Integrand size = 30, antiderivative size = 101 \[ \int (e \sec (c+d x))^{-1-2 n} (a+i a \tan (c+d x))^n \, dx=\frac {i 2^{-\frac {1}{2}-n} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2} (3+2 n),\frac {1}{2},\frac {1}{2} (1+i \tan (c+d x))\right ) (e \sec (c+d x))^{-1-2 n} (1-i \tan (c+d x))^{-1+\frac {1}{2} (3+2 n)} (a+i a \tan (c+d x))^n}{d} \] Output:
I*2^(-1/2-n)*hypergeom([-1/2, 3/2+n],[1/2],1/2+1/2*I*tan(d*x+c))*(e*sec(d* x+c))^(-1-2*n)*(1-I*tan(d*x+c))^(1/2+n)*(a+I*a*tan(d*x+c))^n/d
Time = 9.88 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.55 \[ \int (e \sec (c+d x))^{-1-2 n} (a+i a \tan (c+d x))^n \, dx=\frac {i 2^{-1-n} \left (e^{i d x}\right )^n \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{-1-n} \left (1+e^{2 i (c+d x)}\right )^{-1-n} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-1-n,\frac {1}{2},-e^{2 i (c+d x)}\right ) \sec ^{1+n}(c+d x) (e \sec (c+d x))^{-1-2 n} (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n}{d} \] Input:
Integrate[(e*Sec[c + d*x])^(-1 - 2*n)*(a + I*a*Tan[c + d*x])^n,x]
Output:
(I*2^(-1 - n)*(E^(I*d*x))^n*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(- 1 - n)*(1 + E^((2*I)*(c + d*x)))^(-1 - n)*Hypergeometric2F1[-1/2, -1 - n, 1/2, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(1 + n)*(e*Sec[c + d*x])^(-1 - 2*n )*(a + I*a*Tan[c + d*x])^n)/(d*(Cos[d*x] + I*Sin[d*x])^n)
Time = 0.54 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.36, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3042, 3986, 3042, 4006, 80, 79}
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.
\(\displaystyle \int (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-2 n-1} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-2 n-1}dx\) |
\(\Big \downarrow \) 3986 |
\(\displaystyle (a-i a \tan (c+d x))^{\frac {1}{2} (2 n+1)} (a+i a \tan (c+d x))^{\frac {1}{2} (2 n+1)} (e \sec (c+d x))^{-2 n-1} \int \frac {(a-i a \tan (c+d x))^{\frac {1}{2} (-2 n-1)}}{\sqrt {i \tan (c+d x) a+a}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle (a-i a \tan (c+d x))^{\frac {1}{2} (2 n+1)} (a+i a \tan (c+d x))^{\frac {1}{2} (2 n+1)} (e \sec (c+d x))^{-2 n-1} \int \frac {(a-i a \tan (c+d x))^{\frac {1}{2} (-2 n-1)}}{\sqrt {i \tan (c+d x) a+a}}dx\) |
\(\Big \downarrow \) 4006 |
\(\displaystyle \frac {a^2 (a-i a \tan (c+d x))^{\frac {1}{2} (2 n+1)} (a+i a \tan (c+d x))^{\frac {1}{2} (2 n+1)} (e \sec (c+d x))^{-2 n-1} \int \frac {(a-i a \tan (c+d x))^{\frac {1}{2} (-2 n-3)}}{(i \tan (c+d x) a+a)^{3/2}}d\tan (c+d x)}{d}\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {a 2^{-n-\frac {3}{2}} (1-i \tan (c+d x))^{n+\frac {1}{2}} (a-i a \tan (c+d x))^{-n+\frac {1}{2} (2 n+1)-\frac {1}{2}} (a+i a \tan (c+d x))^{\frac {1}{2} (2 n+1)} (e \sec (c+d x))^{-2 n-1} \int \frac {\left (\frac {1}{2}-\frac {1}{2} i \tan (c+d x)\right )^{\frac {1}{2} (-2 n-3)}}{(i \tan (c+d x) a+a)^{3/2}}d\tan (c+d x)}{d}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {i 2^{-n-\frac {1}{2}} (1-i \tan (c+d x))^{n+\frac {1}{2}} (a-i a \tan (c+d x))^{-n+\frac {1}{2} (2 n+1)-\frac {1}{2}} (a+i a \tan (c+d x))^{\frac {1}{2} (2 n+1)-\frac {1}{2}} (e \sec (c+d x))^{-2 n-1} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2} (2 n+3),\frac {1}{2},\frac {1}{2} (i \tan (c+d x)+1)\right )}{d}\) |
Input:
Int[(e*Sec[c + d*x])^(-1 - 2*n)*(a + I*a*Tan[c + d*x])^n,x]
Output:
(I*2^(-1/2 - n)*Hypergeometric2F1[-1/2, (3 + 2*n)/2, 1/2, (1 + I*Tan[c + d *x])/2]*(e*Sec[c + d*x])^(-1 - 2*n)*(1 - I*Tan[c + d*x])^(1/2 + n)*(a - I* a*Tan[c + d*x])^(-1/2 - n + (1 + 2*n)/2)*(a + I*a*Tan[c + d*x])^(-1/2 + (1 + 2*n)/2))/d
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) ^FracPart[n]) Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) ), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !Integ erQ[n] && (RationalQ[m] || !SimplerQ[n + 1, m + 1])
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)])^(n_.), x_Symbol] :> Simp[(d*Sec[e + f*x])^m/((a + b*Tan[e + f*x])^(m/ 2)*(a - b*Tan[e + f*x])^(m/2)) Int[(a + b*Tan[e + f*x])^(m/2 + n)*(a - b* Tan[e + f*x])^(m/2), x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(c/f) Subst[Int[(a + b*x)^(m - 1)*( c + d*x)^(n - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n }, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]
\[\int \left (e \sec \left (d x +c \right )\right )^{-1-2 n} \left (a +i a \tan \left (d x +c \right )\right )^{n}d x\]
Input:
int((e*sec(d*x+c))^(-1-2*n)*(a+I*a*tan(d*x+c))^n,x)
Output:
int((e*sec(d*x+c))^(-1-2*n)*(a+I*a*tan(d*x+c))^n,x)
\[ \int (e \sec (c+d x))^{-1-2 n} (a+i a \tan (c+d x))^n \, dx=\int { \left (e \sec \left (d x + c\right )\right )^{-2 \, n - 1} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \,d x } \] Input:
integrate((e*sec(d*x+c))^(-1-2*n)*(a+I*a*tan(d*x+c))^n,x, algorithm="frica s")
Output:
integral((2*e*e^(I*d*x + I*c)/(e^(2*I*d*x + 2*I*c) + 1))^(-2*n - 1)*e^(I*d *n*x + I*c*n + n*log(2*e*e^(I*d*x + I*c)/(e^(2*I*d*x + 2*I*c) + 1)) + n*lo g(a/e)), x)
\[ \int (e \sec (c+d x))^{-1-2 n} (a+i a \tan (c+d x))^n \, dx=\int \left (e \sec {\left (c + d x \right )}\right )^{- 2 n - 1} \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{n}\, dx \] Input:
integrate((e*sec(d*x+c))**(-1-2*n)*(a+I*a*tan(d*x+c))**n,x)
Output:
Integral((e*sec(c + d*x))**(-2*n - 1)*(I*a*(tan(c + d*x) - I))**n, x)
\[ \int (e \sec (c+d x))^{-1-2 n} (a+i a \tan (c+d x))^n \, dx=\int { \left (e \sec \left (d x + c\right )\right )^{-2 \, n - 1} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \,d x } \] Input:
integrate((e*sec(d*x+c))^(-1-2*n)*(a+I*a*tan(d*x+c))^n,x, algorithm="maxim a")
Output:
integrate((e*sec(d*x + c))^(-2*n - 1)*(I*a*tan(d*x + c) + a)^n, x)
\[ \int (e \sec (c+d x))^{-1-2 n} (a+i a \tan (c+d x))^n \, dx=\int { \left (e \sec \left (d x + c\right )\right )^{-2 \, n - 1} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \,d x } \] Input:
integrate((e*sec(d*x+c))^(-1-2*n)*(a+I*a*tan(d*x+c))^n,x, algorithm="giac" )
Output:
integrate((e*sec(d*x + c))^(-2*n - 1)*(I*a*tan(d*x + c) + a)^n, x)
Timed out. \[ \int (e \sec (c+d x))^{-1-2 n} (a+i a \tan (c+d x))^n \, dx=\int \frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^n}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{2\,n+1}} \,d x \] Input:
int((a + a*tan(c + d*x)*1i)^n/(e/cos(c + d*x))^(2*n + 1),x)
Output:
int((a + a*tan(c + d*x)*1i)^n/(e/cos(c + d*x))^(2*n + 1), x)
\[ \int (e \sec (c+d x))^{-1-2 n} (a+i a \tan (c+d x))^n \, dx=\frac {\int \frac {\left (\tan \left (d x +c \right ) a i +a \right )^{n}}{\sec \left (d x +c \right )^{2 n} \sec \left (d x +c \right )}d x}{e^{2 n} e} \] Input:
int((e*sec(d*x+c))^(-1-2*n)*(a+I*a*tan(d*x+c))^n,x)
Output:
int((tan(c + d*x)*a*i + a)**n/(sec(c + d*x)**(2*n)*sec(c + d*x)),x)/(e**(2 *n)*e)