Optimal. Leaf size=25 \[ -\frac{A \sin (c+d x) \sec ^{m+1}(c+d x)}{d m} \]
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Rubi [A] time = 0.0406406, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {4043} \[ -\frac{A \sin (c+d x) \sec ^{m+1}(c+d x)}{d m} \]
Antiderivative was successfully verified.
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Rule 4043
Rubi steps
\begin{align*} \int \sec ^m(c+d x) \left (A-\frac{A (1+m) \sec ^2(c+d x)}{m}\right ) \, dx &=-\frac{A \sec ^{1+m}(c+d x) \sin (c+d x)}{d m}\\ \end{align*}
Mathematica [A] time = 0.284062, size = 25, normalized size = 1. \[ -\frac{A \sin (c+d x) \sec ^{m+1}(c+d x)}{d m} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.222, size = 510, normalized size = 20.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.98019, size = 400, normalized size = 16. \begin{align*} \frac{2^{m} A \cos \left (-{\left (d x + c\right )}{\left (m + 2\right )} + m \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right ) \sin \left (2 \, d x + 2 \, c\right ) - 2^{m} A \cos \left (-{\left (d x + c\right )} m + m \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right ) \sin \left (2 \, d x + 2 \, c\right ) +{\left (2^{m} A \cos \left (2 \, d x + 2 \, c\right ) + 2^{m} A\right )} \sin \left (-{\left (d x + c\right )}{\left (m + 2\right )} + m \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right ) -{\left (2^{m} A \cos \left (2 \, d x + 2 \, c\right ) + 2^{m} A\right )} \sin \left (-{\left (d x + c\right )} m + m \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right )}{{\left (m \cos \left (2 \, d x + 2 \, c\right )^{2} + m \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, m \cos \left (2 \, d x + 2 \, c\right ) + m\right )}{\left (\cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )}^{\frac{1}{2} \, m} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.496179, size = 74, normalized size = 2.96 \begin{align*} -\frac{A \frac{1}{\cos \left (d x + c\right )}^{m} \sin \left (d x + c\right )}{d m \cos \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{A \left (\int - m \sec ^{m}{\left (c + d x \right )}\, dx + \int \sec ^{2}{\left (c + d x \right )} \sec ^{m}{\left (c + d x \right )}\, dx + \int m \sec ^{2}{\left (c + d x \right )} \sec ^{m}{\left (c + d x \right )}\, dx\right )}{m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -{\left (\frac{A{\left (m + 1\right )} \sec \left (d x + c\right )^{2}}{m} - A\right )} \sec \left (d x + c\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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