Optimal. Leaf size=75 \[ -\frac {b \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )}-\frac {\log (1-\sin (c+d x))}{2 d (a+b)}+\frac {\log (\sin (c+d x)+1)}{2 d (a-b)} \]
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Rubi [A] time = 0.08, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2668, 706, 31, 633} \[ -\frac {b \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )}-\frac {\log (1-\sin (c+d x))}{2 d (a+b)}+\frac {\log (\sin (c+d x)+1)}{2 d (a-b)} \]
Antiderivative was successfully verified.
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Rule 31
Rule 633
Rule 706
Rule 2668
Rubi steps
\begin {align*} \int \frac {\sec (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {b \operatorname {Subst}\left (\int \frac {1}{(a+x) \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac {b \operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \sin (c+d x)\right )}{\left (a^2-b^2\right ) d}-\frac {b \operatorname {Subst}\left (\int \frac {-a+x}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{\left (a^2-b^2\right ) d}\\ &=-\frac {b \log (a+b \sin (c+d x))}{\left (a^2-b^2\right ) d}-\frac {\operatorname {Subst}\left (\int \frac {1}{-b-x} \, dx,x,b \sin (c+d x)\right )}{2 (a-b) d}+\frac {\operatorname {Subst}\left (\int \frac {1}{b-x} \, dx,x,b \sin (c+d x)\right )}{2 (a+b) d}\\ &=-\frac {\log (1-\sin (c+d x))}{2 (a+b) d}+\frac {\log (1+\sin (c+d x))}{2 (a-b) d}-\frac {b \log (a+b \sin (c+d x))}{\left (a^2-b^2\right ) d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 64, normalized size = 0.85 \[ \frac {(b-a) \log (1-\sin (c+d x))+(a+b) \log (\sin (c+d x)+1)-2 b \log (a+b \sin (c+d x))}{2 d (a-b) (a+b)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 62, normalized size = 0.83 \[ -\frac {2 \, b \log \left (b \sin \left (d x + c\right ) + a\right ) - {\left (a + b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left (a - b\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, {\left (a^{2} - b^{2}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.09, size = 71, normalized size = 0.95 \[ -\frac {\frac {2 \, b^{2} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{2} b - b^{3}} - \frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a - b} + \frac {\log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a + b}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 76, normalized size = 1.01 \[ -\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{d \left (2 a +2 b \right )}-\frac {b \ln \left (a +b \sin \left (d x +c \right )\right )}{d \left (a +b \right ) \left (a -b \right )}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{d \left (2 a -2 b \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 64, normalized size = 0.85 \[ -\frac {\frac {2 \, b \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{2} - b^{2}} - \frac {\log \left (\sin \left (d x + c\right ) + 1\right )}{a - b} + \frac {\log \left (\sin \left (d x + c\right ) - 1\right )}{a + b}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 69, normalized size = 0.92 \[ \frac {\ln \left (\sin \left (c+d\,x\right )+1\right )}{2\,d\,\left (a-b\right )}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )}{2\,d\,\left (a+b\right )}-\frac {b\,\ln \left (a+b\,\sin \left (c+d\,x\right )\right )}{d\,\left (a^2-b^2\right )} \]
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 \[ \int \frac {\sec {\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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