Optimal. Leaf size=61 \[ \frac {1}{8} (4 A+3 C) x+\frac {(4 A+3 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {C \cos ^3(c+d x) \sin (c+d x)}{4 d} \]
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Rubi [A]
time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3093, 2715, 8}
\begin {gather*} \frac {(4 A+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} x (4 A+3 C)+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 3093
Rubi steps
\begin {align*} \int \cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {C \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{4} (4 A+3 C) \int \cos ^2(c+d x) \, dx\\ &=\frac {(4 A+3 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {C \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{8} (4 A+3 C) \int 1 \, dx\\ &=\frac {1}{8} (4 A+3 C) x+\frac {(4 A+3 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {C \cos ^3(c+d x) \sin (c+d x)}{4 d}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 45, normalized size = 0.74 \begin {gather*} \frac {4 (4 A+3 C) (c+d x)+8 (A+C) \sin (2 (c+d x))+C \sin (4 (c+d x))}{32 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 65, normalized size = 1.07
method | result | size |
risch | \(\frac {A x}{2}+\frac {3 C x}{8}+\frac {\sin \left (4 d x +4 c \right ) C}{32 d}+\frac {\sin \left (2 d x +2 c \right ) A}{4 d}+\frac {\sin \left (2 d x +2 c \right ) C}{4 d}\) | \(55\) |
derivativedivides | \(\frac {C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+A \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(65\) |
default | \(\frac {C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+A \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(65\) |
norman | \(\frac {\left (\frac {A}{2}+\frac {3 C}{8}\right ) x +\left (2 A +\frac {3 C}{2}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2 A +\frac {3 C}{2}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3 A +\frac {9 C}{4}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {A}{2}+\frac {3 C}{8}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\left (4 A -3 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {\left (4 A -3 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {\left (4 A +5 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {\left (4 A +5 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}\) | \(197\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 73, normalized size = 1.20 \begin {gather*} \frac {{\left (d x + c\right )} {\left (4 \, A + 3 \, C\right )} + \frac {{\left (4 \, A + 3 \, C\right )} \tan \left (d x + c\right )^{3} + {\left (4 \, A + 5 \, C\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1}}{8 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 49, normalized size = 0.80 \begin {gather*} \frac {{\left (4 \, A + 3 \, C\right )} d x + {\left (2 \, C \cos \left (d x + c\right )^{3} + {\left (4 \, A + 3 \, C\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, 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. 158 vs.
\(2 (53) = 106\).
time = 0.18, size = 158, normalized size = 2.59 \begin {gather*} \begin {cases} \frac {A x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {A x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {A \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {3 C x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 C x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 C x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 C \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 C \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\left (c \right )}\right ) \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 43, normalized size = 0.70 \begin {gather*} \frac {1}{8} \, {\left (4 \, A + 3 \, C\right )} x + \frac {C \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {{\left (A + C\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.78, size = 67, normalized size = 1.10 \begin {gather*} x\,\left (\frac {A}{2}+\frac {3\,C}{8}\right )+\frac {\left (\frac {A}{2}+\frac {3\,C}{8}\right )\,{\mathrm {tan}\left (c+d\,x\right )}^3+\left (\frac {A}{2}+\frac {5\,C}{8}\right )\,\mathrm {tan}\left (c+d\,x\right )}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^4+2\,{\mathrm {tan}\left (c+d\,x\right )}^2+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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