Optimal. Leaf size=193 \[ \frac{\tan (e+f x) (d \cot (e+f x))^n (a+b \tan (e+f x))^m \left (\frac{b \tan (e+f x)}{a}+1\right )^{-m} F_1\left (1-n;-m,1;2-n;-\frac{b \tan (e+f x)}{a},-i \tan (e+f x)\right )}{2 f (1-n)}+\frac{\tan (e+f x) (d \cot (e+f x))^n (a+b \tan (e+f x))^m \left (\frac{b \tan (e+f x)}{a}+1\right )^{-m} F_1\left (1-n;-m,1;2-n;-\frac{b \tan (e+f x)}{a},i \tan (e+f x)\right )}{2 f (1-n)} \]
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Rubi [A] time = 0.254485, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {4241, 3575, 912, 135, 133} \[ \frac{\tan (e+f x) (d \cot (e+f x))^n (a+b \tan (e+f x))^m \left (\frac{b \tan (e+f x)}{a}+1\right )^{-m} F_1\left (1-n;-m,1;2-n;-\frac{b \tan (e+f x)}{a},-i \tan (e+f x)\right )}{2 f (1-n)}+\frac{\tan (e+f x) (d \cot (e+f x))^n (a+b \tan (e+f x))^m \left (\frac{b \tan (e+f x)}{a}+1\right )^{-m} F_1\left (1-n;-m,1;2-n;-\frac{b \tan (e+f x)}{a},i \tan (e+f x)\right )}{2 f (1-n)} \]
Antiderivative was successfully verified.
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Rule 4241
Rule 3575
Rule 912
Rule 135
Rule 133
Rubi steps
\begin{align*} \int (d \cot (e+f x))^n (a+b \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+b \tan (e+f x))^m \, dx\\ &=\frac{\left ((d \cot (e+f x))^n (d \tan (e+f x))^n\right ) \operatorname{Subst}\left (\int \frac{(d x)^{-n} (a+b x)^m}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\left ((d \cot (e+f x))^n (d \tan (e+f x))^n\right ) \operatorname{Subst}\left (\int \left (\frac{i (d x)^{-n} (a+b x)^m}{2 (i-x)}+\frac{i (d x)^{-n} (a+b x)^m}{2 (i+x)}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\left (i (d \cot (e+f x))^n (d \tan (e+f x))^n\right ) \operatorname{Subst}\left (\int \frac{(d x)^{-n} (a+b x)^m}{i-x} \, dx,x,\tan (e+f x)\right )}{2 f}+\frac{\left (i (d \cot (e+f x))^n (d \tan (e+f x))^n\right ) \operatorname{Subst}\left (\int \frac{(d x)^{-n} (a+b x)^m}{i+x} \, dx,x,\tan (e+f x)\right )}{2 f}\\ &=\frac{\left (i (d \cot (e+f x))^n (d \tan (e+f x))^n (a+b \tan (e+f x))^m \left (1+\frac{b \tan (e+f x)}{a}\right )^{-m}\right ) \operatorname{Subst}\left (\int \frac{(d x)^{-n} \left (1+\frac{b x}{a}\right )^m}{i-x} \, dx,x,\tan (e+f x)\right )}{2 f}+\frac{\left (i (d \cot (e+f x))^n (d \tan (e+f x))^n (a+b \tan (e+f x))^m \left (1+\frac{b \tan (e+f x)}{a}\right )^{-m}\right ) \operatorname{Subst}\left (\int \frac{(d x)^{-n} \left (1+\frac{b x}{a}\right )^m}{i+x} \, dx,x,\tan (e+f x)\right )}{2 f}\\ &=\frac{F_1\left (1-n;-m,1;2-n;-\frac{b \tan (e+f x)}{a},-i \tan (e+f x)\right ) (d \cot (e+f x))^n \tan (e+f x) (a+b \tan (e+f x))^m \left (1+\frac{b \tan (e+f x)}{a}\right )^{-m}}{2 f (1-n)}+\frac{F_1\left (1-n;-m,1;2-n;-\frac{b \tan (e+f x)}{a},i \tan (e+f x)\right ) (d \cot (e+f x))^n \tan (e+f x) (a+b \tan (e+f x))^m \left (1+\frac{b \tan (e+f x)}{a}\right )^{-m}}{2 f (1-n)}\\ \end{align*}
Mathematica [F] time = 2.90663, size = 0, normalized size = 0. \[ \int (d \cot (e+f x))^n (a+b \tan (e+f x))^m \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.304, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cot \left ( fx+e \right ) \right ) ^{n} \left ( a+b\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 (b \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 (d \cot \left (f x + e\right )\right )^{n}{\left (b \tan \left (f x + e\right ) + a\right )}^{m}, 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 (b \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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