\(\int (a+a \sec (c+d x))^n \tan ^4(c+d x) \, dx\) [225]

Optimal result

Integrand size = 21, antiderivative size = 106 \[ \int (a+a \sec (c+d x))^n \tan ^4(c+d x) \, dx=\frac {2^{5+n} \operatorname {AppellF1}\left (\frac {5}{2},4+n,1,\frac {7}{2},-\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)},\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)}\right ) \left (\frac {1}{1+\sec (c+d x)}\right )^{5+n} (a+a \sec (c+d x))^n \tan ^5(c+d x)}{5 d} \]

[Out]

1/5*2^(5+n)*AppellF1(5/2,4+n,1,7/2,(-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)))^(5+n)*(a+a*sec(d*x+c))^n*tan(d*x+c)^5/d

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {3974} \[ \int (a+a \sec (c+d x))^n \tan ^4(c+d x) \, dx=\frac {2^{n+5} \tan ^5(c+d x) \left (\frac {1}{\sec (c+d x)+1}\right )^{n+5} (a \sec (c+d x)+a)^n \operatorname {AppellF1}\left (\frac {5}{2},n+4,1,\frac {7}{2},-\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a}\right )}{5 d} \]

[In]

Int[(a + a*Sec[c + d*x])^n*Tan[c + d*x]^4,x]

[Out]

(2^(5 + n)*AppellF1[5/2, 4 + n, 1, 7/2, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a
+ a*Sec[c + d*x])]*((1 + Sec[c + d*x])^(-1))^(5 + n)*(a + a*Sec[c + d*x])^n*Tan[c + d*x]^5)/(5*d)

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^{5+n} \operatorname {AppellF1}\left (\frac {5}{2},4+n,1,\frac {7}{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 )^{5+n} (a+a \sec (c+d x))^n \tan ^5(c+d x)}{5 d} \\ \end{align*}

Mathematica [F]

\[ \int (a+a \sec (c+d x))^n \tan ^4(c+d x) \, dx=\int (a+a \sec (c+d x))^n \tan ^4(c+d x) \, dx \]

[In]

Integrate[(a + a*Sec[c + d*x])^n*Tan[c + d*x]^4,x]

[Out]

Integrate[(a + a*Sec[c + d*x])^n*Tan[c + d*x]^4, x]

Maple [F]

\[\int \left (a +a \sec \left (d x +c \right )\right )^{n} \tan \left (d x +c \right )^{4}d x\]

[In]

int((a+a*sec(d*x+c))^n*tan(d*x+c)^4,x)

[Out]

int((a+a*sec(d*x+c))^n*tan(d*x+c)^4,x)

Fricas [F]

\[ \int (a+a \sec (c+d x))^n \tan ^4(c+d x) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{4} \,d x } \]

[In]

integrate((a+a*sec(d*x+c))^n*tan(d*x+c)^4,x, algorithm="fricas")

[Out]

integral((a*sec(d*x + c) + a)^n*tan(d*x + c)^4, x)

Sympy [F]

\[ \int (a+a \sec (c+d x))^n \tan ^4(c+d x) \, dx=\int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{n} \tan ^{4}{\left (c + d x \right )}\, dx \]

[In]

integrate((a+a*sec(d*x+c))**n*tan(d*x+c)**4,x)

[Out]

Integral((a*(sec(c + d*x) + 1))**n*tan(c + d*x)**4, x)

Maxima [F]

\[ \int (a+a \sec (c+d x))^n \tan ^4(c+d x) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{4} \,d x } \]

[In]

integrate((a+a*sec(d*x+c))^n*tan(d*x+c)^4,x, algorithm="maxima")

[Out]

integrate((a*sec(d*x + c) + a)^n*tan(d*x + c)^4, x)

Giac [F]

\[ \int (a+a \sec (c+d x))^n \tan ^4(c+d x) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{4} \,d x } \]

[In]

integrate((a+a*sec(d*x+c))^n*tan(d*x+c)^4,x, algorithm="giac")

[Out]

integrate((a*sec(d*x + c) + a)^n*tan(d*x + c)^4, x)

Mupad [F(-1)]

Timed out. \[ \int (a+a \sec (c+d x))^n \tan ^4(c+d x) \, dx=\int {\mathrm {tan}\left (c+d\,x\right )}^4\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n \,d x \]

[In]

int(tan(c + d*x)^4*(a + a/cos(c + d*x))^n,x)

[Out]

int(tan(c + d*x)^4*(a + a/cos(c + d*x))^n, x)