3.3.98 \(\int (d \sec (e+f x))^m (a+b \sec ^2(e+f x))^p \, dx\) [298]

3.3.98.1 Optimal result

Integrand size = 25, antiderivative size = 111 \[ \int (d \sec (e+f x))^m \left (a+b \sec ^2(e+f x)\right )^p \, dx=\frac {\operatorname {AppellF1}\left (\frac {m}{2},\frac {1}{2},-p,\frac {2+m}{2},\sec ^2(e+f x),-\frac {b \sec ^2(e+f x)}{a}\right ) \cot (e+f x) (d \sec (e+f x))^m \left (a+b \sec ^2(e+f x)\right )^p \left (1+\frac {b \sec ^2(e+f x)}{a}\right )^{-p} \sqrt {-\tan ^2(e+f x)}}{f m} \]

output

AppellF1(1/2*m,1/2,-p,1+1/2*m,sec(f*x+e)^2,-b*sec(f*x+e)^2/a)*cos(f*x+e)*( 
d*sec(f*x+e))^m*(a+b*sec(f*x+e)^2)^p*(-tan(f*x+e)^2)^(1/2)/f/m/((1+b*sec(f 
*x+e)^2/a)^p)/sin(f*x+e)
 

3.3.98.2 Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(2195\) vs. \(2(111)=222\).

Time = 20.08 (sec) , antiderivative size = 2195, normalized size of antiderivative = 19.77 \[ \int (d \sec (e+f x))^m \left (a+b \sec ^2(e+f x)\right )^p \, dx=\text {Result too large to show} \]

input

Integrate[(d*Sec[e + f*x])^m*(a + b*Sec[e + f*x]^2)^p,x]
 

output

(3*(a + b)*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f 
*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(d*Sec[e + f*x])^m*(Sec[ 
e + f*x]^2)^(-1 + m/2 + p)*(a + b*Sec[e + f*x]^2)^p*Tan[e + f*x])/(f*(3*(a 
 + b)*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2 
)/(a + b))] + (2*b*p*AppellF1[3/2, 1 - m/2, 1 - p, 5/2, -Tan[e + f*x]^2, - 
((b*Tan[e + f*x]^2)/(a + b))] + (a + b)*(-2 + m)*AppellF1[3/2, 2 - m/2, -p 
, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2)*(( 
3*(a + b)*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f* 
x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(m/2 + p 
))/(3*(a + b)*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e 
+ f*x]^2)/(a + b))] + (2*b*p*AppellF1[3/2, 1 - m/2, 1 - p, 5/2, -Tan[e + f 
*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (a + b)*(-2 + m)*AppellF1[3/2, 2 - 
 m/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f* 
x]^2) - (6*a*(a + b)*p*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -( 
(b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e + f*x)])^(-1 + p)*(Sec[ 
e + f*x]^2)^(-1 + m/2 + p)*Sin[2*(e + f*x)]*Tan[e + f*x])/(3*(a + b)*Appel 
lF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] 
 + (2*b*p*AppellF1[3/2, 1 - m/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + 
 f*x]^2)/(a + b))] + (a + b)*(-2 + m)*AppellF1[3/2, 2 - m/2, -p, 5/2, -Tan 
[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) + (6*(a + ...
 

3.3.98.3 Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (d \sec (e+f x))^m \left (a+b \sec ^2(e+f x)\right )^p \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (d \sec (e+f x))^m \left (a+b \sec (e+f x)^2\right )^pdx\)

\(\Big \downarrow \) 4638

\(\displaystyle \int (d \sec (e+f x))^m \left (a+b \sec ^2(e+f x)\right )^pdx\)

input

Int[(d*Sec[e + f*x])^m*(a + b*Sec[e + f*x]^2)^p,x]
 

output

$Aborted
 

3.3.98.3.1 Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4638

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*sec[(e_.) + 
(f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> Unintegrable[(d*Sec[e + f*x])^m*(a + 
 b*(c*Sec[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
 

3.3.98.4 Maple [F]

\[\int \left (d \sec \left (f x +e \right )\right )^{m} \left (a +b \sec \left (f x +e \right )^{2}\right )^{p}d x\]

input

int((d*sec(f*x+e))^m*(a+b*sec(f*x+e)^2)^p,x)
 

output

int((d*sec(f*x+e))^m*(a+b*sec(f*x+e)^2)^p,x)
 

3.3.98.5 Fricas [F]

\[ \int (d \sec (e+f x))^m \left (a+b \sec ^2(e+f x)\right )^p \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \left (d \sec \left (f x + e\right )\right )^{m} \,d x } \]

input

integrate((d*sec(f*x+e))^m*(a+b*sec(f*x+e)^2)^p,x, algorithm="fricas")
 

output

integral((b*sec(f*x + e)^2 + a)^p*(d*sec(f*x + e))^m, x)
 

3.3.98.6 Sympy [F]

\[ \int (d \sec (e+f x))^m \left (a+b \sec ^2(e+f x)\right )^p \, dx=\int \left (d \sec {\left (e + f x \right )}\right )^{m} \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{p}\, dx \]

input

integrate((d*sec(f*x+e))**m*(a+b*sec(f*x+e)**2)**p,x)
 

output

Integral((d*sec(e + f*x))**m*(a + b*sec(e + f*x)**2)**p, x)
 

3.3.98.7 Maxima [F]

\[ \int (d \sec (e+f x))^m \left (a+b \sec ^2(e+f x)\right )^p \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \left (d \sec \left (f x + e\right )\right )^{m} \,d x } \]

input

integrate((d*sec(f*x+e))^m*(a+b*sec(f*x+e)^2)^p,x, algorithm="maxima")
 

output

integrate((b*sec(f*x + e)^2 + a)^p*(d*sec(f*x + e))^m, x)
 

3.3.98.8 Giac [F]

\[ \int (d \sec (e+f x))^m \left (a+b \sec ^2(e+f x)\right )^p \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \left (d \sec \left (f x + e\right )\right )^{m} \,d x } \]

input

integrate((d*sec(f*x+e))^m*(a+b*sec(f*x+e)^2)^p,x, algorithm="giac")
 

output

integrate((b*sec(f*x + e)^2 + a)^p*(d*sec(f*x + e))^m, x)
 

3.3.98.9 Mupad [F(-1)]

Timed out. \[ \int (d \sec (e+f x))^m \left (a+b \sec ^2(e+f x)\right )^p \, dx=\int {\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^p\,{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^m \,d x \]

input

int((a + b/cos(e + f*x)^2)^p*(d/cos(e + f*x))^m,x)
 

output

int((a + b/cos(e + f*x)^2)^p*(d/cos(e + f*x))^m, x)