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 = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.34, size = 25, normalized size = 1.00 \[ -\frac {A \sin (c+d x) \sec ^{m+1}(c+d x)}{d m} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 33, normalized size = 1.32 \[ -\frac {A \frac {1}{\cos \left (d x + c\right )}^{m} \sin \left (d x + c\right )}{d m \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -{\left (\frac {A {\left (m + 1\right )} \sec \left (d x + c\right )^{2}}{m} - A\right )} \sec \left (d x + c\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.42, size = 510, normalized size = 20.40 \[ \frac {i A \left (\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{-m} \left ({\mathrm e}^{i \left (\Re \left (d x \right )+\Re \relax (c )\right )}\right )^{m} 2^{m} {\mathrm e}^{-m \Im \left (d x \right )-m \Im \relax (c )} {\mathrm e}^{-\frac {i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}\right )^{3} m}{2}} {\mathrm e}^{\frac {i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{i \left (d x +c \right )}\right ) m}{2}} {\mathrm e}^{\frac {i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}\right )^{2} \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 i \left (d x +c \right )}+1}\right ) m}{2}} {\mathrm e}^{-\frac {i \pi \,\mathrm {csgn}\left (\frac {i {\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}\right ) \mathrm {csgn}\left (i {\mathrm e}^{i \left (d x +c \right )}\right ) \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 i \left (d x +c \right )}+1}\right ) m}{2}} {\mathrm e}^{2 i d x} {\mathrm e}^{2 i c}-\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{-m} \left ({\mathrm e}^{i \left (\Re \left (d x \right )+\Re \relax (c )\right )}\right )^{m} 2^{m} {\mathrm e}^{-\frac {m \left (i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}\right )^{3}-i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{i \left (d x +c \right )}\right )-i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}\right )^{2} \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 i \left (d x +c \right )}+1}\right )+i \pi \,\mathrm {csgn}\left (\frac {i {\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}\right ) \mathrm {csgn}\left (i {\mathrm e}^{i \left (d x +c \right )}\right ) \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 i \left (d x +c \right )}+1}\right )+2 \Im \left (d x \right )+2 \Im \relax (c )\right )}{2}}\right )}{d m \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.83, size = 296, normalized size = 11.84 \[ \frac {2^{m} A \cos \left (-{\left (d x + c\right )} {\left (m + 2\right )} + m \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right ) \sin \left (2 \, d x + 2 \, c\right ) - 2^{m} A \cos \left (-{\left (d x + c\right )} m + m \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right ) \sin \left (2 \, d x + 2 \, c\right ) + {\left (2^{m} A \cos \left (2 \, d x + 2 \, c\right ) + 2^{m} A\right )} \sin \left (-{\left (d x + c\right )} {\left (m + 2\right )} + m \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right ) - {\left (2^{m} A \cos \left (2 \, d x + 2 \, c\right ) + 2^{m} A\right )} \sin \left (-{\left (d x + c\right )} m + m \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right )}{{\left (m \cos \left (2 \, d x + 2 \, c\right )^{2} + m \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, m \cos \left (2 \, d x + 2 \, c\right ) + m\right )} {\left (\cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )}^{\frac {1}{2} \, m} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.67, size = 41, normalized size = 1.64 \[ -\frac {A\,\sin \left (2\,c+2\,d\,x\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^m}{d\,m\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {A \left (\int \left (- m \sec ^{m}{\left (c + d x \right )}\right )\, dx + \int \sec ^{2}{\left (c + d x \right )} \sec ^{m}{\left (c + d x \right )}\, dx + \int m \sec ^{2}{\left (c + d x \right )} \sec ^{m}{\left (c + d x \right )}\, dx\right )}{m} \]
Verification of antiderivative is not currently implemented for this CAS.
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