Optimal. Leaf size=44 \[ -\frac {x}{2 a}+\frac {\sin (c+d x)}{a d}-\frac {\cos (c+d x) \sin (c+d x)}{2 a d} \]
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Rubi [A]
time = 0.08, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3957, 2918,
2717, 2715, 8} \begin {gather*} \frac {\sin (c+d x)}{a d}-\frac {\sin (c+d x) \cos (c+d x)}{2 a d}-\frac {x}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 2717
Rule 2918
Rule 3957
Rubi steps
\begin {align*} \int \frac {\sin ^2(c+d x)}{a+a \sec (c+d x)} \, dx &=-\int \frac {\cos (c+d x) \sin ^2(c+d x)}{-a-a \cos (c+d x)} \, dx\\ &=\frac {\int \cos (c+d x) \, dx}{a}-\frac {\int \cos ^2(c+d x) \, dx}{a}\\ &=\frac {\sin (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 {\sin (c+d x)}{a d}-\frac {\cos (c+d x) \sin (c+d x)}{2 a d}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 68, normalized size = 1.55 \begin {gather*} -\frac {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \left (-c+2 d x-4 \sin (c+d x)+\sin (2 (c+d x))+\tan \left (\frac {c}{2}\right )\right )}{2 a d (1+\sec (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 64, normalized size = 1.45
method | result | size |
risch | \(-\frac {x}{2 a}+\frac {\sin \left (d x +c \right )}{a d}-\frac {\sin \left (2 d x +2 c \right )}{4 a d}\) | \(38\) |
derivativedivides | \(\frac {-\frac {4 \left (-\frac {3 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}\) | \(64\) |
default | \(\frac {-\frac {4 \left (-\frac {3 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}\) | \(64\) |
norman | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {x}{2 a}+\frac {3 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}\) | \(93\) |
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. 112 vs.
\(2 (40) = 80\).
time = 0.48, size = 112, normalized size = 2.55 \begin {gather*} \frac {\frac {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.46, size = 27, normalized size = 0.61 \begin {gather*} -\frac {d x + {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right )}{2 \, a d} \end {gather*}
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 \begin {gather*} \frac {\int \frac {\sin ^{2}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 58, normalized size = 1.32 \begin {gather*} -\frac {\frac {d x + c}{a} - \frac {2 \, {\left (3 \, \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 )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.96, size = 30, normalized size = 0.68 \begin {gather*} -\frac {\sin \left (2\,c+2\,d\,x\right )-4\,\sin \left (c+d\,x\right )+2\,d\,x}{4\,a\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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