Optimal. Leaf size=21 \[ -(x (a-b))-\frac {a \cot (e+f x)}{f} \]
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Rubi [A] time = 0.03, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3629, 8} \[ x (-(a-b))-\frac {a \cot (e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3629
Rubi steps
\begin {align*} \int \cot ^2(e+f x) \left (a+b \tan ^2(e+f x)\right ) \, dx &=-\frac {a \cot (e+f x)}{f}+\int (-a+b) \, dx\\ &=-(a-b) x-\frac {a \cot (e+f x)}{f}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 34, normalized size = 1.62 \[ b x-\frac {a \cot (e+f x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2(e+f x)\right )}{f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 29, normalized size = 1.38 \[ -\frac {{\left (a - b\right )} f x \tan \left (f x + e\right ) + a}{f \tan \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.32, size = 46, normalized size = 2.19 \[ -\frac {2 \, {\left (f x + e\right )} {\left (a - b\right )} - a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + \frac {a}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.51, size = 31, normalized size = 1.48 \[ \frac {\left (f x +e \right ) b +a \left (-\cot \left (f x +e \right )-f x -e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 27, normalized size = 1.29 \[ -\frac {{\left (f x + e\right )} {\left (a - b\right )} + \frac {a}{\tan \left (f x + e\right )}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.47, size = 21, normalized size = 1.00 \[ -x\,\left (a-b\right )-\frac {a\,\mathrm {cot}\left (e+f\,x\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.78, size = 46, normalized size = 2.19 \[ \begin {cases} \tilde {\infty } a x & \text {for}\: \left (e = 0 \vee e = - f x\right ) \wedge \left (e = - f x \vee f = 0\right ) \\x \left (a + b \tan ^{2}{\relax (e )}\right ) \cot ^{2}{\relax (e )} & \text {for}\: f = 0 \\- a x - \frac {a}{f \tan {\left (e + f x \right )}} + b x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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