Optimal. Leaf size=66 \[ \frac{i (a+i a \tan (e+f x))^{-n} (d \sec (e+f x))^{2 n} \text{Hypergeometric2F1}\left (3,n,n+1,\frac{1}{2} (1-i \tan (e+f x))\right )}{8 a^2 f n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.189005, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3505, 3522, 3487, 68} \[ \frac{i (a+i a \tan (e+f x))^{-n} (d \sec (e+f x))^{2 n} \text{Hypergeometric2F1}\left (3,n,n+1,\frac{1}{2} (1-i \tan (e+f x))\right )}{8 a^2 f n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3505
Rule 3522
Rule 3487
Rule 68
Rubi steps
\begin{align*} \int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{-2-n} \, dx &=\left ((d \sec (e+f x))^{2 n} (a-i a \tan (e+f x))^{-n} (a+i a \tan (e+f x))^{-n}\right ) \int \frac{(a-i a \tan (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx\\ &=\frac{\left ((d \sec (e+f x))^{2 n} (a-i a \tan (e+f x))^{-n} (a+i a \tan (e+f x))^{-n}\right ) \int \cos ^4(e+f x) (a-i a \tan (e+f x))^{2+n} \, dx}{a^4}\\ &=\frac{\left (i a (d \sec (e+f x))^{2 n} (a-i a \tan (e+f x))^{-n} (a+i a \tan (e+f x))^{-n}\right ) \operatorname{Subst}\left (\int \frac{(a+x)^{-1+n}}{(a-x)^3} \, dx,x,-i a \tan (e+f x)\right )}{f}\\ &=\frac{i \, _2F_1\left (3,n;1+n;\frac{1}{2} (1-i \tan (e+f x))\right ) (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{-n}}{8 a^2 f n}\\ \end{align*}
Mathematica [B] time = 118.315, size = 165, normalized size = 2.5 \[ -\frac{i e^{2 i e} 2^{n-3} \left (1+e^{2 i (e+f x)}\right )^3 \left (e^{i f x}\right )^{-n} \left (\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^n \sec ^{2-n}(e+f x) (\cos (f x)+i \sin (f x))^{n+2} \text{Hypergeometric2F1}\left (3,3-n,4-n,1+e^{2 i (e+f x)}\right ) (a+i a \tan (e+f x))^{-n-2} (d \sec (e+f x))^{2 n}}{f (n-3)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 1.473, size = 0, normalized size = 0. \begin{align*} \int \left ( d\sec \left ( fx+e \right ) \right ) ^{2\,n} \left ( a+ia\tan \left ( fx+e \right ) \right ) ^{-2-n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (\frac{2 \, a e^{\left (2 i \, f x + 2 i \, e\right )}}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{-n - 2} \left (\frac{2 \, d e^{\left (i \, f x + i \, e\right )}}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{2 \, n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \sec \left (f x + e\right )\right )^{2 \, n}{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{-n - 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]