\(\int \frac {\sec ^2(a+b \log (c x^n))}{x^3} \, dx\) [248]

Optimal result

Integrand size = 17, antiderivative size = 79 \[ \int \frac {\sec ^2\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {2 e^{2 i a} \left (c x^n\right )^{2 i b} \operatorname {Hypergeometric2F1}\left (2,1+\frac {i}{b n},2+\frac {i}{b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(1-i b n) x^2} \]

[Out]

-2*exp(2*I*a)*(c*x^n)^(2*I*b)*hypergeom([2, 1+I/b/n],[2+I/b/n],-exp(2*I*a)*(c*x^n)^(2*I*b))/(1-I*b*n)/x^2

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 79, 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, 371} \[ \int \frac {\sec ^2\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {2 e^{2 i a} \left (c x^n\right )^{2 i b} \operatorname {Hypergeometric2F1}\left (2,1+\frac {i}{b n},2+\frac {i}{b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{x^2 (1-i b n)} \]

[In]

Int[Sec[a + b*Log[c*x^n]]^2/x^3,x]

[Out]

(-2*E^((2*I)*a)*(c*x^n)^((2*I)*b)*Hypergeometric2F1[2, 1 + I/(b*n), 2 + I/(b*n), -(E^((2*I)*a)*(c*x^n)^((2*I)*
b))])/((1 - I*b*n)*x^2)

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

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}& = \frac {\left (c x^n\right )^{2/n} \text {Subst}\left (\int x^{-1-\frac {2}{n}} \sec ^2(a+b \log (x)) \, dx,x,c x^n\right )}{n x^2} \\ & = \frac {\left (4 e^{2 i a} \left (c x^n\right )^{2/n}\right ) \text {Subst}\left (\int \frac {x^{-1+2 i b-\frac {2}{n}}}{\left (1+e^{2 i a} x^{2 i b}\right )^2} \, dx,x,c x^n\right )}{n x^2} \\ & = -\frac {2 e^{2 i a} \left (c x^n\right )^{2 i b} \operatorname {Hypergeometric2F1}\left (2,1+\frac {i}{b n},2+\frac {i}{b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(1-i b n) x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.81 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.90 \[ \int \frac {\sec ^2\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {-e^{2 i a} \left (c x^n\right )^{2 i b} \operatorname {Hypergeometric2F1}\left (1,1+\frac {i}{b n},2+\frac {i}{b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )+(i+b n) \left (-i \operatorname {Hypergeometric2F1}\left (1,\frac {i}{b n},1+\frac {i}{b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )+\tan \left (a+b \log \left (c x^n\right )\right )\right )}{b n (i+b n) x^2} \]

[In]

Integrate[Sec[a + b*Log[c*x^n]]^2/x^3,x]

[Out]

(-(E^((2*I)*a)*(c*x^n)^((2*I)*b)*Hypergeometric2F1[1, 1 + I/(b*n), 2 + I/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))]
) + (I + b*n)*((-I)*Hypergeometric2F1[1, I/(b*n), 1 + I/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] + Tan[a + b*Log[
c*x^n]]))/(b*n*(I + b*n)*x^2)

Maple [F]

\[\int \frac {{\sec \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{x^{3}}d x\]

[In]

int(sec(a+b*ln(c*x^n))^2/x^3,x)

[Out]

int(sec(a+b*ln(c*x^n))^2/x^3,x)

Fricas [F]

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

[In]

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

[Out]

integral(sec(b*log(c*x^n) + a)^2/x^3, x)

Sympy [F]

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

[In]

integrate(sec(a+b*ln(c*x**n))**2/x**3,x)

[Out]

Integral(sec(a + b*log(c*x**n))**2/x**3, x)

Maxima [F]

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

[In]

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

[Out]

2*(2*(2*b^2*n^2*x^2*cos(2*b*log(c))*cos(2*b*log(x^n) + 2*a) - 2*b^2*n^2*x^2*sin(2*b*log(c))*sin(2*b*log(x^n) +
2*a) + (b^2*cos(2*b*log(c))^2 + b^2*sin(2*b*log(c))^2)*n^2*x^2*cos(2*b*log(x^n) + 2*a)^2 + (b^2*cos(2*b*log(c
))^2 + b^2*sin(2*b*log(c))^2)*n^2*x^2*sin(2*b*log(x^n) + 2*a)^2 + b^2*n^2*x^2)*integrate((cos(2*b*log(x^n) + 2
*a)*sin(2*b*log(c)) + cos(2*b*log(c))*sin(2*b*log(x^n) + 2*a))/(2*b^2*n^2*x^3*cos(2*b*log(c))*cos(2*b*log(x^n)
+ 2*a) - 2*b^2*n^2*x^3*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) + (b^2*cos(2*b*log(c))^2 + b^2*sin(2*b*log(c))
^2)*n^2*x^3*cos(2*b*log(x^n) + 2*a)^2 + (b^2*cos(2*b*log(c))^2 + b^2*sin(2*b*log(c))^2)*n^2*x^3*sin(2*b*log(x^
n) + 2*a)^2 + b^2*n^2*x^3), x) + cos(2*b*log(x^n) + 2*a)*sin(2*b*log(c)) + cos(2*b*log(c))*sin(2*b*log(x^n) +
2*a))/(2*b*n*x^2*cos(2*b*log(c))*cos(2*b*log(x^n) + 2*a) + (b*cos(2*b*log(c))^2 + b*sin(2*b*log(c))^2)*n*x^2*c
os(2*b*log(x^n) + 2*a)^2 - 2*b*n*x^2*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) + (b*cos(2*b*log(c))^2 + b*sin(2*
b*log(c))^2)*n*x^2*sin(2*b*log(x^n) + 2*a)^2 + b*n*x^2)

Giac [F]

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

[In]

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

[Out]

integrate(sec(b*log(c*x^n) + a)^2/x^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sec ^2\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {1}{x^3\,{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]

[In]

int(1/(x^3*cos(a + b*log(c*x^n))^2),x)

[Out]

int(1/(x^3*cos(a + b*log(c*x^n))^2), x)