Optimal. Leaf size=66 \[ -\frac{i \cos ^{-n}(c+d x) \text{Hypergeometric2F1}\left (1,n,n+1,\frac{1}{2} (1+i \tan (c+d x))\right ) (a \cos (c+d x)+i a \sin (c+d x))^n}{2 d n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0611622, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03, Rules used = {3084} \[ -\frac{i \cos ^{-n}(c+d x) \, _2F_1\left (1,n;n+1;\frac{1}{2} (i \tan (c+d x)+1)\right ) (a \cos (c+d x)+i a \sin (c+d x))^n}{2 d n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3084
Rubi steps
\begin{align*} \int \cos ^{-n}(c+d x) (a \cos (c+d x)+i a \sin (c+d x))^n \, dx &=-\frac{i \cos ^{-n}(c+d x) \, _2F_1\left (1,n;1+n;\frac{1}{2} (1+i \tan (c+d x))\right ) (a \cos (c+d x)+i a \sin (c+d x))^n}{2 d n}\\ \end{align*}
Mathematica [A] time = 2.07308, size = 90, normalized size = 1.36 \[ \frac{\cos ^{-n}(c+d x) \left (n (\tan (c+d x)-i) \text{Hypergeometric2F1}\left (1,n+1,n+2,\frac{1}{2} (1+i \tan (c+d x))\right )-2 i (n+1)\right ) (a (\cos (c+d x)+i \sin (c+d x)))^n}{4 d n (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.586, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a\cos \left ( dx+c \right ) +ia\sin \left ( dx+c \right ) \right ) ^{n}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{n}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \cos \left (d x + c\right ) + i \, a \sin \left (d x + c\right )\right )}^{n} \cos \left (d x + c\right )^{-n}\,{d x} \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 (\frac{\left (a e^{\left (i \, d x + i \, c\right )}\right )^{n}}{\left (\frac{1}{2} \,{\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (-i \, d x - i \, c\right )}\right )^{n}}, 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 \frac{{\left (a \cos \left (d x + c\right ) + i \, a \sin \left (d x + c\right )\right )}^{n}}{\cos \left (d x + c\right )^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]