Optimal. Leaf size=34 \[ \frac {b \log (a \cos (c+d x)+b)}{a^2 d}-\frac {\cos (c+d x)}{a d} \]
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Rubi [A] time = 0.08, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3872, 2833, 12, 43} \[ \frac {b \log (a \cos (c+d x)+b)}{a^2 d}-\frac {\cos (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
Rule 3872
Rubi steps
\begin {align*} \int \frac {\sin (c+d x)}{a+b \sec (c+d x)} \, dx &=-\int \frac {\cos (c+d x) \sin (c+d x)}{-b-a \cos (c+d x)} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{a (-b+x)} \, dx,x,-a \cos (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{-b+x} \, dx,x,-a \cos (c+d x)\right )}{a^2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (1-\frac {b}{b-x}\right ) \, dx,x,-a \cos (c+d x)\right )}{a^2 d}\\ &=-\frac {\cos (c+d x)}{a d}+\frac {b \log (b+a \cos (c+d x))}{a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 30, normalized size = 0.88 \[ \frac {b \log (a \cos (c+d x)+b)-a \cos (c+d x)}{a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 31, normalized size = 0.91 \[ -\frac {a \cos \left (d x + c\right ) - b \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 38, normalized size = 1.12 \[ -\frac {\cos \left (d x + c\right )}{a d} + \frac {b \log \left ({\left | -a \cos \left (d x + c\right ) - b \right |}\right )}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 53, normalized size = 1.56 \[ \frac {b \ln \left (a +b \sec \left (d x +c \right )\right )}{d \,a^{2}}-\frac {1}{d a \sec \left (d x +c \right )}-\frac {b \ln \left (\sec \left (d x +c \right )\right )}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 33, normalized size = 0.97 \[ -\frac {\frac {\cos \left (d x + c\right )}{a} - \frac {b \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 30, normalized size = 0.88 \[ \frac {b\,\ln \left (b+a\,\cos \left (c+d\,x\right )\right )-a\,\cos \left (c+d\,x\right )}{a^2\,d} \]
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 \[ \int \frac {\sin {\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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