Optimal. Leaf size=12 \[ \frac {B \tanh ^{-1}(\sin (c+d x))}{d} \]
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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {21, 3855}
\begin {gather*} \frac {B \tanh ^{-1}(\sin (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 3855
Rubi steps
\begin {align*} \int \frac {(a B+b B \cos (c+d x)) \sec (c+d x)}{a+b \cos (c+d x)} \, dx &=B \int \sec (c+d x) \, dx\\ &=\frac {B \tanh ^{-1}(\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} \frac {B \tanh ^{-1}(\sin (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 20, normalized size = 1.67
method | result | size |
derivativedivides | \(\frac {B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(20\) |
default | \(\frac {B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(20\) |
norman | \(\frac {B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(37\) |
risch | \(\frac {B \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {B \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 31 vs.
\(2 (12) = 24\).
time = 0.39, size = 31, normalized size = 2.58 \begin {gather*} \frac {B \log \left (\sin \left (d x + c\right ) + 1\right ) - B \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs.
\(2 (10) = 20\).
time = 2.26, size = 39, normalized size = 3.25 \begin {gather*} \begin {cases} \frac {B \log {\left (\tan {\left (c + d x \right )} + \sec {\left (c + d x \right )} \right )}}{d} & \text {for}\: d \neq 0 \\\frac {x \left (B a + B b \cos {\left (c \right )}\right ) \sec {\left (c \right )}}{a + b \cos {\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. 47 vs.
\(2 (12) = 24\).
time = 0.47, size = 47, normalized size = 3.92 \begin {gather*} \frac {B \log \left ({\left | \frac {1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right ) + 2 \right |}\right ) - B \log \left ({\left | \frac {1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right ) - 2 \right |}\right )}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.49, size = 16, normalized size = 1.33 \begin {gather*} \frac {2\,B\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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