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 = 0.05, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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 8
Rule 2635
Rule 3962
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.49, size = 26, normalized size = 0.90 \[ -\frac {A a d x + A a \cos \left (d x + c\right ) \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 35, normalized size = 1.21 \[ -\frac {{\left (d x + c\right )} A a + \frac {A a \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.50, size = 40, normalized size = 1.38 \[ \frac {a A \left (-\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-a A \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 37, normalized size = 1.28 \[ \frac {{\left (2 \, d x + 2 \, c - \sin \left (2 \, d x + 2 \, c\right )\right )} A a - 4 \, {\left (d x + c\right )} A a}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.66, size = 53, normalized size = 1.83 \[ \frac {A\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-A\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2}-\frac {A\,a\,x}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.53, size = 54, normalized size = 1.86 \[ - A a x + A a \left (\begin {cases} \frac {x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {x \cos ^{2}{\left (c + d x \right )}}{2} - \frac {\sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \sin ^{2}{\relax (c )} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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