left (\frac{1}{4} c^3 \operatorname{Subst}\left (\int \frac{\cos (x)}{a+b x} \, dx,x,\csc ^{-1}(c x)\right )\right )+\frac{1}{4} c^3 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{a+b x} \, dx,x,\csc ^{-1}(c x)\right )\\ &=-\left (\frac{1}{4} \left (c^3 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\csc ^{-1}(c x)\right )\right )+\frac{1}{4} \left (c^3 \cos \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,\csc ^{-1}(c x)\right )-\frac{1}{4} \left (c^3 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\csc ^{-1}(c x)\right )+\frac{1}{4} \left (c^3 \sin \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,\csc ^{-1}(c x)\right )\\ &=-\frac{c^3 \cos \left (\frac{a}{b}\right ) \text{Ci}\left (\frac{a}{b}+\csc ^{-1}(c x)\right )}{4 b}+\frac{c^3 \cos \left (\frac{3 a}{b}\right ) \text{Ci}\left (\frac{3 a}{b}+3 \csc ^{-1}(c x)\right )}{4 b}-\frac{c^3 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\csc ^{-1}(c x)\right )}{4 b}+\frac{c^3 \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \csc ^{-1}(c x)\right )}{4 b}\\ \end{align*}

Mathematica [A]  time = 0.171639, size = 91, normalized size = 0.78 \[ -\frac{c^3 \left (\cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\csc ^{-1}(c x)\right )-\cos \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (3 \left (\frac{a}{b}+\csc ^{-1}(c x)\right )\right )+\sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\csc ^{-1}(c x)\right )-\sin \left (\frac{3 a}{b}\right ) \text{Si}\left (3 \left (\frac{a}{b}+\csc ^{-1}(c x)\right )\right )\right )}{4 b} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*ArcCsc[c*x])),x]

[Out]

-(c^3*(Cos[a/b]*CosIntegral[a/b + ArcCsc[c*x]] - Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcCsc[c*x])] + Sin[a/b]*Si
nIntegral[a/b + ArcCsc[c*x]] - Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcCsc[c*x])]))/(4*b)

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Maple [A]  time = 0.218, size = 102, normalized size = 0.9 \begin{align*}{c}^{3} \left ( -{\frac{1}{4\,b}{\it Si} \left ({\frac{a}{b}}+{\rm arccsc} \left (cx\right ) \right ) \sin \left ({\frac{a}{b}} \right ) }-{\frac{1}{4\,b}{\it Ci} \left ({\frac{a}{b}}+{\rm arccsc} \left (cx\right ) \right ) \cos \left ({\frac{a}{b}} \right ) }+{\frac{1}{4\,b}{\it Si} \left ( 3\,{\frac{a}{b}}+3\,{\rm arccsc} \left (cx\right ) \right ) \sin \left ( 3\,{\frac{a}{b}} \right ) }+{\frac{1}{4\,b}{\it Ci} \left ( 3\,{\frac{a}{b}}+3\,{\rm arccsc} \left (cx\right ) \right ) \cos \left ( 3\,{\frac{a}{b}} \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(a+b*arccsc(c*x)),x)

[Out]

c^3*(-1/4*Si(a/b+arccsc(c*x))*sin(a/b)/b-1/4*Ci(a/b+arccsc(c*x))*cos(a/b)/b+1/4*Si(3*a/b+3*arccsc(c*x))*sin(3*
a/b)/b+1/4*Ci(3*a/b+3*arccsc(c*x))*cos(3*a/b)/b)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a+b*arccsc(c*x)),x, algorithm="maxima")

[Out]

integrate(1/((b*arccsc(c*x) + a)*x^4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a+b*arccsc(c*x)),x, algorithm="fricas")

[Out]

integral(1/(b*x^4*arccsc(c*x) + a*x^4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(a+b*acsc(c*x)),x)

[Out]

Integral(1/(x**4*(a + b*acsc(c*x))), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a+b*arccsc(c*x)),x, algorithm="giac")

[Out]

integrate(1/((b*arccsc(c*x) + a)*x^4), x)