Optimal. Leaf size=19 \[ -\frac {(a+b) \cot (e+f x)}{f}-a x \]
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Rubi [A] time = 0.05, 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 = {4141, 1802, 203} \[ -\frac {(a+b) \cot (e+f x)}{f}-a x \]
Antiderivative was successfully verified.
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Rule 203
Rule 1802
Rule 4141
Rubi steps
\begin {align*} \int \cot ^2(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx &=\frac {\operatorname {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 {\operatorname {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 \operatorname {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] time = 0.03, size = 43, normalized size = 2.26 \[ -\frac {a \cot (e+f x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2(e+f x)\right )}{f}-\frac {b \cot (e+f x)}{f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 34, normalized size = 1.79 \[ -\frac {a f x \sin \left (f x + e\right ) + {\left (a + b\right )} \cos \left (f x + e\right )}{f \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 57, normalized size = 3.00 \[ -\frac {2 \, {\left (f x + e\right )} a - a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + \frac {a + b}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.89, size = 33, normalized size = 1.74 \[ \frac {a \left (-\cot \left (f x +e \right )-f x -e \right )-b \cot \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 25, normalized size = 1.32 \[ -\frac {{\left (f x + e\right )} a + \frac {a + b}{\tan \left (f x + e\right )}}{f} \]
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 \[ -a\,x-\frac {\mathrm {cot}\left (e+f\,x\right )\,\left (a+b\right )}{f} \]
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 \[ \int \left (a + b \sec ^{2}{\left (e + f x \right )}\right ) \cot ^{2}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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