Optimal. Leaf size=45 \[ -\frac {\cos (c+d x)}{a d}+\frac {\sin (c+d x) \cos (c+d x)}{2 a d}-\frac {x}{2 a} \]
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Rubi [A] time = 0.07, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2839, 2638, 2635, 8} \[ -\frac {\cos (c+d x)}{a d}+\frac {\sin (c+d x) \cos (c+d x)}{2 a d}-\frac {x}{2 a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2638
Rule 2839
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int \sin (c+d x) \, dx}{a}-\frac {\int \sin ^2(c+d x) \, dx}{a}\\ &=-\frac {\cos (c+d x)}{a d}+\frac {\cos (c+d x) \sin (c+d x)}{2 a d}-\frac {\int 1 \, dx}{2 a}\\ &=-\frac {x}{2 a}-\frac {\cos (c+d x)}{a d}+\frac {\cos (c+d x) \sin (c+d x)}{2 a d}\\ \end {align*}
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Mathematica [B] time = 0.59, size = 161, normalized size = 3.58 \[ \frac {-4 d x \sin \left (\frac {c}{2}\right )+4 \sin \left (\frac {c}{2}+d x\right )-4 \sin \left (\frac {3 c}{2}+d x\right )+\sin \left (\frac {3 c}{2}+2 d x\right )+\sin \left (\frac {5 c}{2}+2 d x\right )+2 \cos \left (\frac {c}{2}\right ) (c-2 d x)-4 \cos \left (\frac {c}{2}+d x\right )-4 \cos \left (\frac {3 c}{2}+d x\right )+\cos \left (\frac {3 c}{2}+2 d x\right )-\cos \left (\frac {5 c}{2}+2 d x\right )+2 c \sin \left (\frac {c}{2}\right )-4 \sin \left (\frac {c}{2}\right )}{8 a d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 34, normalized size = 0.76 \[ -\frac {d x - \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 2 \, \cos \left (d x + c\right )}{2 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 72, normalized size = 1.60 \[ -\frac {\frac {d x + c}{a} + \frac {2 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.20, size = 142, normalized size = 3.16 \[ -\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {2}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 133, normalized size = 2.96 \[ \frac {\frac {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 2}{a + \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac {\arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.69, size = 33, normalized size = 0.73 \[ -\frac {x}{2\,a}-\frac {\cos \left (c+d\,x\right )-\frac {\sin \left (2\,c+2\,d\,x\right )}{4}}{a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.18, size = 366, normalized size = 8.13 \[ \begin {cases} - \frac {d x \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d} - \frac {2 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d} - \frac {d x}{2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d} - \frac {2 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d} - \frac {4 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d} + \frac {2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d} - \frac {4}{2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d} & \text {for}\: d \neq 0 \\\frac {x \sin {\relax (c )} \cos ^{2}{\relax (c )}}{a \sin {\relax (c )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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