Optimal. Leaf size=24 \[ \frac {B \tanh ^{-1}(\sin (c+d x))}{d}+\frac {C \tan (c+d x)}{d} \]
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Rubi [A] time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3770, 3767, 8} \[ \frac {B \tanh ^{-1}(\sin (c+d x))}{d}+\frac {C \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rubi steps
\begin {align*} \int \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=B \int \sec (c+d x) \, dx+C \int \sec ^2(c+d x) \, dx\\ &=\frac {B \tanh ^{-1}(\sin (c+d x))}{d}-\frac {C \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=\frac {B \tanh ^{-1}(\sin (c+d x))}{d}+\frac {C \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 1.00 \[ \frac {B \tanh ^{-1}(\sin (c+d x))}{d}+\frac {C \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 60, normalized size = 2.50 \[ \frac {B \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - B \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, C \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.35, size = 57, normalized size = 2.38 \[ \frac {B {\left (\log \left ({\left | \frac {1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right ) + 2 \right |}\right ) - \log \left ({\left | \frac {1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right ) - 2 \right |}\right )\right )}}{4 \, d} + \frac {C \tan \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.94, size = 32, normalized size = 1.33 \[ \frac {B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {C \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 31, normalized size = 1.29 \[ \frac {B \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right )}{d} + \frac {C \tan \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.41, size = 47, normalized size = 1.96 \[ \frac {2\,B\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {2\,C\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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 \[ \int \left (B + C \sec {\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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