Optimal. Leaf size=16 \[ \frac {\log (1+\sin (c+d x))}{a d} \]
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Rubi [A]
time = 0.02, antiderivative size = 16, 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 = {2746, 31}
\begin {gather*} \frac {\log (\sin (c+d x)+1)}{a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 2746
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{a+x} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\log (1+\sin (c+d x))}{a d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {\log (1+\sin (c+d x))}{a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.00, size = 19, normalized size = 1.19
method | result | size |
derivativedivides | \(\frac {\ln \left (a +a \sin \left (d x +c \right )\right )}{d a}\) | \(19\) |
default | \(\frac {\ln \left (a +a \sin \left (d x +c \right )\right )}{d a}\) | \(19\) |
risch | \(-\frac {i x}{a}-\frac {2 i c}{a d}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a d}\) | \(40\) |
norman | \(\frac {2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a d}-\frac {\ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 18, normalized size = 1.12 \begin {gather*} \frac {\log \left (a \sin \left (d x + c\right ) + a\right )}{a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 16, normalized size = 1.00 \begin {gather*} \frac {\log \left (\sin \left (d x + c\right ) + 1\right )}{a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.26, size = 24, normalized size = 1.50 \begin {gather*} \begin {cases} \frac {\log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{a d} & \text {for}\: d \neq 0 \\\frac {x \cos {\left (c \right )}}{a \sin {\left (c \right )} + a} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.50, size = 19, normalized size = 1.19 \begin {gather*} \frac {\log \left ({\left | a \sin \left (d x + c\right ) + a \right |}\right )}{a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 16, normalized size = 1.00 \begin {gather*} \frac {\ln \left (\sin \left (c+d\,x\right )+1\right )}{a\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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