Optimal. Leaf size=37 \[ \frac {\tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac {1}{2 d (a+a \sin (c+d x))} \]
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Rubi [A]
time = 0.04, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2746, 46, 212}
\begin {gather*} \frac {\tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac {1}{2 d (a \sin (c+d x)+a)} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 212
Rule 2746
Rubi steps
\begin {align*} \int \frac {\sec (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {a \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a \text {Subst}\left (\int \left (\frac {1}{2 a (a+x)^2}+\frac {1}{2 a \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {1}{2 d (a+a \sin (c+d x))}+\frac {\text {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{2 d}\\ &=\frac {\tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac {1}{2 d (a+a \sin (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 30, normalized size = 0.81 \begin {gather*} \frac {\tanh ^{-1}(\sin (c+d x))-\frac {1}{1+\sin (c+d x)}}{2 a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.00, size = 43, normalized size = 1.16
method | result | size |
derivativedivides | \(\frac {-\frac {1}{2 \left (1+\sin \left (d x +c \right )\right )}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{4}-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{4}}{d a}\) | \(43\) |
default | \(\frac {-\frac {1}{2 \left (1+\sin \left (d x +c \right )\right )}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{4}-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{4}}{d a}\) | \(43\) |
norman | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a d}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a d}\) | \(71\) |
risch | \(-\frac {i {\mathrm e}^{i \left (d x +c \right )}}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{2}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 a d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 a d}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 47, normalized size = 1.27 \begin {gather*} \frac {\frac {\log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {\log \left (\sin \left (d x + c\right ) - 1\right )}{a} - \frac {2}{a \sin \left (d x + c\right ) + a}}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 58, normalized size = 1.57 \begin {gather*} \frac {{\left (\sin \left (d x + c\right ) + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (\sin \left (d x + c\right ) + 1\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2}{4 \, {\left (a d \sin \left (d x + c\right ) + a d\right )}} \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*} \frac {\int \frac {\sec {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.43, size = 58, normalized size = 1.57 \begin {gather*} \frac {\frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {\log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} - \frac {\sin \left (d x + c\right ) + 3}{a {\left (\sin \left (d x + c\right ) + 1\right )}}}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 33, normalized size = 0.89 \begin {gather*} \frac {\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )}{2\,a\,d}-\frac {1}{2\,d\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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