Optimal. Leaf size=40 \[ \frac {(2 A+C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {C \tan (c+d x) \sec (c+d x)}{2 d} \]
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Rubi [A] time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {4046, 3770} \[ \frac {(2 A+C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {C \tan (c+d x) \sec (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3770
Rule 4046
Rubi steps
\begin {align*} \int \sec (c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {C \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} (2 A+C) \int \sec (c+d x) \, dx\\ &=\frac {(2 A+C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {C \sec (c+d x) \tan (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 48, normalized size = 1.20 \[ \frac {A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {C \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {C \tan (c+d x) \sec (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 72, normalized size = 1.80 \[ \frac {{\left (2 \, A + C\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (2 \, A + C\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, C \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 60, normalized size = 1.50 \[ \frac {{\left (2 \, A + C\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - {\left (2 \, A + C\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {2 \, C \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.90, size = 59, normalized size = 1.48 \[ \frac {A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {C \tan \left (d x +c \right ) \sec \left (d x +c \right )}{2 d}+\frac {C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 58, normalized size = 1.45 \[ \frac {{\left (2 \, A + C\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (2 \, A + C\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, C \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.41, size = 41, normalized size = 1.02 \[ \frac {\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )\,\left (A+\frac {C}{2}\right )}{d}-\frac {C\,\sin \left (c+d\,x\right )}{2\,d\,\left ({\sin \left (c+d\,x\right )}^2-1\right )} \]
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 \[ \int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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