Optimal. Leaf size=121 \[ \frac{i a 2^{\frac{n+3}{2}} (1+i \tan (c+d x))^{\frac{1}{2} (-n-1)} (a+i a \tan (c+d x))^{n-1} (e \sec (c+d x))^{3-n} \text{Hypergeometric2F1}\left (\frac{1}{2} (-n-1),\frac{3-n}{2},\frac{5-n}{2},\frac{1}{2} (1-i \tan (c+d x))\right )}{d (3-n)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.235446, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3505, 3523, 70, 69} \[ \frac{i a 2^{\frac{n+3}{2}} (1+i \tan (c+d x))^{\frac{1}{2} (-n-1)} (a+i a \tan (c+d x))^{n-1} (e \sec (c+d x))^{3-n} \text{Hypergeometric2F1}\left (\frac{1}{2} (-n-1),\frac{3-n}{2},\frac{5-n}{2},\frac{1}{2} (1-i \tan (c+d x))\right )}{d (3-n)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3505
Rule 3523
Rule 70
Rule 69
Rubi steps
\begin{align*} \int (e \sec (c+d x))^{3-n} (a+i a \tan (c+d x))^n \, dx &=\left ((e \sec (c+d x))^{3-n} (a-i a \tan (c+d x))^{\frac{1}{2} (-3+n)} (a+i a \tan (c+d x))^{\frac{1}{2} (-3+n)}\right ) \int (a-i a \tan (c+d x))^{\frac{3-n}{2}} (a+i a \tan (c+d x))^{\frac{3-n}{2}+n} \, dx\\ &=\frac{\left (a^2 (e \sec (c+d x))^{3-n} (a-i a \tan (c+d x))^{\frac{1}{2} (-3+n)} (a+i a \tan (c+d x))^{\frac{1}{2} (-3+n)}\right ) \operatorname{Subst}\left (\int (a-i a x)^{-1+\frac{3-n}{2}} (a+i a x)^{-1+\frac{3-n}{2}+n} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\left (2^{\frac{1}{2}+\frac{n}{2}} a^2 (e \sec (c+d x))^{3-n} (a-i a \tan (c+d x))^{\frac{1}{2} (-3+n)} (a+i a \tan (c+d x))^{\frac{1}{2}+\frac{1}{2} (-3+n)+\frac{n}{2}} \left (\frac{a+i a \tan (c+d x)}{a}\right )^{-\frac{1}{2}-\frac{n}{2}}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2}+\frac{i x}{2}\right )^{-1+\frac{3-n}{2}+n} (a-i a x)^{-1+\frac{3-n}{2}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{i 2^{\frac{3+n}{2}} a \, _2F_1\left (\frac{1}{2} (-1-n),\frac{3-n}{2};\frac{5-n}{2};\frac{1}{2} (1-i \tan (c+d x))\right ) (e \sec (c+d x))^{3-n} (1+i \tan (c+d x))^{\frac{1}{2} (-1-n)} (a+i a \tan (c+d x))^{-1+n}}{d (3-n)}\\ \end{align*}
Mathematica [A] time = 11.0644, size = 116, normalized size = 0.96 \[ \frac{8 e^3 (\tan (d x)+i) \sec (d x) (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n} \text{Hypergeometric2F1}\left (3,\frac{3-n}{2},\frac{5-n}{2},-\cos (2 (c+d x))+i \sin (2 (c+d x))\right )}{d (n-3) (\cos (c)+i \sin (c))^3 (\tan (d x)-i)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.886, size = 0, normalized size = 0. \begin{align*} \int \left ( e\sec \left ( dx+c \right ) \right ) ^{3-n} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \, d x + 2 i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{n} \left (\frac{2 \, e e^{\left (i \, d x + i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{-n + 3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (e \sec \left (d x + c\right )\right )^{-n + 3}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]