Optimal. Leaf size=19 \[ \frac {\cot (c+d x)}{a d}+\frac {x}{a} \]
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Rubi [A] time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4120, 3473, 8} \[ \frac {\cot (c+d x)}{a d}+\frac {x}{a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rule 4120
Rubi steps
\begin {align*} \int \frac {1}{a-a \sec ^2(c+d x)} \, dx &=-\frac {\int \cot ^2(c+d x) \, dx}{a}\\ &=\frac {\cot (c+d x)}{a d}+\frac {\int 1 \, dx}{a}\\ &=\frac {x}{a}+\frac {\cot (c+d x)}{a d}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 31, normalized size = 1.63 \[ \frac {\cot (c+d x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2(c+d x)\right )}{a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 31, normalized size = 1.63 \[ \frac {d x \sin \left (d x + c\right ) + \cos \left (d x + c\right )}{a d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 45, normalized size = 2.37 \[ \frac {\frac {2 \, {\left (d x + c\right )}}{a} - \frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a} + \frac {1}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.52, size = 31, normalized size = 1.63 \[ \frac {1}{a d \tan \left (d x +c \right )}+\frac {\arctan \left (\tan \left (d x +c \right )\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 26, normalized size = 1.37 \[ \frac {\frac {d x + c}{a} + \frac {1}{a \tan \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.70, size = 19, normalized size = 1.00 \[ \frac {x}{a}+\frac {\mathrm {cot}\left (c+d\,x\right )}{a\,d} \]
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 \[ - \frac {\int \frac {1}{\sec ^{2}{\left (c + d x \right )} - 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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