Integrand size = 19, antiderivative size = 75 \[ \int \frac {\sec (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\log (1-\sin (c+d x))}{2 (a+b) d}+\frac {\log (1+\sin (c+d x))}{2 (a-b) d}-\frac {b \log (a+b \sin (c+d x))}{\left (a^2-b^2\right ) d} \]
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Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2747, 720, 31, 647} \[ \int \frac {\sec (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {b \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )}-\frac {\log (1-\sin (c+d x))}{2 d (a+b)}+\frac {\log (\sin (c+d x)+1)}{2 d (a-b)} \]
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Rule 31
Rule 647
Rule 720
Rule 2747
Rubi steps \begin{align*} \text {integral}& = \frac {b \text {Subst}\left (\int \frac {1}{(a+x) \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{d} \\ & = -\frac {b \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \sin (c+d x)\right )}{\left (a^2-b^2\right ) d}-\frac {b \text {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 {\text {Subst}\left (\int \frac {1}{-b-x} \, dx,x,b \sin (c+d x)\right )}{2 (a-b) d}+\frac {\text {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*}
Time = 0.01 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.85 \[ \int \frac {\sec (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {(-a+b) \log (1-\sin (c+d x))+(a+b) \log (1+\sin (c+d x))-2 b \log (a+b \sin (c+d x))}{2 (a-b) (a+b) d} \]
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Time = 0.66 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {-\frac {b \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a -b \right ) \left (a +b \right )}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{2 a -2 b}-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{2 a +2 b}}{d}\) | \(71\) |
default | \(\frac {-\frac {b \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a -b \right ) \left (a +b \right )}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{2 a -2 b}-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{2 a +2 b}}{d}\) | \(71\) |
parallelrisch | \(\frac {-b \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )+\left (-a +b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \left (a +b \right )}{d \left (a^{2}-b^{2}\right )}\) | \(81\) |
norman | \(\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \left (a -b \right )}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d \left (a +b \right )}-\frac {b \ln \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}{d \left (a^{2}-b^{2}\right )}\) | \(92\) |
risch | \(\frac {i x}{a +b}+\frac {i c}{d \left (a +b \right )}-\frac {i x}{a -b}-\frac {i c}{d \left (a -b \right )}+\frac {2 i b x}{a^{2}-b^{2}}+\frac {2 i b c}{d \left (a^{2}-b^{2}\right )}-\frac {\ln \left (-i+{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left (a +b \right )}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \left (a -b \right )}-\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}-1\right )}{d \left (a^{2}-b^{2}\right )}\) | \(175\) |
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Time = 0.31 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.83 \[ \int \frac {\sec (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {2 \, b \log \left (b \sin \left (d x + c\right ) + a\right ) - {\left (a + b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left (a - b\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, {\left (a^{2} - b^{2}\right )} d} \]
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\[ \int \frac {\sec (c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\sec {\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
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Time = 0.18 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.85 \[ \int \frac {\sec (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\frac {2 \, b \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{2} - b^{2}} - \frac {\log \left (\sin \left (d x + c\right ) + 1\right )}{a - b} + \frac {\log \left (\sin \left (d x + c\right ) - 1\right )}{a + b}}{2 \, d} \]
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Time = 0.30 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.95 \[ \int \frac {\sec (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\frac {2 \, b^{2} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{2} b - b^{3}} - \frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a - b} + \frac {\log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a + b}}{2 \, d} \]
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Time = 4.51 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.92 \[ \int \frac {\sec (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )}{2\,d\,\left (a-b\right )}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )}{2\,d\,\left (a+b\right )}-\frac {b\,\ln \left (a+b\,\sin \left (c+d\,x\right )\right )}{d\,\left (a^2-b^2\right )} \]
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