Optimal. Leaf size=44 \[ \frac {2 \sin (c+d x)}{a d}-\frac {\sin (c+d x)}{d (a \sec (c+d x)+a)}-\frac {x}{a} \]
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Rubi [A] time = 0.06, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3819, 3787, 2637, 8} \[ \frac {2 \sin (c+d x)}{a d}-\frac {\sin (c+d x)}{d (a \sec (c+d x)+a)}-\frac {x}{a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2637
Rule 3787
Rule 3819
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{a+a \sec (c+d x)} \, dx &=-\frac {\sin (c+d x)}{d (a+a \sec (c+d x))}-\frac {\int \cos (c+d x) (-2 a+a \sec (c+d x)) \, dx}{a^2}\\ &=-\frac {\sin (c+d x)}{d (a+a \sec (c+d x))}-\frac {\int 1 \, dx}{a}+\frac {2 \int \cos (c+d x) \, dx}{a}\\ &=-\frac {x}{a}+\frac {2 \sin (c+d x)}{a d}-\frac {\sin (c+d x)}{d (a+a \sec (c+d x))}\\ \end {align*}
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Mathematica [B] time = 0.23, size = 89, normalized size = 2.02 \[ \frac {\sec \left (\frac {c}{2}\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \left (\sin \left (c+\frac {d x}{2}\right )+\sin \left (c+\frac {3 d x}{2}\right )+\sin \left (2 c+\frac {3 d x}{2}\right )-2 d x \cos \left (c+\frac {d x}{2}\right )+5 \sin \left (\frac {d x}{2}\right )-2 d x \cos \left (\frac {d x}{2}\right )\right )}{4 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 46, normalized size = 1.05 \[ -\frac {d x \cos \left (d x + c\right ) + d x - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right )}{a d \cos \left (d x + c\right ) + a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 58, normalized size = 1.32 \[ -\frac {\frac {d x + c}{a} - \frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a} - \frac {2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.55, size = 68, normalized size = 1.55 \[ \frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.61, size = 92, normalized size = 2.09 \[ -\frac {\frac {2 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {2 \, \sin \left (d x + c\right )}{{\left (a + \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} - \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.66, size = 66, normalized size = 1.50 \[ \frac {2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\left (-c-d\,x\right )\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,d\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )} \]
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 \[ \frac {\int \frac {\cos {\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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