3.592 \(\int \frac{x^3 \csc (a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx\)

Optimal. Leaf size=104 \[ \frac{i \text{PolyLog}\left (2,-e^{i a x}\right )}{a^4}-\frac{i \text{PolyLog}\left (2,e^{i a x}\right )}{a^4}+\frac{x^2 \csc ^2(a x)}{a^2 (a x \cos (a x)-\sin (a x))}-\frac{\csc (a x)}{a^4}-\frac{2 x \tanh ^{-1}\left (e^{i a x}\right )}{a^3}-\frac{x \cot (a x) \csc (a x)}{a^3} \]

[Out]

(-2*x*ArcTanh[E^(I*a*x)])/a^3 - Csc[a*x]/a^4 - (x*Cot[a*x]*Csc[a*x])/a^3 + (I*PolyLog[2, -E^(I*a*x)])/a^4 - (I
*PolyLog[2, E^(I*a*x)])/a^4 + (x^2*Csc[a*x]^2)/(a^2*(a*x*Cos[a*x] - Sin[a*x]))

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Rubi [A]  time = 0.0912974, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4600, 4185, 4183, 2279, 2391} \[ \frac{i \text{PolyLog}\left (2,-e^{i a x}\right )}{a^4}-\frac{i \text{PolyLog}\left (2,e^{i a x}\right )}{a^4}+\frac{x^2 \csc ^2(a x)}{a^2 (a x \cos (a x)-\sin (a x))}-\frac{\csc (a x)}{a^4}-\frac{2 x \tanh ^{-1}\left (e^{i a x}\right )}{a^3}-\frac{x \cot (a x) \csc (a x)}{a^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*Csc[a*x])/(a*x*Cos[a*x] - Sin[a*x])^2,x]

[Out]

(-2*x*ArcTanh[E^(I*a*x)])/a^3 - Csc[a*x]/a^4 - (x*Cot[a*x]*Csc[a*x])/a^3 + (I*PolyLog[2, -E^(I*a*x)])/a^4 - (I
*PolyLog[2, E^(I*a*x)])/a^4 + (x^2*Csc[a*x]^2)/(a^2*(a*x*Cos[a*x] - Sin[a*x]))

Rule 4600

Int[(Csc[(a_.)*(x_)]^(n_.)*((b_.)*(x_))^(m_.))/(Cos[(a_.)*(x_)]*(d_.)*(x_) + (c_.)*Sin[(a_.)*(x_)])^2, x_Symbo
l] :> Simp[(b*(b*x)^(m - 1)*Csc[a*x]^(n + 1))/(a*d*(c*Sin[a*x] + d*x*Cos[a*x])), x] + Dist[(b^2*(n + 1))/d^2,
Int[(b*x)^(m - 2)*Csc[a*x]^(n + 2), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[a*c + d, 0] && EqQ[m, n + 2]

Rule 4185

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> -Simp[(b^2*(c + d*x)*Cot[e + f*x]*
(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2
), x], x] - Simp[(b^2*d*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[{b, c, d, e, f}, x] && G
tQ[n, 1] && NeQ[n, 2]

Rule 4183

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E^(I*(e + f*
x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[(d*m)/f, Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{x^3 \csc (a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx &=\frac{x^2 \csc ^2(a x)}{a^2 (a x \cos (a x)-\sin (a x))}+\frac{2 \int x \csc ^3(a x) \, dx}{a^2}\\ &=-\frac{\csc (a x)}{a^4}-\frac{x \cot (a x) \csc (a x)}{a^3}+\frac{x^2 \csc ^2(a x)}{a^2 (a x \cos (a x)-\sin (a x))}+\frac{\int x \csc (a x) \, dx}{a^2}\\ &=-\frac{2 x \tanh ^{-1}\left (e^{i a x}\right )}{a^3}-\frac{\csc (a x)}{a^4}-\frac{x \cot (a x) \csc (a x)}{a^3}+\frac{x^2 \csc ^2(a x)}{a^2 (a x \cos (a x)-\sin (a x))}-\frac{\int \log \left (1-e^{i a x}\right ) \, dx}{a^3}+\frac{\int \log \left (1+e^{i a x}\right ) \, dx}{a^3}\\ &=-\frac{2 x \tanh ^{-1}\left (e^{i a x}\right )}{a^3}-\frac{\csc (a x)}{a^4}-\frac{x \cot (a x) \csc (a x)}{a^3}+\frac{x^2 \csc ^2(a x)}{a^2 (a x \cos (a x)-\sin (a x))}+\frac{i \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i a x}\right )}{a^4}-\frac{i \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i a x}\right )}{a^4}\\ &=-\frac{2 x \tanh ^{-1}\left (e^{i a x}\right )}{a^3}-\frac{\csc (a x)}{a^4}-\frac{x \cot (a x) \csc (a x)}{a^3}+\frac{i \text{Li}_2\left (-e^{i a x}\right )}{a^4}-\frac{i \text{Li}_2\left (e^{i a x}\right )}{a^4}+\frac{x^2 \csc ^2(a x)}{a^2 (a x \cos (a x)-\sin (a x))}\\ \end{align*}

Mathematica [A]  time = 1.01867, size = 157, normalized size = 1.51 \[ \frac{i (a x \cot (a x)-1) \text{PolyLog}\left (2,-e^{i a x}\right )-i (a x \cot (a x)-1) \text{PolyLog}\left (2,e^{i a x}\right )+a^2 x^2 \csc (a x)+a^2 x^2 \log \left (1-e^{i a x}\right ) \cot (a x)-a^2 x^2 \log \left (1+e^{i a x}\right ) \cot (a x)-a x \log \left (1-e^{i a x}\right )+a x \log \left (1+e^{i a x}\right )+\csc (a x)}{a^4 (a x \cot (a x)-1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*Csc[a*x])/(a*x*Cos[a*x] - Sin[a*x])^2,x]

[Out]

(Csc[a*x] + a^2*x^2*Csc[a*x] - a*x*Log[1 - E^(I*a*x)] + a^2*x^2*Cot[a*x]*Log[1 - E^(I*a*x)] + a*x*Log[1 + E^(I
*a*x)] - a^2*x^2*Cot[a*x]*Log[1 + E^(I*a*x)] + I*(-1 + a*x*Cot[a*x])*PolyLog[2, -E^(I*a*x)] - I*(-1 + a*x*Cot[
a*x])*PolyLog[2, E^(I*a*x)])/(a^4*(-1 + a*x*Cot[a*x]))

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Maple [F]  time = 1.249, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}\csc \left ( ax \right ) }{ \left ( ax\cos \left ( ax \right ) -\sin \left ( ax \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*csc(a*x)/(a*x*cos(a*x)-sin(a*x))^2,x)

[Out]

int(x^3*csc(a*x)/(a*x*cos(a*x)-sin(a*x))^2,x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*csc(a*x)/(a*x*cos(a*x)-sin(a*x))^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [B]  time = 2.33218, size = 771, normalized size = 7.41 \begin{align*} \frac{2 \, a^{2} x^{2} -{\left (i \, a x \cos \left (a x\right ) - i \, \sin \left (a x\right )\right )}{\rm Li}_2\left (\cos \left (a x\right ) + i \, \sin \left (a x\right )\right ) -{\left (-i \, a x \cos \left (a x\right ) + i \, \sin \left (a x\right )\right )}{\rm Li}_2\left (\cos \left (a x\right ) - i \, \sin \left (a x\right )\right ) -{\left (i \, a x \cos \left (a x\right ) - i \, \sin \left (a x\right )\right )}{\rm Li}_2\left (-\cos \left (a x\right ) + i \, \sin \left (a x\right )\right ) -{\left (-i \, a x \cos \left (a x\right ) + i \, \sin \left (a x\right )\right )}{\rm Li}_2\left (-\cos \left (a x\right ) - i \, \sin \left (a x\right )\right ) -{\left (a^{2} x^{2} \cos \left (a x\right ) - a x \sin \left (a x\right )\right )} \log \left (\cos \left (a x\right ) + i \, \sin \left (a x\right ) + 1\right ) -{\left (a^{2} x^{2} \cos \left (a x\right ) - a x \sin \left (a x\right )\right )} \log \left (\cos \left (a x\right ) - i \, \sin \left (a x\right ) + 1\right ) +{\left (a^{2} x^{2} \cos \left (a x\right ) - a x \sin \left (a x\right )\right )} \log \left (-\cos \left (a x\right ) + i \, \sin \left (a x\right ) + 1\right ) +{\left (a^{2} x^{2} \cos \left (a x\right ) - a x \sin \left (a x\right )\right )} \log \left (-\cos \left (a x\right ) - i \, \sin \left (a x\right ) + 1\right ) + 2}{2 \,{\left (a^{5} x \cos \left (a x\right ) - a^{4} \sin \left (a x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*csc(a*x)/(a*x*cos(a*x)-sin(a*x))^2,x, algorithm="fricas")

[Out]

1/2*(2*a^2*x^2 - (I*a*x*cos(a*x) - I*sin(a*x))*dilog(cos(a*x) + I*sin(a*x)) - (-I*a*x*cos(a*x) + I*sin(a*x))*d
ilog(cos(a*x) - I*sin(a*x)) - (I*a*x*cos(a*x) - I*sin(a*x))*dilog(-cos(a*x) + I*sin(a*x)) - (-I*a*x*cos(a*x) +
I*sin(a*x))*dilog(-cos(a*x) - I*sin(a*x)) - (a^2*x^2*cos(a*x) - a*x*sin(a*x))*log(cos(a*x) + I*sin(a*x) + 1)
- (a^2*x^2*cos(a*x) - a*x*sin(a*x))*log(cos(a*x) - I*sin(a*x) + 1) + (a^2*x^2*cos(a*x) - a*x*sin(a*x))*log(-co
s(a*x) + I*sin(a*x) + 1) + (a^2*x^2*cos(a*x) - a*x*sin(a*x))*log(-cos(a*x) - I*sin(a*x) + 1) + 2)/(a^5*x*cos(a
*x) - a^4*sin(a*x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \csc{\left (a x \right )}}{\left (a x \cos{\left (a x \right )} - \sin{\left (a x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*csc(a*x)/(a*x*cos(a*x)-sin(a*x))**2,x)

[Out]

Integral(x**3*csc(a*x)/(a*x*cos(a*x) - sin(a*x))**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*csc(a*x)/(a*x*cos(a*x)-sin(a*x))^2,x, algorithm="giac")

[Out]

integrate(x^3*csc(a*x)/(a*x*cos(a*x) - sin(a*x))^2, x)