Optimal. Leaf size=47 \[ \frac {\cos (c+d x)}{a^2 d}+\frac {2 \cos (c+d x)}{d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {2 x}{a^2} \]
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Rubi [A] time = 0.07, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2857, 2638} \[ \frac {\cos (c+d x)}{a^2 d}+\frac {2 \cos (c+d x)}{d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {2 x}{a^2} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 2857
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {2 \cos (c+d x)}{d \left (a^2+a^2 \sin (c+d x)\right )}-\frac {\int (-2 a+a \sin (c+d x)) \, dx}{a^3}\\ &=\frac {2 x}{a^2}+\frac {2 \cos (c+d x)}{d \left (a^2+a^2 \sin (c+d x)\right )}-\frac {\int \sin (c+d x) \, dx}{a^2}\\ &=\frac {2 x}{a^2}+\frac {\cos (c+d x)}{a^2 d}+\frac {2 \cos (c+d x)}{d \left (a^2+a^2 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [B] time = 0.33, size = 117, normalized size = 2.49 \[ \frac {12 d x \sin \left (c+\frac {d x}{2}\right )+3 \sin \left (2 c+\frac {3 d x}{2}\right )+2 \cos \left (c+\frac {d x}{2}\right )+3 \cos \left (c+\frac {3 d x}{2}\right )-28 \sin \left (\frac {d x}{2}\right )+12 d x \cos \left (\frac {d x}{2}\right )}{6 a^2 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 77, normalized size = 1.64 \[ \frac {2 \, d x + {\left (2 \, d x + 3\right )} \cos \left (d x + c\right ) + \cos \left (d x + c\right )^{2} + {\left (2 \, d x + \cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + 2}{a^{2} d \cos \left (d x + c\right ) + a^{2} d \sin \left (d x + c\right ) + a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 78, normalized size = 1.66 \[ \frac {2 \, {\left (\frac {d x + c}{a^{2}} + \frac {2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} a^{2}}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 64, normalized size = 1.36 \[ \frac {2}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {4 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}}+\frac {4}{d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 139, normalized size = 2.96 \[ \frac {2 \, {\left (\frac {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {2 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 3}{a^{2} + \frac {a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}} + \frac {2 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.86, size = 68, normalized size = 1.45 \[ \frac {2\,x}{a^2}+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+6}{a^2\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.90, size = 479, normalized size = 10.19 \[ \begin {cases} \frac {2 d x \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a^{2} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d} + \frac {2 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a^{2} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d} + \frac {2 d x \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a^{2} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d} + \frac {2 d x}{a^{2} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d} + \frac {4 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a^{2} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d} + \frac {2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a^{2} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d} + \frac {6}{a^{2} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{2} d} & \text {for}\: d \neq 0 \\\frac {x \sin {\relax (c )} \cos ^{2}{\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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