3.263 \(\int \sec ^3(a+2 \log (c x^{-\frac{i}{2}})) \, dx\)

Optimal. Leaf size=48 \[ \frac{2 e^{3 i a} x \left (c x^{-\frac{i}{2}}\right )^{6 i}}{\left (1+e^{2 i a} \left (c x^{-\frac{i}{2}}\right )^{4 i}\right )^2} \]

[Out]

(2*E^((3*I)*a)*(c/x^(I/2))^(6*I)*x)/(1 + E^((2*I)*a)*(c/x^(I/2))^(4*I))^2

________________________________________________________________________________________

Rubi [A]  time = 0.0406423, antiderivative size = 48, normalized size of antiderivative = 1., 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 = {4503, 4505, 264} \[ \frac{2 e^{3 i a} x \left (c x^{-\frac{i}{2}}\right )^{6 i}}{\left (1+e^{2 i a} \left (c x^{-\frac{i}{2}}\right )^{4 i}\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[Sec[a + 2*Log[c/x^(I/2)]]^3,x]

[Out]

(2*E^((3*I)*a)*(c/x^(I/2))^(6*I)*x)/(1 + E^((2*I)*a)*(c/x^(I/2))^(4*I))^2

Rule 4503

Int[Sec[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[x
^(1/n - 1)*Sec[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])

Rule 4505

Int[((e_.)*(x_))^(m_.)*Sec[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[2^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 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \sec ^3\left (a+2 \log \left (c x^{-\frac{i}{2}}\right )\right ) \, dx &=\left (2 i \left (c x^{-\frac{i}{2}}\right )^{-2 i} x\right ) \operatorname{Subst}\left (\int x^{-1+2 i} \sec ^3(a+2 \log (x)) \, dx,x,c x^{-\frac{i}{2}}\right )\\ &=\left (16 i e^{3 i a} \left (c x^{-\frac{i}{2}}\right )^{-2 i} x\right ) \operatorname{Subst}\left (\int \frac{x^{-1+8 i}}{\left (1+e^{2 i a} x^{4 i}\right )^3} \, dx,x,c x^{-\frac{i}{2}}\right )\\ &=\frac{2 e^{3 i a} \left (c x^{-\frac{i}{2}}\right )^{6 i} x}{\left (1+e^{2 i a} \left (c x^{-\frac{i}{2}}\right )^{4 i}\right )^2}\\ \end{align*}

Mathematica [B]  time = 0.139061, size = 139, normalized size = 2.9 \[ \frac{\sec ^2\left (a+2 \log \left (c x^{-\frac{i}{2}}\right )\right ) \left (i \left (2 x^2-1\right ) \sin \left (a+2 \log \left (c x^{-\frac{i}{2}}\right )+i \log (x)\right )+\left (2 x^2+1\right ) \cos \left (a+2 \log \left (c x^{-\frac{i}{2}}\right )+i \log (x)\right )\right ) \left (2 i \sin \left (2 \left (a+2 \log \left (c x^{-\frac{i}{2}}\right )+i \log (x)\right )\right )-2 \cos \left (2 \left (a+2 \log \left (c x^{-\frac{i}{2}}\right )+i \log (x)\right )\right )\right )}{4 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 0.204, size = 238, normalized size = 5. \begin{align*} 2\,{x{{\rm e}^{-3\,i \left ( i\pi \, \left ({\it csgn} \left ({\frac{ic}{{x}^{i/2}}} \right ) \right ) ^{3}-i\pi \, \left ({\it csgn} \left ({\frac{ic}{{x}^{i/2}}} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -i\pi \, \left ({\it csgn} \left ({\frac{ic}{{x}^{i/2}}} \right ) \right ) ^{2}{\it csgn} \left ({\frac{i}{{x}^{i/2}}} \right ) +i\pi \,{\it csgn} \left ({\frac{ic}{{x}^{i/2}}} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ({\frac{i}{{x}^{i/2}}} \right ) -2\,\ln \left ( c \right ) +2\,\ln \left ({x}^{i/2} \right ) -a \right ) }} \left ({{\rm e}^{-2\,i \left ( i\pi \, \left ({\it csgn} \left ({\frac{ic}{{x}^{i/2}}} \right ) \right ) ^{3}-i\pi \, \left ({\it csgn} \left ({\frac{ic}{{x}^{i/2}}} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -i\pi \, \left ({\it csgn} \left ({\frac{ic}{{x}^{i/2}}} \right ) \right ) ^{2}{\it csgn} \left ({\frac{i}{{x}^{i/2}}} \right ) +i\pi \,{\it csgn} \left ({\frac{ic}{{x}^{i/2}}} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ({\frac{i}{{x}^{i/2}}} \right ) -2\,\ln \left ( c \right ) +2\,\ln \left ({x}^{i/2} \right ) -a \right ) }}+1 \right ) ^{-2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(a+2*ln(c/(x^(1/2*I))))^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.23325, size = 224, normalized size = 4.67 \begin{align*} \frac{{\left ({\left (2 \, \cos \left (3 \, a\right ) + 2 i \, \sin \left (3 \, a\right )\right )} \cos \left (6 \, \log \left (c\right )\right ) + 2 \,{\left (i \, \cos \left (3 \, a\right ) - \sin \left (3 \, a\right )\right )} \sin \left (6 \, \log \left (c\right )\right )\right )} x e^{\left (6 \, \arctan \left (\sin \left (\frac{1}{2} \, \log \left (x\right )\right ), \cos \left (\frac{1}{2} \, \log \left (x\right )\right )\right )\right )}}{{\left ({\left (\cos \left (4 \, a\right ) + i \, \sin \left (4 \, a\right )\right )} \cos \left (8 \, \log \left (c\right )\right ) -{\left (-i \, \cos \left (4 \, a\right ) + \sin \left (4 \, a\right )\right )} \sin \left (8 \, \log \left (c\right )\right )\right )} e^{\left (8 \, \arctan \left (\sin \left (\frac{1}{2} \, \log \left (x\right )\right ), \cos \left (\frac{1}{2} \, \log \left (x\right )\right )\right )\right )} +{\left ({\left (2 \, \cos \left (2 \, a\right ) + 2 i \, \sin \left (2 \, a\right )\right )} \cos \left (4 \, \log \left (c\right )\right ) + 2 \,{\left (i \, \cos \left (2 \, a\right ) - \sin \left (2 \, a\right )\right )} \sin \left (4 \, \log \left (c\right )\right )\right )} e^{\left (4 \, \arctan \left (\sin \left (\frac{1}{2} \, \log \left (x\right )\right ), \cos \left (\frac{1}{2} \, \log \left (x\right )\right )\right )\right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*cos(3*a) + 2*I*sin(3*a))*cos(6*log(c)) + 2*(I*cos(3*a) - sin(3*a))*sin(6*log(c)))*x*e^(6*arctan2(sin(1/2*l
og(x)), cos(1/2*log(x))))/(((cos(4*a) + I*sin(4*a))*cos(8*log(c)) - (-I*cos(4*a) + sin(4*a))*sin(8*log(c)))*e^
(8*arctan2(sin(1/2*log(x)), cos(1/2*log(x)))) + ((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(1/2*log(x)), cos(1/2*log(x)))) + 1)

________________________________________________________________________________________

Fricas [A]  time = 0.449647, size = 158, normalized size = 3.29 \begin{align*} \frac{2 \, x e^{\left (3 i \, a + 6 i \, \log \left (c x^{-\frac{1}{2} i}\right )\right )}}{e^{\left (4 i \, a + 8 i \, \log \left (c x^{-\frac{1}{2} i}\right )\right )} + 2 \, e^{\left (2 i \, a + 4 i \, \log \left (c x^{-\frac{1}{2} i}\right )\right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x*e^(3*I*a + 6*I*log(c*x^(-1/2*I)))/(e^(4*I*a + 8*I*log(c*x^(-1/2*I))) + 2*e^(2*I*a + 4*I*log(c*x^(-1/2*I)))
+ 1)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(a+2*ln(c/(x**(1/2*I))))**3,x)

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 4.55561, size = 112, normalized size = 2.33 \begin{align*} -\frac{4 \, c^{4 i} x^{2} e^{\left (2 i \, a\right )}}{c^{10 i} x^{4} e^{\left (5 i \, a\right )} + 2 \, c^{6 i} x^{2} e^{\left (3 i \, a\right )} + c^{2 i} e^{\left (i \, a\right )}} - \frac{2}{c^{10 i} x^{4} e^{\left (5 i \, a\right )} + 2 \, c^{6 i} x^{2} e^{\left (3 i \, a\right )} + c^{2 i} e^{\left (i \, a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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