Optimal. Leaf size=52 \[ -\frac{i (a+i a \tan (e+f x))^m \, _2F_1\left (2,m;m+1;\frac{1}{2} (i \tan (e+f x)+1)\right )}{4 c f m} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.127838, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {3522, 3487, 68} \[ -\frac{i (a+i a \tan (e+f x))^m \, _2F_1\left (2,m;m+1;\frac{1}{2} (i \tan (e+f x)+1)\right )}{4 c f m} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3522
Rule 3487
Rule 68
Rubi steps
\begin{align*} \int \frac{(a+i a \tan (e+f x))^m}{c-i c \tan (e+f x)} \, dx &=\frac{\int \cos ^2(e+f x) (a+i a \tan (e+f x))^{1+m} \, dx}{a c}\\ &=-\frac{\left (i a^2\right ) \operatorname{Subst}\left (\int \frac{(a+x)^{-1+m}}{(a-x)^2} \, dx,x,i a \tan (e+f x)\right )}{c f}\\ &=-\frac{i \, _2F_1\left (2,m;1+m;\frac{1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m}{4 c f m}\\ \end{align*}
Mathematica [B] time = 15.8075, size = 133, normalized size = 2.56 \[ -\frac{i 2^{m-2} \left (1+e^{2 i (e+f x)}\right )^2 \left (e^{i f x}\right )^m \left (\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^m \, _2F_1\left (1,2;m+1;-e^{2 i (e+f x)}\right ) \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m}{c f m} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.815, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+ia\tan \left ( fx+e \right ) \right ) ^{m}}{c-ic\tan \left ( fx+e \right ) }}\, 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 (\frac{\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 )^{m}{\left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}}{2 \, c}, 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 \frac{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}}{-i \, c \tan \left (f x + e\right ) + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]