\(\int (e x)^m \sec ^p(d (a+b \log (c x^n))) \, dx\) [286]

Optimal result

Integrand size = 21, antiderivative size = 139 \[ \int (e x)^m \sec ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(e x)^{1+m} \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \operatorname {Hypergeometric2F1}\left (p,-\frac {i+i m-b d n p}{2 b d n},\frac {1}{2} \left (2-\frac {i (1+m)}{b d n}+p\right ),-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \sec ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m+i b d n p)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4605, 4603, 371} \[ \int (e x)^m \sec ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(e x)^{m+1} \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \operatorname {Hypergeometric2F1}\left (p,\frac {1}{2} \left (p-\frac {i (m+1)}{b d n}\right ),\frac {1}{2} \left (-\frac {i (m+1)}{b d n}+p+2\right ),-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \sec ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (i b d n p+m+1)} \]

[In]

Int[(e*x)^m*Sec[d*(a + b*Log[c*x^n])]^p,x]

[Out]

((e*x)^(1 + m)*(1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))^p*Hypergeometric2F1[p, (((-I)*(1 + m))/(b*d*n) + p)/2,
(2 - (I*(1 + m))/(b*d*n) + p)/2, -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))]*Sec[d*(a + b*Log[c*x^n])]^p)/(e*(1 + m
+ I*b*d*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 ((e x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int x^{-1+\frac {1+m}{n}} \sec ^p(d (a+b \log (x))) \, dx,x,c x^n\right )}{e n} \\ & = \frac {\left ((e x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}-i b d p} \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \sec ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right )\right ) \text {Subst}\left (\int x^{-1+\frac {1+m}{n}+i b d p} \left (1+e^{2 i a d} x^{2 i b d}\right )^{-p} \, dx,x,c x^n\right )}{e n} \\ & = \frac {(e x)^{1+m} \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \operatorname {Hypergeometric2F1}\left (p,\frac {1}{2} \left (-\frac {i (1+m)}{b d n}+p\right ),\frac {1}{2} \left (2-\frac {i (1+m)}{b d n}+p\right ),-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \sec ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m+i b d n p)} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.18 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.22 \[ \int (e x)^m \sec ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {2^p x (e x)^m \left (\frac {e^{i a d} \left (c x^n\right )^{i b d}}{1+e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \operatorname {Hypergeometric2F1}\left (p,-\frac {i (1+m+i b d n p)}{2 b d n},\frac {1}{2} \left (2-\frac {i (1+m)}{b d n}+p\right ),-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{1+m+i b d n p} \]

[In]

Integrate[(e*x)^m*Sec[d*(a + b*Log[c*x^n])]^p,x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((e*x)^m*(1/cos(d*(a + b*log(c*x^n))))^p,x)

[Out]

int((e*x)^m*(1/cos(d*(a + b*log(c*x^n))))^p, x)