Optimal. Leaf size=43 \[ \frac {(a-b) \log (\sin (c+d x)+1)}{2 d}-\frac {(a+b) \log (1-\sin (c+d x))}{2 d} \]
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Rubi [A] time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2668, 633, 31} \[ \frac {(a-b) \log (\sin (c+d x)+1)}{2 d}-\frac {(a+b) \log (1-\sin (c+d x))}{2 d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 633
Rule 2668
Rubi steps
\begin {align*} \int \sec (c+d x) (a+b \sin (c+d x)) \, dx &=\frac {b \operatorname {Subst}\left (\int \frac {a+x}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac {(a-b) \operatorname {Subst}\left (\int \frac {1}{-b-x} \, dx,x,b \sin (c+d x)\right )}{2 d}+\frac {(a+b) \operatorname {Subst}\left (\int \frac {1}{b-x} \, dx,x,b \sin (c+d x)\right )}{2 d}\\ &=-\frac {(a+b) \log (1-\sin (c+d x))}{2 d}+\frac {(a-b) \log (1+\sin (c+d x))}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.60 \[ \frac {a \tanh ^{-1}(\sin (c+d x))}{d}-\frac {b \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 37, normalized size = 0.86 \[ \frac {{\left (a - b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (a + b\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.75, size = 37, normalized size = 0.86 \[ \frac {{\left (a - b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - {\left (a + b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 34, normalized size = 0.79 \[ -\frac {b \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 35, normalized size = 0.81 \[ \frac {{\left (a - b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (a + b\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 54, normalized size = 1.26 \[ -\frac {\frac {a\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{2}-\frac {a\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{2}+\frac {b\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{2}+\frac {b\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{2}}{d} \]
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 \[ \int \left (a + b \sin {\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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