3.303 \(\int x \csc ^3(a+2 \log (c x^i)) \, dx\)

Optimal. Leaf size=49 \[ -\frac {i e^{i a} x^2 \left (c x^i\right )^{2 i}}{\left (1-e^{2 i a} \left (c x^i\right )^{4 i}\right )^2} \]

[Out]

-I*exp(I*a)*(c*x^I)^(2*I)*x^2/(1-exp(2*I*a)*(c*x^I)^(4*I))^2

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Rubi [A]  time = 0.04, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4510, 4506, 261} \[ -\frac {i e^{i a} x^2 \left (c x^i\right )^{2 i}}{\left (1-e^{2 i a} \left (c x^i\right )^{4 i}\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[x*Csc[a + 2*Log[c*x^I]]^3,x]

[Out]

((-I)*E^(I*a)*(c*x^I)^(2*I)*x^2)/(1 - E^((2*I)*a)*(c*x^I)^(4*I))^2

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 4506

Int[Csc[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(-2*I)^p*E^(I*a*d*p), Int[(
(e*x)^m*x^(I*b*d*p))/(1 - E^(2*I*a*d)*x^(2*I*b*d))^p, x], x] /; FreeQ[{a, b, d, e, m}, x] && IntegerQ[p]

Rule 4510

Int[Csc[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(e*x)^(m + 1)
/(e*n*(c*x^n)^((m + 1)/n)), Subst[Int[x^((m + 1)/n - 1)*Csc[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a
, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])

Rubi steps

\begin {align*} \int x \csc ^3\left (a+2 \log \left (c x^i\right )\right ) \, dx &=-\left (\left (i \left (c x^i\right )^{2 i} x^2\right ) \operatorname {Subst}\left (\int x^{-1-2 i} \csc ^3(a+2 \log (x)) \, dx,x,c x^i\right )\right )\\ &=\left (8 e^{3 i a} \left (c x^i\right )^{2 i} x^2\right ) \operatorname {Subst}\left (\int \frac {x^{-1+4 i}}{\left (1-e^{2 i a} x^{4 i}\right )^3} \, dx,x,c x^i\right )\\ &=-\frac {i e^{i a} \left (c x^i\right )^{2 i} x^2}{\left (1-e^{2 i a} \left (c x^i\right )^{4 i}\right )^2}\\ \end {align*}

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Mathematica [B]  time = 0.21, size = 127, normalized size = 2.59 \[ \frac {\csc ^2\left (a+2 \log \left (c x^i\right )\right ) \left (\left (2 x^4+1\right ) \sin \left (a+2 \log \left (c x^i\right )-2 i \log (x)\right )+i \left (2 x^4-1\right ) \cos \left (a+2 \log \left (c x^i\right )-2 i \log (x)\right )\right ) \left (i \sin \left (2 \left (a+2 \log \left (c x^i\right )-2 i \log (x)\right )\right )+\cos \left (2 \left (a+2 \log \left (c x^i\right )-2 i \log (x)\right )\right )\right )}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Csc[a + 2*Log[c*x^I]]^3,x]

[Out]

(Csc[a + 2*Log[c*x^I]]^2*(I*(-1 + 2*x^4)*Cos[a + 2*Log[c*x^I] - (2*I)*Log[x]] + (1 + 2*x^4)*Sin[a + 2*Log[c*x^
I] - (2*I)*Log[x]])*(Cos[2*(a + 2*Log[c*x^I] - (2*I)*Log[x])] + I*Sin[2*(a + 2*Log[c*x^I] - (2*I)*Log[x])]))/(
4*x^4)

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fricas [A]  time = 0.88, size = 56, normalized size = 1.14 \[ \frac {-2 i \, x^{4} e^{\left (3 i \, a + 6 i \, \log \relax (c)\right )} + i \, e^{\left (5 i \, a + 10 i \, \log \relax (c)\right )}}{x^{8} - 2 \, x^{4} e^{\left (2 i \, a + 4 i \, \log \relax (c)\right )} + e^{\left (4 i \, a + 8 i \, \log \relax (c)\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*csc(a+2*log(c*x^I))^3,x, algorithm="fricas")

[Out]

(-2*I*x^4*e^(3*I*a + 6*I*log(c)) + I*e^(5*I*a + 10*I*log(c)))/(x^8 - 2*x^4*e^(2*I*a + 4*I*log(c)) + e^(4*I*a +
8*I*log(c)))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \csc \left (a + 2 \, \log \left (c x^{i}\right )\right )^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*csc(a+2*log(c*x^I))^3,x, algorithm="giac")

[Out]

integrate(x*csc(a + 2*log(c*x^I))^3, x)

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maple [C]  time = 0.18, size = 211, normalized size = 4.31 \[ -\frac {i x^{2} c^{2 i} \left (x^{i}\right )^{2 i} {\mathrm e}^{\pi \mathrm {csgn}\left (i c \,x^{i}\right )^{3}-\pi \mathrm {csgn}\left (i c \,x^{i}\right )^{2} \mathrm {csgn}\left (i c \right )-\pi \mathrm {csgn}\left (i c \,x^{i}\right )^{2} \mathrm {csgn}\left (i x^{i}\right )+\pi \,\mathrm {csgn}\left (i c \,x^{i}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{i}\right )+i a}}{\left (\left (x^{i}\right )^{4 i} c^{4 i} {\mathrm e}^{2 \pi \mathrm {csgn}\left (i c \,x^{i}\right )^{3}} {\mathrm e}^{-2 \pi \mathrm {csgn}\left (i c \,x^{i}\right )^{2} \mathrm {csgn}\left (i c \right )} {\mathrm e}^{-2 \pi \mathrm {csgn}\left (i c \,x^{i}\right )^{2} \mathrm {csgn}\left (i x^{i}\right )} {\mathrm e}^{2 \pi \,\mathrm {csgn}\left (i c \,x^{i}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{i}\right )} {\mathrm e}^{2 i a}-1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*csc(a+2*ln(c*x^I))^3,x)

[Out]

-I*x^2*c^(2*I)*(x^I)^(2*I)*exp(Pi*csgn(I*c*x^I)^3-Pi*csgn(I*c*x^I)^2*csgn(I*c)-Pi*csgn(I*c*x^I)^2*csgn(I*x^I)+
Pi*csgn(I*c*x^I)*csgn(I*c)*csgn(I*x^I)+I*a)/(((x^I)^(2*I))^2*(c^(2*I))^2*exp(2*Pi*csgn(I*c*x^I)^3)*exp(-2*Pi*c
sgn(I*c*x^I)^2*csgn(I*c))*exp(-2*Pi*csgn(I*c*x^I)^2*csgn(I*x^I))*exp(2*Pi*csgn(I*c*x^I)*csgn(I*c)*csgn(I*x^I))
*exp(2*I*a)-1)^2

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maxima [B]  time = 0.39, size = 142, normalized size = 2.90 \[ \frac {{\left ({\left (-i \, \cos \relax (a) + \sin \relax (a)\right )} \cos \left (2 \, \log \relax (c)\right ) + {\left (\cos \relax (a) + i \, \sin \relax (a)\right )} \sin \left (2 \, \log \relax (c)\right )\right )} x^{2} e^{\left (6 \, \arctan \left (\sin \left (\log \relax (x)\right ), \cos \left (\log \relax (x)\right )\right )\right )}}{{\left (\cos \left (4 \, a\right ) + i \, \sin \left (4 \, a\right )\right )} \cos \left (8 \, \log \relax (c)\right ) - {\left ({\left (2 \, \cos \left (2 \, a\right ) + 2 i \, \sin \left (2 \, a\right )\right )} \cos \left (4 \, \log \relax (c)\right ) + 2 \, {\left (i \, \cos \left (2 \, a\right ) - \sin \left (2 \, a\right )\right )} \sin \left (4 \, \log \relax (c)\right )\right )} e^{\left (4 \, \arctan \left (\sin \left (\log \relax (x)\right ), \cos \left (\log \relax (x)\right )\right )\right )} + {\left (i \, \cos \left (4 \, a\right ) - \sin \left (4 \, a\right )\right )} \sin \left (8 \, \log \relax (c)\right ) + e^{\left (8 \, \arctan \left (\sin \left (\log \relax (x)\right ), \cos \left (\log \relax (x)\right )\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*csc(a+2*log(c*x^I))^3,x, algorithm="maxima")

[Out]

((-I*cos(a) + sin(a))*cos(2*log(c)) + (cos(a) + I*sin(a))*sin(2*log(c)))*x^2*e^(6*arctan2(sin(log(x)), cos(log
(x))))/((cos(4*a) + I*sin(4*a))*cos(8*log(c)) - ((2*cos(2*a) + 2*I*sin(2*a))*cos(4*log(c)) + 2*(I*cos(2*a) - s
in(2*a))*sin(4*log(c)))*e^(4*arctan2(sin(log(x)), cos(log(x)))) + (I*cos(4*a) - sin(4*a))*sin(8*log(c)) + e^(8
*arctan2(sin(log(x)), cos(log(x)))))

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mupad [B]  time = 4.41, size = 45, normalized size = 0.92 \[ -\frac {x^2\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^{1{}\mathrm {i}}\right )}^{2{}\mathrm {i}}\,1{}\mathrm {i}}{1+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^{1{}\mathrm {i}}\right )}^{8{}\mathrm {i}}-2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^{1{}\mathrm {i}}\right )}^{4{}\mathrm {i}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/sin(a + 2*log(c*x^1i))^3,x)

[Out]

-(x^2*exp(a*1i)*(c*x^1i)^2i*1i)/(exp(a*4i)*(c*x^1i)^8i - 2*exp(a*2i)*(c*x^1i)^4i + 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \csc ^{3}{\left (a + 2 \log {\left (c x^{i} \right )} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*csc(a+2*ln(c*x**I))**3,x)

[Out]

Integral(x*csc(a + 2*log(c*x**I))**3, x)

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