Integrand size = 19, antiderivative size = 43 \[ \int (a+a \sec (c+d x))^n \tan (c+d x) \, dx=-\frac {\operatorname {Hypergeometric2F1}(1,1+n,2+n,1+\sec (c+d x)) (a+a \sec (c+d x))^{1+n}}{a d (1+n)} \] Output:
-hypergeom([1, 1+n],[2+n],1+sec(d*x+c))*(a+a*sec(d*x+c))^(1+n)/a/d/(1+n)
Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int (a+a \sec (c+d x))^n \tan (c+d x) \, dx=-\frac {\operatorname {Hypergeometric2F1}(1,1+n,2+n,1+\sec (c+d x)) (a (1+\sec (c+d x)))^{1+n}}{a d (1+n)} \] Input:
Integrate[(a + a*Sec[c + d*x])^n*Tan[c + d*x],x]
Output:
-((Hypergeometric2F1[1, 1 + n, 2 + n, 1 + Sec[c + d*x]]*(a*(1 + Sec[c + d* x]))^(1 + n))/(a*d*(1 + n)))
Time = 0.23 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3042, 25, 4368, 75}
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.
\(\displaystyle \int \tan (c+d x) (a \sec (c+d x)+a)^n \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\cot \left (c+d x+\frac {\pi }{2}\right ) \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^ndx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \cot \left (\frac {1}{2} (2 c+\pi )+d x\right ) \left (\csc \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )^ndx\) |
\(\Big \downarrow \) 4368 |
\(\displaystyle \frac {\int \cos (c+d x) (\sec (c+d x) a+a)^nd\sec (c+d x)}{d}\) |
\(\Big \downarrow \) 75 |
\(\displaystyle -\frac {(a \sec (c+d x)+a)^{n+1} \operatorname {Hypergeometric2F1}(1,n+1,n+2,\sec (c+d x)+1)}{a d (n+1)}\) |
Input:
Int[(a + a*Sec[c + d*x])^n*Tan[c + d*x],x]
Output:
-((Hypergeometric2F1[1, 1 + n, 2 + n, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x ])^(1 + n))/(a*d*(1 + n)))
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x )^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[n] && (IntegerQ[m] || GtQ[-d/(b*c), 0])
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _), x_Symbol] :> Simp[-(d*b^(m - 1))^(-1) Subst[Int[(-a + b*x)^((m - 1)/2 )*((a + b*x)^((m - 1)/2 + n)/x), x], x, Csc[c + d*x]], x] /; FreeQ[{a, b, c , d, n}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && !IntegerQ[n]
\[\int \left (a +a \sec \left (d x +c \right )\right )^{n} \tan \left (d x +c \right )d x\]
Input:
int((a+a*sec(d*x+c))^n*tan(d*x+c),x)
Output:
int((a+a*sec(d*x+c))^n*tan(d*x+c),x)
\[ \int (a+a \sec (c+d x))^n \tan (c+d x) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right ) \,d x } \] Input:
integrate((a+a*sec(d*x+c))^n*tan(d*x+c),x, algorithm="fricas")
Output:
integral((a*sec(d*x + c) + a)^n*tan(d*x + c), x)
\[ \int (a+a \sec (c+d x))^n \tan (c+d x) \, dx=\int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{n} \tan {\left (c + d x \right )}\, dx \] Input:
integrate((a+a*sec(d*x+c))**n*tan(d*x+c),x)
Output:
Integral((a*(sec(c + d*x) + 1))**n*tan(c + d*x), x)
\[ \int (a+a \sec (c+d x))^n \tan (c+d x) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right ) \,d x } \] Input:
integrate((a+a*sec(d*x+c))^n*tan(d*x+c),x, algorithm="maxima")
Output:
integrate((a*sec(d*x + c) + a)^n*tan(d*x + c), x)
\[ \int (a+a \sec (c+d x))^n \tan (c+d x) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right ) \,d x } \] Input:
integrate((a+a*sec(d*x+c))^n*tan(d*x+c),x, algorithm="giac")
Output:
integrate((a*sec(d*x + c) + a)^n*tan(d*x + c), x)
Timed out. \[ \int (a+a \sec (c+d x))^n \tan (c+d x) \, dx=\int \mathrm {tan}\left (c+d\,x\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n \,d x \] Input:
int(tan(c + d*x)*(a + a/cos(c + d*x))^n,x)
Output:
int(tan(c + d*x)*(a + a/cos(c + d*x))^n, x)
\[ \int (a+a \sec (c+d x))^n \tan (c+d x) \, dx=\frac {\left (\sec \left (d x +c \right ) a +a \right )^{n}+\left (\int \frac {\left (\sec \left (d x +c \right ) a +a \right )^{n} \tan \left (d x +c \right )}{\sec \left (d x +c \right )+1}d x \right ) d n}{d n} \] Input:
int((a+a*sec(d*x+c))^n*tan(d*x+c),x)
Output:
((sec(c + d*x)*a + a)**n + int(((sec(c + d*x)*a + a)**n*tan(c + d*x))/(sec (c + d*x) + 1),x)*d*n)/(d*n)