\(\int (a+a \sec (c+d x))^n (e \tan (c+d x))^m \, dx\) [208]

Optimal result

Integrand size = 23, antiderivative size = 125 \[ \int (a+a \sec (c+d x))^n (e \tan (c+d x))^m \, dx=\frac {2^{1+m+n} \operatorname {AppellF1}\left (\frac {1+m}{2},m+n,1,\frac {3+m}{2},-\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)},\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)}\right ) \left (\frac {1}{1+\sec (c+d x)}\right )^{1+m+n} (a+a \sec (c+d x))^n (e \tan (c+d x))^{1+m}}{d e (1+m)} \]

[Out]

2^(1+m+n)*AppellF1(1/2+1/2*m,m+n,1,3/2+1/2*m,(-a+a*sec(d*x+c))/(a+a*sec(d*x+c)),(a-a*sec(d*x+c))/(a+a*sec(d*x+
c)))*(1/(1+sec(d*x+c)))^(1+m+n)*(a+a*sec(d*x+c))^n*(e*tan(d*x+c))^(1+m)/d/e/(1+m)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {3974} \[ \int (a+a \sec (c+d x))^n (e \tan (c+d x))^m \, dx=\frac {2^{m+n+1} (a \sec (c+d x)+a)^n (e \tan (c+d x))^{m+1} \left (\frac {1}{\sec (c+d x)+1}\right )^{m+n+1} \operatorname {AppellF1}\left (\frac {m+1}{2},m+n,1,\frac {m+3}{2},-\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a}\right )}{d e (m+1)} \]

[In]

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

[Out]

(2^(1 + m + n)*AppellF1[(1 + m)/2, m + n, 1, (3 + m)/2, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*S
ec[c + d*x])/(a + a*Sec[c + d*x])]*((1 + Sec[c + d*x])^(-1))^(1 + m + n)*(a + a*Sec[c + d*x])^n*(e*Tan[c + d*x
])^(1 + m))/(d*e*(1 + m))

Rule 3974

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[(-2^(m
+ n + 1))*(e*Cot[c + d*x])^(m + 1)*((a + b*Csc[c + d*x])^n/(d*e*(m + 1)))*(a/(a + b*Csc[c + d*x]))^(m + n + 1
)*AppellF1[(m + 1)/2, m + n, 1, (m + 3)/2, -(a - b*Csc[c + d*x])/(a + b*Csc[c + d*x]), (a - b*Csc[c + d*x])/(a
+ b*Csc[c + d*x])], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = \frac {2^{1+m+n} \operatorname {AppellF1}\left (\frac {1+m}{2},m+n,1,\frac {3+m}{2},-\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)},\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)}\right ) \left (\frac {1}{1+\sec (c+d x)}\right )^{1+m+n} (a+a \sec (c+d x))^n (e \tan (c+d x))^{1+m}}{d e (1+m)} \\ \end{align*}

Mathematica [F]

\[ \int (a+a \sec (c+d x))^n (e \tan (c+d x))^m \, dx=\int (a+a \sec (c+d x))^n (e \tan (c+d x))^m \, dx \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

\[ \int (a+a \sec (c+d x))^n (e \tan (c+d x))^m \, dx=\int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{n} \left (e \tan {\left (c + d x \right )}\right )^{m}\, dx \]

[In]

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

[Out]

Integral((a*(sec(c + d*x) + 1))**n*(e*tan(c + d*x))**m, x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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