Integrand size = 34, antiderivative size = 36 \[ \int \frac {(a B+b B \cos (c+d x)) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx=\frac {B \text {arctanh}(\sin (c+d x))}{2 d}+\frac {B \sec (c+d x) \tan (c+d x)}{2 d} \]
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Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {21, 3853, 3855} \[ \int \frac {(a B+b B \cos (c+d x)) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx=\frac {B \text {arctanh}(\sin (c+d x))}{2 d}+\frac {B \tan (c+d x) \sec (c+d x)}{2 d} \]
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Rule 21
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = B \int \sec ^3(c+d x) \, dx \\ & = \frac {B \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} B \int \sec (c+d x) \, dx \\ & = \frac {B \text {arctanh}(\sin (c+d x))}{2 d}+\frac {B \sec (c+d x) \tan (c+d x)}{2 d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \frac {(a B+b B \cos (c+d x)) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx=B \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\sec (c+d x) \tan (c+d x)}{2 d}\right ) \]
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Time = 1.58 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.03
method | result | size |
derivativedivides | \(\frac {B \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(37\) |
default | \(\frac {B \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(37\) |
parallelrisch | \(-\frac {\left (\left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (-1-\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-2 \sin \left (d x +c \right )\right ) B}{2 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(79\) |
risch | \(-\frac {i B \left ({\mathrm e}^{3 i \left (d x +c \right )}-{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {B \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}-\frac {B \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}\) | \(81\) |
norman | \(\frac {\frac {B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {B \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(117\) |
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Time = 0.32 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.78 \[ \int \frac {(a B+b B \cos (c+d x)) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx=\frac {B \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - B \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, B \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
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\[ \int \frac {(a B+b B \cos (c+d x)) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx=B \int \sec ^{3}{\left (c + d x \right )}\, dx \]
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Exception generated. \[ \int \frac {(a B+b B \cos (c+d x)) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.30 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.44 \[ \int \frac {(a B+b B \cos (c+d x)) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx=\frac {B \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - B \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {2 \, B \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1}}{4 \, d} \]
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Time = 0.99 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.03 \[ \int \frac {(a B+b B \cos (c+d x)) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx=\frac {B\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+B\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {B\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \]
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