Optimal. Leaf size=26 \[ -\frac {\cot (c+d x) (a+b \sec (c+d x))}{d}-a x \]
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Rubi [A] time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3882, 8} \[ -\frac {\cot (c+d x) (a+b \sec (c+d x))}{d}-a x \]
Antiderivative was successfully verified.
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Rule 8
Rule 3882
Rubi steps
\begin {align*} \int \cot ^2(c+d x) (a+b \sec (c+d x)) \, dx &=-\frac {\cot (c+d x) (a+b \sec (c+d x))}{d}-\int a \, dx\\ &=-a x-\frac {\cot (c+d x) (a+b \sec (c+d x))}{d}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 43, normalized size = 1.65 \[ -\frac {a \cot (c+d x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2(c+d x)\right )}{d}-\frac {b \csc (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 33, normalized size = 1.27 \[ -\frac {a d x \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) + b}{d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 52, normalized size = 2.00 \[ -\frac {2 \, {\left (d x + c\right )} a - a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {a + b}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.45, size = 35, normalized size = 1.35 \[ \frac {a \left (-\cot \left (d x +c \right )-d x -c \right )-\frac {b}{\sin \left (d x +c \right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 31, normalized size = 1.19 \[ -\frac {{\left (d x + c + \frac {1}{\tan \left (d x + c\right )}\right )} a + \frac {b}{\sin \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.30, size = 48, normalized size = 1.85 \[ \frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {a}{2}-\frac {b}{2}\right )}{d}-\frac {\frac {a}{2}+\frac {b}{2}}{d\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}-a\,x \]
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 \[ \int \left (a + b \sec {\left (c + d x \right )}\right ) \cot ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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