\(\int x \sec ^p(a+b \log (c x^n)) \, dx\) [287]

Optimal result

Integrand size = 15, antiderivative size = 106 \[ \int x \sec ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^2 \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \operatorname {Hypergeometric2F1}\left (p,\frac {1}{2} \left (-\frac {2 i}{b n}+p\right ),\frac {1}{2} \left (2-\frac {2 i}{b n}+p\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sec ^p\left (a+b \log \left (c x^n\right )\right )}{2+i b n p} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4605, 4603, 371} \[ \int x \sec ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^2 \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \operatorname {Hypergeometric2F1}\left (p,\frac {1}{2} \left (p-\frac {2 i}{b n}\right ),\frac {1}{2} \left (p-\frac {2 i}{b n}+2\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sec ^p\left (a+b \log \left (c x^n\right )\right )}{2+i b n p} \]

[In]

Int[x*Sec[a + b*Log[c*x^n]]^p,x]

[Out]

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

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 4603

Int[((e_.)*(x_))^(m_.)*Sec[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[Sec[d*(a + b*Log[x])]^p*((1
+ E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*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, p}, 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 (x^2 \left (c x^n\right )^{-2/n}\right ) \text {Subst}\left (\int x^{-1+\frac {2}{n}} \sec ^p(a+b \log (x)) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x^2 \left (c x^n\right )^{-\frac {2}{n}-i b p} \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \sec ^p\left (a+b \log \left (c x^n\right )\right )\right ) \text {Subst}\left (\int x^{-1+\frac {2}{n}+i b p} \left (1+e^{2 i a} x^{2 i b}\right )^{-p} \, dx,x,c x^n\right )}{n} \\ & = \frac {x^2 \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \operatorname {Hypergeometric2F1}\left (p,\frac {1}{2} \left (-\frac {2 i}{b n}+p\right ),\frac {1}{2} \left (2-\frac {2 i}{b n}+p\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sec ^p\left (a+b \log \left (c x^n\right )\right )}{2+i b n p} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.83 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.34 \[ \int x \sec ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {i 2^p x^2 \left (\frac {e^{i a} \left (c x^n\right )^{i b}}{1+e^{2 i a} \left (c x^n\right )^{2 i b}}\right )^p \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \operatorname {Hypergeometric2F1}\left (-\frac {i}{b n}+\frac {p}{2},p,1-\frac {i}{b n}+\frac {p}{2},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{-2 i+b n p} \]

[In]

Integrate[x*Sec[a + b*Log[c*x^n]]^p,x]

[Out]

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

Maple [F]

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

[In]

int(x*sec(a+b*ln(c*x^n))^p,x)

[Out]

int(x*sec(a+b*ln(c*x^n))^p,x)

Fricas [F]

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

[In]

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

[Out]

integral(x*sec(b*log(c*x^n) + a)^p, x)

Sympy [F]

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

[In]

integrate(x*sec(a+b*ln(c*x**n))**p,x)

[Out]

Integral(x*sec(a + b*log(c*x**n))**p, x)

Maxima [F]

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

[In]

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

[Out]

integrate(x*sec(b*log(c*x^n) + a)^p, x)

Giac [F]

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

[In]

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

[Out]

integrate(x*sec(b*log(c*x^n) + a)^p, x)

Mupad [F(-1)]

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

[In]

int(x*(1/cos(a + b*log(c*x^n)))^p,x)

[Out]

int(x*(1/cos(a + b*log(c*x^n)))^p, x)