Optimal. Leaf size=122 \[ \frac{(a+i a \tan (e+f x))^m \, _2F_1\left (1,m;m+1;\frac{1}{2} (i \tan (e+f x)+1)\right )}{2 f m (d+i c)}-\frac{d (a+i a \tan (e+f x))^m \, _2F_1\left (1,m;m+1;-\frac{d (i \tan (e+f x)+1)}{i c-d}\right )}{f m \left (c^2+d^2\right )} \]
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Rubi [A] time = 0.259604, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3562, 3481, 68, 3599} \[ \frac{(a+i a \tan (e+f x))^m \, _2F_1\left (1,m;m+1;\frac{1}{2} (i \tan (e+f x)+1)\right )}{2 f m (d+i c)}-\frac{d (a+i a \tan (e+f x))^m \, _2F_1\left (1,m;m+1;-\frac{d (i \tan (e+f x)+1)}{i c-d}\right )}{f m \left (c^2+d^2\right )} \]
Antiderivative was successfully verified.
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Rule 3562
Rule 3481
Rule 68
Rule 3599
Rubi steps
\begin{align*} \int \frac{(a+i a \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx &=\frac{\int (a+i a \tan (e+f x))^m \, dx}{c-i d}-\frac{d \int \frac{(a+i a \tan (e+f x))^m (i a+a \tan (e+f x))}{c+d \tan (e+f x)} \, dx}{a (c-i d)}\\ &=\frac{a \operatorname{Subst}\left (\int \frac{(a+x)^{-1+m}}{a-x} \, dx,x,i a \tan (e+f x)\right )}{(i c+d) f}+\frac{(a d) \operatorname{Subst}\left (\int \frac{(a+i a x)^{-1+m}}{c+d x} \, dx,x,\tan (e+f x)\right )}{(i c+d) f}\\ &=\frac{\, _2F_1\left (1,m;1+m;\frac{1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m}{2 (i c+d) f m}-\frac{d \, _2F_1\left (1,m;1+m;-\frac{d (1+i \tan (e+f x))}{i c-d}\right ) (a+i a \tan (e+f x))^m}{\left (c^2+d^2\right ) f m}\\ \end{align*}
Mathematica [F] time = 13.8814, size = 0, normalized size = 0. \[ \int \frac{(a+i a \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.348, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+ia\tan \left ( fx+e \right ) \right ) ^{m}}{c+d\tan \left ( fx+e \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}}{d \tan \left (f x + e\right ) + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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 (i \, e^{\left (2 i \, f x + 2 i \, e\right )} + i\right )}}{{\left (i \, c + d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c - d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a \left (i \tan{\left (e + f x \right )} + 1\right )\right )^{m}}{c + d \tan{\left (e + f x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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}}{d \tan \left (f x + e\right ) + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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