Optimal. Leaf size=43 \[ -\frac {x}{a}+\frac {\sin (c+d x)}{a d}+\frac {\sin (c+d x)}{a d (1+\cos (c+d x))} \]
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Rubi [A]
time = 0.05, antiderivative size = 43, 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 = {2825, 12, 2814,
2727} \begin {gather*} \frac {\sin (c+d x)}{a d}+\frac {\sin (c+d x)}{a d (\cos (c+d x)+1)}-\frac {x}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2727
Rule 2814
Rule 2825
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx &=\frac {\sin (c+d x)}{a d}-\frac {\int \frac {a \cos (c+d x)}{a+a \cos (c+d x)} \, dx}{a}\\ &=\frac {\sin (c+d x)}{a d}-\int \frac {\cos (c+d x)}{a+a \cos (c+d x)} \, dx\\ &=-\frac {x}{a}+\frac {\sin (c+d x)}{a d}+\int \frac {1}{a+a \cos (c+d x)} \, dx\\ &=-\frac {x}{a}+\frac {\sin (c+d x)}{a d}+\frac {\sin (c+d x)}{d (a+a \cos (c+d x))}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(89\) vs. \(2(43)=86\).
time = 0.22, size = 89, normalized size = 2.07 \begin {gather*} \frac {\sec \left (\frac {c}{2}\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \left (-2 d x \cos \left (\frac {d x}{2}\right )-2 d x \cos \left (c+\frac {d x}{2}\right )+5 \sin \left (\frac {d x}{2}\right )+\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 )\right )}{4 a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 56, normalized size = 1.30
method | result | size |
derivativedivides | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}-2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\) | \(56\) |
default | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}-2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\) | \(56\) |
risch | \(-\frac {x}{a}-\frac {i {\mathrm e}^{i \left (d x +c \right )}}{2 d a}+\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{2 d a}+\frac {2 i}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}\) | \(66\) |
norman | \(\frac {\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {x}{a}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {2 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}\) | \(112\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 92 vs.
\(2 (43) = 86\).
time = 0.51, size = 92, normalized size = 2.14 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.46, size = 46, normalized size = 1.07 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 129 vs.
\(2 (31) = 62\).
time = 0.59, size = 129, normalized size = 3.00 \begin {gather*} \begin {cases} - \frac {d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} - \frac {d x}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} + \frac {\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} + \frac {3 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{2}{\left (c \right )}}{a \cos {\left (c \right )} + a} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 58, normalized size = 1.35 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.40, size = 66, normalized size = 1.53 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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