\(\int x \sec ^3(a+2 \log (c x^i)) \, dx\) [261]

Optimal result

Integrand size = 17, antiderivative size = 45 \[ \int x \sec ^3\left (a+2 \log \left (c x^i\right )\right ) \, dx=\frac {e^{i a} \left (c x^i\right )^{2 i} x^2}{\left (1+e^{2 i a} \left (c x^i\right )^{4 i}\right )^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4605, 4601, 267} \[ \int x \sec ^3\left (a+2 \log \left (c x^i\right )\right ) \, dx=\frac {e^{i a} x^2 \left (c x^i\right )^{2 i}}{\left (1+e^{2 i a} \left (c x^i\right )^{4 i}\right )^2} \]

[In]

Int[x*Sec[a + 2*Log[c*x^I]]^3,x]

[Out]

(E^(I*a)*(c*x^I)^(2*I)*x^2)/(1 + E^((2*I)*a)*(c*x^I)^(4*I))^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 4601

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 4605

Int[((e_.)*(x_))^(m_.)*Sec[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[(e*x)^(m + 1)
/(e*n*(c*x^n)^((m + 1)/n)), Subst[Int[x^((m + 1)/n - 1)*Sec[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*} \text {integral}& = -\left (\left (i \left (c x^i\right )^{2 i} x^2\right ) \text {Subst}\left (\int x^{-1-2 i} \sec ^3(a+2 \log (x)) \, dx,x,c x^i\right )\right ) \\ & = -\left (\left (8 i e^{3 i a} \left (c x^i\right )^{2 i} x^2\right ) \text {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 )\right ) \\ & = \frac {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*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(127\) vs. \(2(45)=90\).

Time = 0.12 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.82 \[ \int x \sec ^3\left (a+2 \log \left (c x^i\right )\right ) \, dx=-\frac {\sec ^2\left (a+2 \log \left (c x^i\right )\right ) \left (\left (1+2 x^4\right ) \cos \left (a+2 \log \left (c x^i\right )-2 i \log (x)\right )+i \left (1-2 x^4\right ) \sin \left (a+2 \log \left (c x^i\right )-2 i \log (x)\right )\right ) \left (\cos \left (2 \left (a+2 \log \left (c x^i\right )-2 i \log (x)\right )\right )+i \sin \left (2 \left (a+2 \log \left (c x^i\right )-2 i \log (x)\right )\right )\right )}{4 x^4} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.17 (sec) , antiderivative size = 209, normalized size of antiderivative = 4.64

\[\frac {x^{2} c^{2 i} \left (x^{i}\right )^{2 i} {\mathrm e}^{-\pi \,\operatorname {csgn}\left (i x^{i}\right ) \operatorname {csgn}\left (i c \,x^{i}\right )^{2}+\pi \,\operatorname {csgn}\left (i x^{i}\right ) \operatorname {csgn}\left (i c \,x^{i}\right ) \operatorname {csgn}\left (i c \right )+\pi \operatorname {csgn}\left (i c \,x^{i}\right )^{3}-\pi \operatorname {csgn}\left (i c \,x^{i}\right )^{2} \operatorname {csgn}\left (i c \right )+i a}}{{\left (\left (x^{i}\right )^{4 i} c^{4 i} {\mathrm e}^{-2 \pi \,\operatorname {csgn}\left (i x^{i}\right ) \operatorname {csgn}\left (i c \,x^{i}\right )^{2}} {\mathrm e}^{2 \pi \,\operatorname {csgn}\left (i x^{i}\right ) \operatorname {csgn}\left (i c \,x^{i}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 \pi \operatorname {csgn}\left (i c \,x^{i}\right )^{3}} {\mathrm e}^{-2 \pi \operatorname {csgn}\left (i c \,x^{i}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}+1\right )}^{2}}\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.22 \[ \int x \sec ^3\left (a+2 \log \left (c x^i\right )\right ) \, dx=-\frac {2 \, x^{4} e^{\left (3 i \, a + 6 i \, \log \left (c\right )\right )} + e^{\left (5 i \, a + 10 i \, \log \left (c\right )\right )}}{x^{8} + 2 \, x^{4} e^{\left (2 i \, a + 4 i \, \log \left (c\right )\right )} + e^{\left (4 i \, a + 8 i \, \log \left (c\right )\right )}} \]

[In]

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

[Out]

-(2*x^4*e^(3*I*a + 6*I*log(c)) + 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)))

Sympy [F]

\[ \int x \sec ^3\left (a+2 \log \left (c x^i\right )\right ) \, dx=\int x \sec ^{3}{\left (a + 2 \log {\left (c x^{i} \right )} \right )}\, dx \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (31) = 62\).

Time = 0.25 (sec) , antiderivative size = 139, normalized size of antiderivative = 3.09 \[ \int x \sec ^3\left (a+2 \log \left (c x^i\right )\right ) \, dx=\frac {{\left ({\left (\cos \left (a\right ) + i \, \sin \left (a\right )\right )} \cos \left (2 \, \log \left (c\right )\right ) - {\left (-i \, \cos \left (a\right ) + \sin \left (a\right )\right )} \sin \left (2 \, \log \left (c\right )\right )\right )} x^{2} e^{\left (6 \, \arctan \left (\sin \left (\log \left (x\right )\right ), \cos \left (\log \left (x\right )\right )\right )\right )}}{{\left (\cos \left (4 \, a\right ) + i \, \sin \left (4 \, a\right )\right )} \cos \left (8 \, \log \left (c\right )\right ) + 2 \, {\left ({\left (\cos \left (2 \, a\right ) + i \, \sin \left (2 \, a\right )\right )} \cos \left (4 \, \log \left (c\right )\right ) - {\left (-i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} \sin \left (4 \, \log \left (c\right )\right )\right )} e^{\left (4 \, \arctan \left (\sin \left (\log \left (x\right )\right ), \cos \left (\log \left (x\right )\right )\right )\right )} + {\left (i \, \cos \left (4 \, a\right ) - \sin \left (4 \, a\right )\right )} \sin \left (8 \, \log \left (c\right )\right ) + e^{\left (8 \, \arctan \left (\sin \left (\log \left (x\right )\right ), \cos \left (\log \left (x\right )\right )\right )\right )}} \]

[In]

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

[Out]

((cos(a) + I*sin(a))*cos(2*log(c)) - (-I*cos(a) + 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) + I*sin(2*a))*cos(4*log(c)) - (-I*cos(2*a) + sin(
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*ar
ctan2(sin(log(x)), cos(log(x)))))

Giac [F]

\[ \int x \sec ^3\left (a+2 \log \left (c x^i\right )\right ) \, dx=\int { x \sec \left (a + 2 \, \log \left (c x^{i}\right )\right )^{3} \,d x } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 29.92 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.02 \[ \int x \sec ^3\left (a+2 \log \left (c x^i\right )\right ) \, dx=\frac {x^2\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^{1{}\mathrm {i}}\right )}^{2{}\mathrm {i}}}{2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^{1{}\mathrm {i}}\right )}^{4{}\mathrm {i}}+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^{1{}\mathrm {i}}\right )}^{8{}\mathrm {i}}+1} \]

[In]

int(x/cos(a + 2*log(c*x^1i))^3,x)

[Out]

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