Optimal. Leaf size=24 \[ \frac {C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {A \sin (c+d x)}{d} \]
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Rubi [A]
time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {4130, 3855}
\begin {gather*} \frac {A \sin (c+d x)}{d}+\frac {C \tanh ^{-1}(\sin (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3855
Rule 4130
Rubi steps
\begin {align*} \int \cos (c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \sin (c+d x)}{d}+C \int \sec (c+d x) \, dx\\ &=\frac {C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {A \sin (c+d x)}{d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 35, normalized size = 1.46 \begin {gather*} \frac {C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {A \cos (d x) \sin (c)}{d}+\frac {A \cos (c) \sin (d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 30, normalized size = 1.25
method | result | size |
derivativedivides | \(\frac {A \sin \left (d x +c \right )+C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(30\) |
default | \(\frac {A \sin \left (d x +c \right )+C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(30\) |
risch | \(-\frac {i A \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {i A \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}\) | \(71\) |
norman | \(\frac {-\frac {2 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 A \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(101\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 38, normalized size = 1.58 \begin {gather*} \frac {C {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, A \sin \left (d x + c\right )}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.44, size = 40, normalized size = 1.67 \begin {gather*} \frac {C \log \left (\sin \left (d x + c\right ) + 1\right ) - C \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, A \sin \left (d x + c\right )}{2 \, d} \end {gather*}
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 \begin {gather*} \int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \cos {\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 40, normalized size = 1.67 \begin {gather*} \frac {C \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - C \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) + 2 \, A \sin \left (d x + c\right )}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 22, normalized size = 0.92 \begin {gather*} \frac {A\,\sin \left (c+d\,x\right )+C\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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