Optimal. Leaf size=27 \[ A x+\frac {B \tanh ^{-1}(\sin (c+d x))}{d}+\frac {C \tan (c+d x)}{d} \]
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Rubi [A] time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3770, 3767, 8} \[ A x+\frac {B \tanh ^{-1}(\sin (c+d x))}{d}+\frac {C \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rubi steps
\begin {align*} \int \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=A x+B \int \sec (c+d x) \, dx+C \int \sec ^2(c+d x) \, dx\\ &=A x+\frac {B \tanh ^{-1}(\sin (c+d x))}{d}-\frac {C \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=A x+\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 = 27, normalized size = 1.00 \[ A x+\frac {B \tanh ^{-1}(\sin (c+d x))}{d}+\frac {C \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 71, normalized size = 2.63 \[ \frac {2 \, A d x \cos \left (d x + c\right ) + B \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - B \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, C \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.40, size = 60, normalized size = 2.22 \[ A x + \frac {B {\left (\log \left ({\left | \frac {1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right ) + 2 \right |}\right ) - \log \left ({\left | \frac {1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right ) - 2 \right |}\right )\right )}}{4 \, d} + \frac {C \tan \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.94, size = 35, normalized size = 1.30 \[ A x +\frac {B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {C \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 34, normalized size = 1.26 \[ A x + \frac {B \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right )}{d} + \frac {C \tan \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.54, size = 161, normalized size = 5.96 \[ \frac {2\,A\,\mathrm {atan}\left (\frac {64\,A^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,A^3+64\,A\,B^2}+\frac {64\,A\,B^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,A^3+64\,A\,B^2}\right )}{d}+\frac {2\,B\,\mathrm {atanh}\left (\frac {64\,B^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,A^2\,B+64\,B^3}+\frac {64\,A^2\,B\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,A^2\,B+64\,B^3}\right )}{d}-\frac {2\,C\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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 \[ \int \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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