Optimal. Leaf size=24 \[ \frac {a \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a \tan (c+d x)}{d} \]
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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 = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3872, 3855,
3852, 8} \begin {gather*} \frac {a \tan (c+d x)}{d}+\frac {a \tanh ^{-1}(\sin (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3852
Rule 3855
Rule 3872
Rubi steps
\begin {align*} \int \sec (c+d x) (a+a \sec (c+d x)) \, dx &=a \int \sec (c+d x) \, dx+a \int \sec ^2(c+d x) \, dx\\ &=\frac {a \tanh ^{-1}(\sin (c+d x))}{d}-\frac {a \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=\frac {a \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 24, normalized size = 1.00 \begin {gather*} \frac {a \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a \tan (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 30, normalized size = 1.25
method | result | size |
derivativedivides | \(\frac {a \tan \left (d x +c \right )+a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(30\) |
default | \(\frac {a \tan \left (d x +c \right )+a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(30\) |
risch | \(\frac {2 i a}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}\) | \(59\) |
norman | \(-\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 29, normalized size = 1.21 \begin {gather*} \frac {a \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + a \tan \left (d x + c\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs.
\(2 (24) = 48\).
time = 2.52, size = 60, normalized size = 2.50 \begin {gather*} \frac {a \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - a \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, a \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.55, size = 37, normalized size = 1.54 \begin {gather*} \begin {cases} \frac {a \log {\left (\tan {\left (c + d x \right )} + \sec {\left (c + d x \right )} \right )} + a \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \sec {\left (c \right )} + a\right ) \sec {\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 63 vs.
\(2 (24) = 48\).
time = 0.44, size = 63, normalized size = 2.62 \begin {gather*} \frac {a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.68, size = 47, normalized size = 1.96 \begin {gather*} \frac {2\,a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {2\,a\,\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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