3.593 \(\int \frac{x^4 \csc ^2(a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx\)

Optimal. Leaf size=127 \[ -\frac{2 i \text{PolyLog}\left (2,e^{2 i a x}\right )}{a^5}-\frac{2 i x^2}{a^3}-\frac{2 x^2 \cot (a x)}{a^3}-\frac{x^2 \cot (a x) \csc ^2(a x)}{a^3}+\frac{x^3 \csc ^3(a x)}{a^2 (a x \cos (a x)-\sin (a x))}+\frac{4 x \log \left (1-e^{2 i a x}\right )}{a^4}-\frac{\cot (a x)}{a^5}-\frac{x \csc ^2(a x)}{a^4} \]

[Out]

((-2*I)*x^2)/a^3 - Cot[a*x]/a^5 - (2*x^2*Cot[a*x])/a^3 - (x*Csc[a*x]^2)/a^4 - (x^2*Cot[a*x]*Csc[a*x]^2)/a^3 +
(4*x*Log[1 - E^((2*I)*a*x)])/a^4 - ((2*I)*PolyLog[2, E^((2*I)*a*x)])/a^5 + (x^3*Csc[a*x]^3)/(a^2*(a*x*Cos[a*x]
- Sin[a*x]))

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Rubi [A]  time = 0.181463, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {4600, 4186, 3767, 8, 4184, 3717, 2190, 2279, 2391} \[ -\frac{2 i \text{PolyLog}\left (2,e^{2 i a x}\right )}{a^5}-\frac{2 i x^2}{a^3}-\frac{2 x^2 \cot (a x)}{a^3}-\frac{x^2 \cot (a x) \csc ^2(a x)}{a^3}+\frac{x^3 \csc ^3(a x)}{a^2 (a x \cos (a x)-\sin (a x))}+\frac{4 x \log \left (1-e^{2 i a x}\right )}{a^4}-\frac{\cot (a x)}{a^5}-\frac{x \csc ^2(a x)}{a^4} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*Csc[a*x]^2)/(a*x*Cos[a*x] - Sin[a*x])^2,x]

[Out]

((-2*I)*x^2)/a^3 - Cot[a*x]/a^5 - (2*x^2*Cot[a*x])/a^3 - (x*Csc[a*x]^2)/a^4 - (x^2*Cot[a*x]*Csc[a*x]^2)/a^3 +
(4*x*Log[1 - E^((2*I)*a*x)])/a^4 - ((2*I)*PolyLog[2, E^((2*I)*a*x)])/a^5 + (x^3*Csc[a*x]^3)/(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 4186

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

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 4184

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

Rule 3717

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[(I*(c + d*x)^(m + 1))/(d*
(m + 1)), x] - Dist[2*I, Int[((c + d*x)^m*E^(2*I*k*Pi)*E^(2*I*(e + f*x)))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, 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^4 \csc ^2(a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx &=\frac{x^3 \csc ^3(a x)}{a^2 (a x \cos (a x)-\sin (a x))}+\frac{3 \int x^2 \csc ^4(a x) \, dx}{a^2}\\ &=-\frac{x \csc ^2(a x)}{a^4}-\frac{x^2 \cot (a x) \csc ^2(a x)}{a^3}+\frac{x^3 \csc ^3(a x)}{a^2 (a x \cos (a x)-\sin (a x))}+\frac{\int \csc ^2(a x) \, dx}{a^4}+\frac{2 \int x^2 \csc ^2(a x) \, dx}{a^2}\\ &=-\frac{2 x^2 \cot (a x)}{a^3}-\frac{x \csc ^2(a x)}{a^4}-\frac{x^2 \cot (a x) \csc ^2(a x)}{a^3}+\frac{x^3 \csc ^3(a x)}{a^2 (a x \cos (a x)-\sin (a x))}-\frac{\operatorname{Subst}(\int 1 \, dx,x,\cot (a x))}{a^5}+\frac{4 \int x \cot (a x) \, dx}{a^3}\\ &=-\frac{2 i x^2}{a^3}-\frac{\cot (a x)}{a^5}-\frac{2 x^2 \cot (a x)}{a^3}-\frac{x \csc ^2(a x)}{a^4}-\frac{x^2 \cot (a x) \csc ^2(a x)}{a^3}+\frac{x^3 \csc ^3(a x)}{a^2 (a x \cos (a x)-\sin (a x))}-\frac{(8 i) \int \frac{e^{2 i a x} x}{1-e^{2 i a x}} \, dx}{a^3}\\ &=-\frac{2 i x^2}{a^3}-\frac{\cot (a x)}{a^5}-\frac{2 x^2 \cot (a x)}{a^3}-\frac{x \csc ^2(a x)}{a^4}-\frac{x^2 \cot (a x) \csc ^2(a x)}{a^3}+\frac{4 x \log \left (1-e^{2 i a x}\right )}{a^4}+\frac{x^3 \csc ^3(a x)}{a^2 (a x \cos (a x)-\sin (a x))}-\frac{4 \int \log \left (1-e^{2 i a x}\right ) \, dx}{a^4}\\ &=-\frac{2 i x^2}{a^3}-\frac{\cot (a x)}{a^5}-\frac{2 x^2 \cot (a x)}{a^3}-\frac{x \csc ^2(a x)}{a^4}-\frac{x^2 \cot (a x) \csc ^2(a x)}{a^3}+\frac{4 x \log \left (1-e^{2 i a x}\right )}{a^4}+\frac{x^3 \csc ^3(a x)}{a^2 (a x \cos (a x)-\sin (a x))}+\frac{(2 i) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i a x}\right )}{a^5}\\ &=-\frac{2 i x^2}{a^3}-\frac{\cot (a x)}{a^5}-\frac{2 x^2 \cot (a x)}{a^3}-\frac{x \csc ^2(a x)}{a^4}-\frac{x^2 \cot (a x) \csc ^2(a x)}{a^3}+\frac{4 x \log \left (1-e^{2 i a x}\right )}{a^4}-\frac{2 i \text{Li}_2\left (e^{2 i a x}\right )}{a^5}+\frac{x^3 \csc ^3(a x)}{a^2 (a x \cos (a x)-\sin (a x))}\\ \end{align*}

Mathematica [A]  time = 1.06375, size = 102, normalized size = 0.8 \[ \frac{-2 i a \left (a^2 x^2+\text{PolyLog}\left (2,e^{2 i a x}\right )\right )+a^3 \left (-x^2\right ) \cot (a x)+\frac{\left (a^2 x^2+1\right )^2 \sin (a x)}{x (a x \cos (a x)-\sin (a x))}+a^2 x+4 a^2 x \log \left (1-e^{2 i a x}\right )+\frac{1}{x}}{a^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*Csc[a*x]^2)/(a*x*Cos[a*x] - Sin[a*x])^2,x]

[Out]

(x^(-1) + a^2*x - a^3*x^2*Cot[a*x] + 4*a^2*x*Log[1 - E^((2*I)*a*x)] - (2*I)*a*(a^2*x^2 + PolyLog[2, E^((2*I)*a
*x)]) + ((1 + a^2*x^2)^2*Sin[a*x])/(x*(a*x*Cos[a*x] - Sin[a*x])))/a^6

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Maple [A]  time = 0.357, size = 172, normalized size = 1.4 \begin{align*}{\frac{-2\,i \left ( 2\,i{a}^{2}{x}^{2}{{\rm e}^{2\,iax}}+2\,{x}^{3}{a}^{3}-2\,i{x}^{2}{a}^{2}-ax{{\rm e}^{2\,iax}}+i{{\rm e}^{2\,iax}}+ax-i \right ) }{ \left ({{\rm e}^{2\,iax}}-1 \right ) \left ( ax{{\rm e}^{2\,iax}}+i{{\rm e}^{2\,iax}}+ax-i \right ){a}^{5}}}-{\frac{4\,i{x}^{2}}{{a}^{3}}}+4\,{\frac{x\ln \left ({{\rm e}^{iax}}+1 \right ) }{{a}^{4}}}-{\frac{4\,i{\it polylog} \left ( 2,-{{\rm e}^{iax}} \right ) }{{a}^{5}}}+4\,{\frac{x\ln \left ( 1-{{\rm e}^{iax}} \right ) }{{a}^{4}}}-{\frac{4\,i{\it polylog} \left ( 2,{{\rm e}^{iax}} \right ) }{{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*csc(a*x)^2/(a*x*cos(a*x)-sin(a*x))^2,x)

[Out]

-2*I*(2*I*a^2*x^2*exp(2*I*a*x)+2*x^3*a^3-2*I*a^2*x^2-a*x*exp(2*I*a*x)+I*exp(2*I*a*x)+a*x-I)/(exp(2*I*a*x)-1)/(
a*x*exp(2*I*a*x)+I*exp(2*I*a*x)+a*x-I)/a^5-4*I/a^3*x^2+4/a^4*x*ln(exp(I*a*x)+1)-4*I/a^5*polylog(2,-exp(I*a*x))
+4/a^4*x*ln(1-exp(I*a*x))-4*I/a^5*polylog(2,exp(I*a*x))

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Maxima [B]  time = 1.2702, size = 821, normalized size = 6.46 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*csc(a*x)^2/(a*x*cos(a*x)-sin(a*x))^2,x, algorithm="maxima")

[Out]

-(2*a*x + (4*a^2*x^2 + 8*I*a*x*cos(2*a*x) - 8*a*x*sin(2*a*x) - 4*I*a*x - (4*a^2*x^2 + 4*I*a*x)*cos(4*a*x) + 4*
(-I*a^2*x^2 + a*x)*sin(4*a*x))*arctan2(sin(a*x), cos(a*x) + 1) - (4*a^2*x^2 + 8*I*a*x*cos(2*a*x) - 8*a*x*sin(2
*a*x) - 4*I*a*x - (4*a^2*x^2 + 4*I*a*x)*cos(4*a*x) - 4*(I*a^2*x^2 - a*x)*sin(4*a*x))*arctan2(sin(a*x), -cos(a*
x) + 1) + 4*(a^3*x^3 + I*a^2*x^2)*cos(4*a*x) - (4*I*a^2*x^2 + 2*a*x - 2*I)*cos(2*a*x) - (4*a*x - (4*a*x + 4*I)
*cos(4*a*x) - 4*(I*a*x - 1)*sin(4*a*x) + 8*I*cos(2*a*x) - 8*sin(2*a*x) - 4*I)*dilog(-e^(I*a*x)) - (4*a*x - (4*
a*x + 4*I)*cos(4*a*x) - 4*(I*a*x - 1)*sin(4*a*x) + 8*I*cos(2*a*x) - 8*sin(2*a*x) - 4*I)*dilog(e^(I*a*x)) - (2*
I*a^2*x^2 - 4*a*x*cos(2*a*x) - 4*I*a*x*sin(2*a*x) + 2*a*x - 2*(I*a^2*x^2 - a*x)*cos(4*a*x) + (2*a^2*x^2 + 2*I*
a*x)*sin(4*a*x))*log(cos(a*x)^2 + sin(a*x)^2 + 2*cos(a*x) + 1) - (2*I*a^2*x^2 - 4*a*x*cos(2*a*x) - 4*I*a*x*sin
(2*a*x) + 2*a*x - 2*(I*a^2*x^2 - a*x)*cos(4*a*x) + (2*a^2*x^2 + 2*I*a*x)*sin(4*a*x))*log(cos(a*x)^2 + sin(a*x)
^2 - 2*cos(a*x) + 1) - (-4*I*a^3*x^3 + 4*a^2*x^2)*sin(4*a*x) + (4*a^2*x^2 - 2*I*a*x - 2)*sin(2*a*x) - 2*I)/((I
*a*x + (-I*a*x + 1)*cos(4*a*x) + (a*x + I)*sin(4*a*x) - 2*cos(2*a*x) - 2*I*sin(2*a*x) + 1)*a^5)

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Fricas [B]  time = 2.43328, size = 1095, normalized size = 8.62 \begin{align*} \frac{a^{3} x^{3} -{\left (2 \, a^{3} x^{3} + a x\right )} \cos \left (a x\right )^{2} +{\left (2 \, a^{2} x^{2} + 1\right )} \cos \left (a x\right ) \sin \left (a x\right ) + a x +{\left (-2 i \, a x \cos \left (a x\right ) \sin \left (a x\right ) - 2 i \, \cos \left (a x\right )^{2} + 2 i\right )}{\rm Li}_2\left (\cos \left (a x\right ) + i \, \sin \left (a x\right )\right ) +{\left (2 i \, a x \cos \left (a x\right ) \sin \left (a x\right ) + 2 i \, \cos \left (a x\right )^{2} - 2 i\right )}{\rm Li}_2\left (\cos \left (a x\right ) - i \, \sin \left (a x\right )\right ) +{\left (2 i \, a x \cos \left (a x\right ) \sin \left (a x\right ) + 2 i \, \cos \left (a x\right )^{2} - 2 i\right )}{\rm Li}_2\left (-\cos \left (a x\right ) + i \, \sin \left (a x\right )\right ) +{\left (-2 i \, a x \cos \left (a x\right ) \sin \left (a x\right ) - 2 i \, \cos \left (a x\right )^{2} + 2 i\right )}{\rm Li}_2\left (-\cos \left (a x\right ) - i \, \sin \left (a x\right )\right ) + 2 \,{\left (a^{2} x^{2} \cos \left (a x\right ) \sin \left (a x\right ) + a x \cos \left (a x\right )^{2} - a x\right )} \log \left (\cos \left (a x\right ) + i \, \sin \left (a x\right ) + 1\right ) + 2 \,{\left (a^{2} x^{2} \cos \left (a x\right ) \sin \left (a x\right ) + a x \cos \left (a x\right )^{2} - a x\right )} \log \left (\cos \left (a x\right ) - i \, \sin \left (a x\right ) + 1\right ) + 2 \,{\left (a^{2} x^{2} \cos \left (a x\right ) \sin \left (a x\right ) + a x \cos \left (a x\right )^{2} - a x\right )} \log \left (-\cos \left (a x\right ) + i \, \sin \left (a x\right ) + 1\right ) + 2 \,{\left (a^{2} x^{2} \cos \left (a x\right ) \sin \left (a x\right ) + a x \cos \left (a x\right )^{2} - a x\right )} \log \left (-\cos \left (a x\right ) - i \, \sin \left (a x\right ) + 1\right )}{a^{6} x \cos \left (a x\right ) \sin \left (a x\right ) + a^{5} \cos \left (a x\right )^{2} - a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*csc(a*x)^2/(a*x*cos(a*x)-sin(a*x))^2,x, algorithm="fricas")

[Out]

(a^3*x^3 - (2*a^3*x^3 + a*x)*cos(a*x)^2 + (2*a^2*x^2 + 1)*cos(a*x)*sin(a*x) + a*x + (-2*I*a*x*cos(a*x)*sin(a*x
) - 2*I*cos(a*x)^2 + 2*I)*dilog(cos(a*x) + I*sin(a*x)) + (2*I*a*x*cos(a*x)*sin(a*x) + 2*I*cos(a*x)^2 - 2*I)*di
log(cos(a*x) - I*sin(a*x)) + (2*I*a*x*cos(a*x)*sin(a*x) + 2*I*cos(a*x)^2 - 2*I)*dilog(-cos(a*x) + I*sin(a*x))
+ (-2*I*a*x*cos(a*x)*sin(a*x) - 2*I*cos(a*x)^2 + 2*I)*dilog(-cos(a*x) - I*sin(a*x)) + 2*(a^2*x^2*cos(a*x)*sin(
a*x) + a*x*cos(a*x)^2 - a*x)*log(cos(a*x) + I*sin(a*x) + 1) + 2*(a^2*x^2*cos(a*x)*sin(a*x) + a*x*cos(a*x)^2 -
a*x)*log(cos(a*x) - I*sin(a*x) + 1) + 2*(a^2*x^2*cos(a*x)*sin(a*x) + a*x*cos(a*x)^2 - a*x)*log(-cos(a*x) + I*s
in(a*x) + 1) + 2*(a^2*x^2*cos(a*x)*sin(a*x) + a*x*cos(a*x)^2 - a*x)*log(-cos(a*x) - I*sin(a*x) + 1))/(a^6*x*co
s(a*x)*sin(a*x) + a^5*cos(a*x)^2 - a^5)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*csc(a*x)**2/(a*x*cos(a*x)-sin(a*x))**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \csc \left (a x\right )^{2}}{{\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^4*csc(a*x)^2/(a*x*cos(a*x)-sin(a*x))^2,x, algorithm="giac")

[Out]

integrate(x^4*csc(a*x)^2/(a*x*cos(a*x) - sin(a*x))^2, x)