3.1.69 \(\int \frac {1}{\sin ^{\frac {3}{2}}(a-2 i \log (c x))} \, dx\) [69]

Optimal. Leaf size=49 \[ \frac {e^{-2 i a} \left (1-c^4 e^{2 i a} x^4\right )}{2 c^4 x^3 \sin ^{\frac {3}{2}}(a-2 i \log (c x))} \]

[Out]

1/2*(1-c^4*exp(2*I*a)*x^4)/c^4/exp(2*I*a)/x^3/sin(a-2*I*ln(c*x))^(3/2)

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Rubi [A]
time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4571, 4569, 267} \begin {gather*} \frac {e^{-2 i a} \left (1-e^{2 i a} c^4 x^4\right )}{2 c^4 x^3 \sin ^{\frac {3}{2}}(a-2 i \log (c x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[a - (2*I)*Log[c*x]]^(-3/2),x]

[Out]

(1 - c^4*E^((2*I)*a)*x^4)/(2*c^4*E^((2*I)*a)*x^3*Sin[a - (2*I)*Log[c*x]]^(3/2))

Rule 267

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 4569

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

Rule 4571

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

Rubi steps

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

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Mathematica [A]
time = 0.14, size = 81, normalized size = 1.65 \begin {gather*} \frac {x (\cos (a)-i \sin (a)) \sqrt {\frac {-2 i \left (-1+c^4 x^4\right ) \cos (a)+2 \left (1+c^4 x^4\right ) \sin (a)}{c^2 x^2}}}{\left (-1+c^4 x^4\right ) \cos (a)+i \left (1+c^4 x^4\right ) \sin (a)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[a - (2*I)*Log[c*x]]^(-3/2),x]

[Out]

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

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{\sin \left (a -2 i \ln \left (c x \right )\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sin(a-2*I*ln(c*x))^(3/2),x)

[Out]

int(1/sin(a-2*I*ln(c*x))^(3/2),x)

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 402 vs. \(2 (36) = 72\).
time = 0.62, size = 402, normalized size = 8.20 \begin {gather*} \frac {{\left ({\left (\cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right )} c^{4} x^{4} + 2 \, c^{2} x^{2} \cos \left (a\right ) + 1\right )}^{\frac {1}{4}} {\left ({\left (\cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right )} c^{4} x^{4} - 2 \, c^{2} x^{2} \cos \left (a\right ) + 1\right )}^{\frac {1}{4}} {\left ({\left ({\left (c^{4} x^{4} {\left (\left (i + 1\right ) \, \cos \left (\frac {3}{2} \, a\right ) + \left (i - 1\right ) \, \sin \left (\frac {3}{2} \, a\right )\right )} - \left (i + 1\right ) \, \cos \left (\frac {1}{2} \, a\right ) + \left (i - 1\right ) \, \sin \left (\frac {1}{2} \, a\right )\right )} \cos \left (\frac {3}{2} \, \arctan \left (c^{2} x^{2} \sin \left (a\right ), -c^{2} x^{2} \cos \left (a\right ) + 1\right )\right ) + {\left (c^{4} x^{4} {\left (\left (i - 1\right ) \, \cos \left (\frac {3}{2} \, a\right ) - \left (i + 1\right ) \, \sin \left (\frac {3}{2} \, a\right )\right )} - \left (i - 1\right ) \, \cos \left (\frac {1}{2} \, a\right ) - \left (i + 1\right ) \, \sin \left (\frac {1}{2} \, a\right )\right )} \sin \left (\frac {3}{2} \, \arctan \left (c^{2} x^{2} \sin \left (a\right ), -c^{2} x^{2} \cos \left (a\right ) + 1\right )\right )\right )} \cos \left (\frac {3}{2} \, \arctan \left (c^{2} x^{2} \sin \left (a\right ), c^{2} x^{2} \cos \left (a\right ) + 1\right )\right ) + {\left ({\left (c^{4} x^{4} {\left (-\left (i - 1\right ) \, \cos \left (\frac {3}{2} \, a\right ) + \left (i + 1\right ) \, \sin \left (\frac {3}{2} \, a\right )\right )} + \left (i - 1\right ) \, \cos \left (\frac {1}{2} \, a\right ) + \left (i + 1\right ) \, \sin \left (\frac {1}{2} \, a\right )\right )} \cos \left (\frac {3}{2} \, \arctan \left (c^{2} x^{2} \sin \left (a\right ), -c^{2} x^{2} \cos \left (a\right ) + 1\right )\right ) + {\left (c^{4} x^{4} {\left (\left (i + 1\right ) \, \cos \left (\frac {3}{2} \, a\right ) + \left (i - 1\right ) \, \sin \left (\frac {3}{2} \, a\right )\right )} - \left (i + 1\right ) \, \cos \left (\frac {1}{2} \, a\right ) + \left (i - 1\right ) \, \sin \left (\frac {1}{2} \, a\right )\right )} \sin \left (\frac {3}{2} \, \arctan \left (c^{2} x^{2} \sin \left (a\right ), -c^{2} x^{2} \cos \left (a\right ) + 1\right )\right )\right )} \sin \left (\frac {3}{2} \, \arctan \left (c^{2} x^{2} \sin \left (a\right ), c^{2} x^{2} \cos \left (a\right ) + 1\right )\right )\right )}}{{\left ({\left (\cos \left (a\right )^{4} + 2 \, \cos \left (a\right )^{2} \sin \left (a\right )^{2} + \sin \left (a\right )^{4}\right )} c^{8} x^{8} - 2 \, {\left (\cos \left (a\right )^{2} - \sin \left (a\right )^{2}\right )} c^{4} x^{4} + 1\right )} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(a-2*I*log(c*x))^(3/2),x, algorithm="maxima")

[Out]

((cos(a)^2 + sin(a)^2)*c^4*x^4 + 2*c^2*x^2*cos(a) + 1)^(1/4)*((cos(a)^2 + sin(a)^2)*c^4*x^4 - 2*c^2*x^2*cos(a)
+ 1)^(1/4)*(((c^4*x^4*((I + 1)*cos(3/2*a) + (I - 1)*sin(3/2*a)) - (I + 1)*cos(1/2*a) + (I - 1)*sin(1/2*a))*co
s(3/2*arctan2(c^2*x^2*sin(a), -c^2*x^2*cos(a) + 1)) + (c^4*x^4*((I - 1)*cos(3/2*a) - (I + 1)*sin(3/2*a)) - (I
- 1)*cos(1/2*a) - (I + 1)*sin(1/2*a))*sin(3/2*arctan2(c^2*x^2*sin(a), -c^2*x^2*cos(a) + 1)))*cos(3/2*arctan2(c
^2*x^2*sin(a), c^2*x^2*cos(a) + 1)) + ((c^4*x^4*(-(I - 1)*cos(3/2*a) + (I + 1)*sin(3/2*a)) + (I - 1)*cos(1/2*a
) + (I + 1)*sin(1/2*a))*cos(3/2*arctan2(c^2*x^2*sin(a), -c^2*x^2*cos(a) + 1)) + (c^4*x^4*((I + 1)*cos(3/2*a) +
(I - 1)*sin(3/2*a)) - (I + 1)*cos(1/2*a) + (I - 1)*sin(1/2*a))*sin(3/2*arctan2(c^2*x^2*sin(a), -c^2*x^2*cos(a
) + 1)))*sin(3/2*arctan2(c^2*x^2*sin(a), c^2*x^2*cos(a) + 1)))/(((cos(a)^4 + 2*cos(a)^2*sin(a)^2 + sin(a)^4)*c
^8*x^8 - 2*(cos(a)^2 - sin(a)^2)*c^4*x^4 + 1)*c)

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Fricas [A]
time = 0.80, size = 43, normalized size = 0.88 \begin {gather*} \frac {2 \, \sqrt {\frac {1}{2}} \sqrt {-i \, c^{4} x^{4} + i \, e^{\left (-2 i \, a\right )}} e^{\left (-\frac {3}{2} i \, a\right )}}{c^{5} x^{4} - c e^{\left (-2 i \, a\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(a-2*I*log(c*x))^(3/2),x, algorithm="fricas")

[Out]

2*sqrt(1/2)*sqrt(-I*c^4*x^4 + I*e^(-2*I*a))*e^(-3/2*I*a)/(c^5*x^4 - c*e^(-2*I*a))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sin ^{\frac {3}{2}}{\left (a - 2 i \log {\left (c x \right )} \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(a-2*I*ln(c*x))**(3/2),x)

[Out]

Integral(sin(a - 2*I*log(c*x))**(-3/2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(a-2*I*log(c*x))^(3/2),x, algorithm="giac")

[Out]

integrate(sin(a - 2*I*log(c*x))^(-3/2), x)

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Mupad [B]
time = 2.94, size = 50, normalized size = 1.02 \begin {gather*} \frac {2\,x\,\sqrt {\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,c^2\,x^2}-\frac {c^2\,x^2\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}}}{c^4\,x^4\,{\mathrm {e}}^{a\,2{}\mathrm {i}}-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sin(a - log(c*x)*2i)^(3/2),x)

[Out]

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

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