Optimal. Leaf size=45 \[ -\frac {\cos (c+d x)}{a d}-\frac {\cos (c+d x)}{a d (\sin (c+d x)+1)}-\frac {x}{a} \]
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Rubi [A] time = 0.08, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2746, 12, 2735, 2648} \[ -\frac {\cos (c+d x)}{a d}-\frac {\cos (c+d x)}{a d (\sin (c+d x)+1)}-\frac {x}{a} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2648
Rule 2735
Rule 2746
Rubi steps
\begin {align*} \int \frac {\sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac {\cos (c+d x)}{a d}-\frac {\int \frac {a \sin (c+d x)}{a+a \sin (c+d x)} \, dx}{a}\\ &=-\frac {\cos (c+d x)}{a d}-\int \frac {\sin (c+d x)}{a+a \sin (c+d x)} \, dx\\ &=-\frac {x}{a}-\frac {\cos (c+d x)}{a d}+\int \frac {1}{a+a \sin (c+d x)} \, dx\\ &=-\frac {x}{a}-\frac {\cos (c+d x)}{a d}-\frac {\cos (c+d x)}{d (a+a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 85, normalized size = 1.89 \[ -\frac {\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right ) (\cos (c+d x)+c+d x)+\sin \left (\frac {1}{2} (c+d x)\right ) (\cos (c+d x)+c+d x-2)\right )}{a d (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 69, normalized size = 1.53 \[ -\frac {d x + {\left (d x + 2\right )} \cos \left (d x + c\right ) + \cos \left (d x + c\right )^{2} + {\left (d x + \cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right ) + 1}{a d \cos \left (d x + c\right ) + a d \sin \left (d x + c\right ) + a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 77, normalized size = 1.71 \[ -\frac {\frac {d x + c}{a} + \frac {2 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 64, normalized size = 1.42 \[ -\frac {2}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {2}{a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.94, size = 129, normalized size = 2.87 \[ -\frac {2 \, {\left (\frac {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 2}{a + \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}} + \frac {\arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 69, normalized size = 1.53 \[ -\frac {x}{a}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+4}{a\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.05, size = 422, normalized size = 9.38 \[ \begin {cases} - \frac {d x \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} - \frac {d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} - \frac {d x \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} - \frac {d x}{a d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} - \frac {2 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} - \frac {2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} - \frac {4}{a d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{2}{\relax (c )}}{a \sin {\relax (c )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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