Optimal. Leaf size=19 \[ -a x-\frac {(a+b) \cot (e+f x)}{f} \]
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Rubi [A]
time = 0.04, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4226, 1816,
209} \begin {gather*} -\frac {(a+b) \cot (e+f x)}{f}-a x \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 1816
Rule 4226
Rubi steps
\begin {align*} \int \cot ^2(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx &=\frac {\text {Subst}\left (\int \frac {a+b \left (1+x^2\right )}{x^2 \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a+b}{x^2}-\frac {a}{1+x^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {(a+b) \cot (e+f x)}{f}-\frac {a \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-a x-\frac {(a+b) \cot (e+f x)}{f}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.04, size = 43, normalized size = 2.26 \begin {gather*} -\frac {b \cot (e+f x)}{f}-\frac {a \cot (e+f x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2(e+f x)\right )}{f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 33, normalized size = 1.74
method | result | size |
derivativedivides | \(\frac {a \left (-\cot \left (f x +e \right )-f x -e \right )-b \cot \left (f x +e \right )}{f}\) | \(33\) |
default | \(\frac {a \left (-\cot \left (f x +e \right )-f x -e \right )-b \cot \left (f x +e \right )}{f}\) | \(33\) |
risch | \(-a x -\frac {2 i a}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )}-\frac {2 i b}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 27, normalized size = 1.42 \begin {gather*} -\frac {{\left (f x + e\right )} a + \frac {a + b}{\tan \left (f x + e\right )}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.54, size = 37, normalized size = 1.95 \begin {gather*} -\frac {a f x \sin \left (f x + e\right ) + {\left (a + b\right )} \cos \left (f x + e\right )}{f \sin \left (f x + e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \sec ^{2}{\left (e + f x \right )}\right ) \cot ^{2}{\left (e + f x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs.
\(2 (19) = 38\).
time = 0.46, size = 53, normalized size = 2.79 \begin {gather*} -\frac {2 \, {\left (f x + e\right )} a - a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + \frac {a + b}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}}{2 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.50, size = 19, normalized size = 1.00 \begin {gather*} -a\,x-\frac {\mathrm {cot}\left (e+f\,x\right )\,\left (a+b\right )}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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