Optimal. Leaf size=29 \[ -\frac{a A \sin (c+d x) \cos (c+d x)}{2 d}-\frac{1}{2} a A x \]
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Rubi [A] time = 0.0549121, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3962, 2635, 8} \[ -\frac{a A \sin (c+d x) \cos (c+d x)}{2 d}-\frac{1}{2} a A x \]
Antiderivative was successfully verified.
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Rule 3962
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int (a-a \csc (c+d x)) (A+A \csc (c+d x)) \sin ^2(c+d x) \, dx &=-\left ((a A) \int \cos ^2(c+d x) \, dx\right )\\ &=-\frac{a A \cos (c+d x) \sin (c+d x)}{2 d}-\frac{1}{2} (a A) \int 1 \, dx\\ &=-\frac{1}{2} a A x-\frac{a A \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.022163, size = 25, normalized size = 0.86 \[ -\frac{a A (2 (c+d x)+\sin (2 (c+d x)))}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 40, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ( Aa \left ( -{\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) -Aa \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03968, size = 50, normalized size = 1.72 \begin{align*} \frac{{\left (2 \, d x + 2 \, c - \sin \left (2 \, d x + 2 \, c\right )\right )} A a - 4 \,{\left (d x + c\right )} A a}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.475762, size = 68, normalized size = 2.34 \begin{align*} -\frac{A a d x + A a \cos \left (d x + c\right ) \sin \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.3814, size = 54, normalized size = 1.86 \begin{align*} - A a x + A 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 \sin ^{2}{\left (c \right )} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25946, size = 47, normalized size = 1.62 \begin{align*} -\frac{{\left (d x + c\right )} A a + \frac{A a \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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