\(\int (d \sec (e+f x))^m (a+b \tan (e+f x))^n \, dx\) [645]

Optimal result

Integrand size = 23, antiderivative size = 181 \[ \int (d \sec (e+f x))^m (a+b \tan (e+f x))^n \, dx=\frac {b \operatorname {AppellF1}\left (1+n,1-\frac {m}{2},1-\frac {m}{2},2+n,\frac {a+b \tan (e+f x)}{a+\sqrt {-b^2}},\frac {a+b \tan (e+f x)}{a-\sqrt {-b^2}}\right ) (d \sec (e+f x))^m (a+b \tan (e+f x))^{1+n} \left (1+\frac {a+b \tan (e+f x)}{-a+\sqrt {-b^2}}\right )^{-m/2} \left (1-\frac {a+b \tan (e+f x)}{a+\sqrt {-b^2}}\right )^{-m/2}}{\left (a^2+b^2\right ) f (1+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3593, 774, 138} \[ \int (d \sec (e+f x))^m (a+b \tan (e+f x))^n \, dx=\frac {\cos ^2(e+f x) (d \sec (e+f x))^m \left (1-\frac {a+b \tan (e+f x)}{a-\sqrt {-b^2}}\right )^{1-\frac {m}{2}} \left (1-\frac {a+b \tan (e+f x)}{a+\sqrt {-b^2}}\right )^{1-\frac {m}{2}} (a+b \tan (e+f x))^{n+1} \operatorname {AppellF1}\left (n+1,1-\frac {m}{2},1-\frac {m}{2},n+2,\frac {a+b \tan (e+f x)}{a-\sqrt {-b^2}},\frac {a+b \tan (e+f x)}{a+\sqrt {-b^2}}\right )}{b f (n+1)} \]

[In]

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

[Out]

(AppellF1[1 + n, 1 - m/2, 1 - m/2, 2 + n, (a + b*Tan[e + f*x])/(a - Sqrt[-b^2]), (a + b*Tan[e + f*x])/(a + Sqr
t[-b^2])]*Cos[e + f*x]^2*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(1 + n)*(1 - (a + b*Tan[e + f*x])/(a - Sqrt[-
b^2]))^(1 - m/2)*(1 - (a + b*Tan[e + f*x])/(a + Sqrt[-b^2]))^(1 - m/2))/(b*f*(1 + n))

Rule 138

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[c^n*e^p*((b*x)^(m +
1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rule 774

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[(a + c*x^
2)^p/(e*(1 - (d + e*x)/(d + e*(q/c)))^p*(1 - (d + e*x)/(d - e*(q/c)))^p), Subst[Int[x^m*Simp[1 - x/(d + e*(q/c
)), x]^p*Simp[1 - x/(d - e*(q/c)), x]^p, x], x, d + e*x], x]] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a
*e^2, 0] &&  !IntegerQ[p]

Rule 3593

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[d^(2*
IntPart[m/2])*((d*Sec[e + f*x])^(2*FracPart[m/2])/(b*f*(Sec[e + f*x]^2)^FracPart[m/2])), Subst[Int[(a + x)^n*(
1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 + b^2, 0] &&
!IntegerQ[m/2]

Rubi steps \begin{align*} \text {integral}& = \frac {\left ((d \sec (e+f x))^m \sec ^2(e+f x)^{-m/2}\right ) \text {Subst}\left (\int (a+x)^n \left (1+\frac {x^2}{b^2}\right )^{-1+\frac {m}{2}} \, dx,x,b \tan (e+f x)\right )}{b f} \\ & = \frac {\left (\cos ^2(e+f x) (d \sec (e+f x))^m \left (1-\frac {a+b \tan (e+f x)}{a-\frac {b^2}{\sqrt {-b^2}}}\right )^{1-\frac {m}{2}} \left (1-\frac {a+b \tan (e+f x)}{a+\frac {b^2}{\sqrt {-b^2}}}\right )^{1-\frac {m}{2}}\right ) \text {Subst}\left (\int x^n \left (1-\frac {x}{a-\sqrt {-b^2}}\right )^{-1+\frac {m}{2}} \left (1-\frac {x}{a+\sqrt {-b^2}}\right )^{-1+\frac {m}{2}} \, dx,x,a+b \tan (e+f x)\right )}{b f} \\ & = \frac {\operatorname {AppellF1}\left (1+n,1-\frac {m}{2},1-\frac {m}{2},2+n,\frac {a+b \tan (e+f x)}{a-\sqrt {-b^2}},\frac {a+b \tan (e+f x)}{a+\sqrt {-b^2}}\right ) \cos ^2(e+f x) (d \sec (e+f x))^m (a+b \tan (e+f x))^{1+n} \left (1-\frac {a+b \tan (e+f x)}{a-\sqrt {-b^2}}\right )^{1-\frac {m}{2}} \left (1-\frac {a+b \tan (e+f x)}{a+\sqrt {-b^2}}\right )^{1-\frac {m}{2}}}{b f (1+n)} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 10.00 (sec) , antiderivative size = 699, normalized size of antiderivative = 3.86 \[ \int (d \sec (e+f x))^m (a+b \tan (e+f x))^n \, dx=\frac {2 \operatorname {AppellF1}\left (1+n,1-\frac {m}{2},1-\frac {m}{2},2+n,\frac {a+b \tan (e+f x)}{a-i b},\frac {a+b \tan (e+f x)}{a+i b}\right ) (d \sec (e+f x))^m (a+b \tan (e+f x))^{1+n}}{f \left (2 b \operatorname {AppellF1}\left (1+n,1-\frac {m}{2},1-\frac {m}{2},2+n,\frac {a+b \tan (e+f x)}{a-i b},\frac {a+b \tan (e+f x)}{a+i b}\right ) \sec ^2(e+f x)+2 n \operatorname {AppellF1}\left (1+n,1-\frac {m}{2},1-\frac {m}{2},2+n,\frac {a+b \tan (e+f x)}{a-i b},\frac {a+b \tan (e+f x)}{a+i b}\right ) (b-a \tan (e+f x))-\frac {b (-2+m) (1+n) \left ((a-i b) \operatorname {AppellF1}\left (2+n,1-\frac {m}{2},2-\frac {m}{2},3+n,\frac {a+b \tan (e+f x)}{a-i b},\frac {a+b \tan (e+f x)}{a+i b}\right )+(a+i b) \operatorname {AppellF1}\left (2+n,2-\frac {m}{2},1-\frac {m}{2},3+n,\frac {a+b \tan (e+f x)}{a-i b},\frac {a+b \tan (e+f x)}{a+i b}\right )\right ) \sec ^2(e+f x) (a+b \tan (e+f x))}{(a-i b) (a+i b) (2+n)}+2 (m+n) \operatorname {AppellF1}\left (1+n,1-\frac {m}{2},1-\frac {m}{2},2+n,\frac {a+b \tan (e+f x)}{a-i b},\frac {a+b \tan (e+f x)}{a+i b}\right ) \tan (e+f x) (a+b \tan (e+f x))-\frac {m \operatorname {AppellF1}\left (1+n,1-\frac {m}{2},1-\frac {m}{2},2+n,\frac {a+b \tan (e+f x)}{a-i b},\frac {a+b \tan (e+f x)}{a+i b}\right ) \sec ^2(e+f x) (a+b \tan (e+f x))}{-i+\tan (e+f x)}-\frac {m \operatorname {AppellF1}\left (1+n,1-\frac {m}{2},1-\frac {m}{2},2+n,\frac {a+b \tan (e+f x)}{a-i b},\frac {a+b \tan (e+f x)}{a+i b}\right ) \sec ^2(e+f x) (a+b \tan (e+f x))}{i+\tan (e+f x)}\right )} \]

[In]

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

[Out]

(2*AppellF1[1 + n, 1 - m/2, 1 - m/2, 2 + n, (a + b*Tan[e + f*x])/(a - I*b), (a + b*Tan[e + f*x])/(a + I*b)]*(d
*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(1 + n))/(f*(2*b*AppellF1[1 + n, 1 - m/2, 1 - m/2, 2 + n, (a + b*Tan[e +
f*x])/(a - I*b), (a + b*Tan[e + f*x])/(a + I*b)]*Sec[e + f*x]^2 + 2*n*AppellF1[1 + n, 1 - m/2, 1 - m/2, 2 + n
, (a + b*Tan[e + f*x])/(a - I*b), (a + b*Tan[e + f*x])/(a + I*b)]*(b - a*Tan[e + f*x]) - (b*(-2 + m)*(1 + n)*(
(a - I*b)*AppellF1[2 + n, 1 - m/2, 2 - m/2, 3 + n, (a + b*Tan[e + f*x])/(a - I*b), (a + b*Tan[e + f*x])/(a + I
*b)] + (a + I*b)*AppellF1[2 + n, 2 - m/2, 1 - m/2, 3 + n, (a + b*Tan[e + f*x])/(a - I*b), (a + b*Tan[e + f*x])
/(a + I*b)])*Sec[e + f*x]^2*(a + b*Tan[e + f*x]))/((a - I*b)*(a + I*b)*(2 + n)) + 2*(m + n)*AppellF1[1 + n, 1
- m/2, 1 - m/2, 2 + n, (a + b*Tan[e + f*x])/(a - I*b), (a + b*Tan[e + f*x])/(a + I*b)]*Tan[e + f*x]*(a + b*Tan
[e + f*x]) - (m*AppellF1[1 + n, 1 - m/2, 1 - m/2, 2 + n, (a + b*Tan[e + f*x])/(a - I*b), (a + b*Tan[e + f*x])/
(a + I*b)]*Sec[e + f*x]^2*(a + b*Tan[e + f*x]))/(-I + Tan[e + f*x]) - (m*AppellF1[1 + n, 1 - m/2, 1 - m/2, 2 +
n, (a + b*Tan[e + f*x])/(a - I*b), (a + b*Tan[e + f*x])/(a + I*b)]*Sec[e + f*x]^2*(a + b*Tan[e + f*x]))/(I +
Tan[e + f*x])))

Maple [F]

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

[In]

int((d*sec(f*x+e))^m*(a+b*tan(f*x+e))^n,x)

[Out]

int((d*sec(f*x+e))^m*(a+b*tan(f*x+e))^n,x)

Fricas [F]

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

[In]

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

[Out]

integral((d*sec(f*x + e))^m*(b*tan(f*x + e) + a)^n, x)

Sympy [F]

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

[In]

integrate((d*sec(f*x+e))**m*(a+b*tan(f*x+e))**n,x)

[Out]

Integral((d*sec(e + f*x))**m*(a + b*tan(e + f*x))**n, x)

Maxima [F]

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

[In]

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

[Out]

integrate((d*sec(f*x + e))^m*(b*tan(f*x + e) + a)^n, x)

Giac [F]

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

[In]

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

[Out]

integrate((d*sec(f*x + e))^m*(b*tan(f*x + e) + a)^n, x)

Mupad [F(-1)]

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

[In]

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

[Out]

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