Optimal. Leaf size=31 \[ \frac {A \sin (c+d x) \cos (c+d x)}{2 d}+\frac {1}{2} x (A+2 C) \]
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Rubi [A] time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {4045, 8} \[ \frac {A \sin (c+d x) \cos (c+d x)}{2 d}+\frac {1}{2} x (A+2 C) \]
Antiderivative was successfully verified.
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Rule 8
Rule 4045
Rubi steps
\begin {align*} \int \cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} (A+2 C) \int 1 \, dx\\ &=\frac {1}{2} (A+2 C) x+\frac {A \cos (c+d x) \sin (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 33, normalized size = 1.06 \[ \frac {A (c+d x)}{2 d}+\frac {A \sin (2 (c+d x))}{4 d}+C x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 28, normalized size = 0.90 \[ \frac {{\left (A + 2 \, C\right )} d x + A \cos \left (d x + c\right ) \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 37, normalized size = 1.19 \[ \frac {{\left (d x + c\right )} {\left (A + 2 \, C\right )} + \frac {A \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.08, size = 37, normalized size = 1.19 \[ \frac {A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 37, normalized size = 1.19 \[ \frac {{\left (d x + c\right )} {\left (A + 2 \, C\right )} + \frac {A \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.36, size = 25, normalized size = 0.81 \[ \frac {\frac {A\,\sin \left (2\,c+2\,d\,x\right )}{4}+d\,x\,\left (\frac {A}{2}+C\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.94, size = 51, normalized size = 1.65 \[ A \left (\begin {cases} \frac {x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {\sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \cos ^{2}{\relax (c )} & \text {otherwise} \end {cases}\right ) + C x \]
Verification of antiderivative is not currently implemented for this CAS.
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