Optimal. Leaf size=95 \[ \frac{\tan (e+f x) (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m (d \cot (e+f x))^n F_1(1-n;1-m,1;2-n;-i \tan (e+f x),i \tan (e+f x))}{f (1-n)} \]
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Rubi [A] time = 0.164415, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {4241, 3564, 135, 133} \[ \frac{\tan (e+f x) (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m (d \cot (e+f x))^n F_1(1-n;1-m,1;2-n;-i \tan (e+f x),i \tan (e+f x))}{f (1-n)} \]
Antiderivative was successfully verified.
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Rule 4241
Rule 3564
Rule 135
Rule 133
Rubi steps
\begin{align*} \int (d \cot (e+f x))^n (a+i a \tan (e+f x))^m \, dx &=\left ((d \cot (e+f x))^n (d \tan (e+f x))^n\right ) \int (d \tan (e+f x))^{-n} (a+i a \tan (e+f x))^m \, dx\\ &=\frac{\left (i a^2 (d \cot (e+f x))^n (d \tan (e+f x))^n\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{i d x}{a}\right )^{-n} (a+x)^{-1+m}}{-a^2+a x} \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=\frac{\left (i a (d \cot (e+f x))^n (1+i \tan (e+f x))^{-m} (d \tan (e+f x))^n (a+i a \tan (e+f x))^m\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{i d x}{a}\right )^{-n} \left (1+\frac{x}{a}\right )^{-1+m}}{-a^2+a x} \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=\frac{F_1(1-n;1-m,1;2-n;-i \tan (e+f x),i \tan (e+f x)) (d \cot (e+f x))^n (1+i \tan (e+f x))^{-m} \tan (e+f x) (a+i a \tan (e+f x))^m}{f (1-n)}\\ \end{align*}
Mathematica [F] time = 5.34427, size = 0, normalized size = 0. \[ \int (d \cot (e+f x))^n (a+i a \tan (e+f x))^m \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.792, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cot \left ( fx+e \right ) \right ) ^{n} \left ( a+ia\tan \left ( fx+e \right ) \right ) ^{m}\, 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 \left (d \cot \left (f x + e\right )\right )^{n}{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}\,{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 (\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 (\frac{i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} - 1}\right )^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \cot \left (f x + e\right )\right )^{n}{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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