Integrand size = 26, antiderivative size = 24 \[ \int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx=\frac {a \text {arctanh}(\sin (c+d x))}{d}+\frac {b \sec (c+d x)}{d} \]
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Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3169, 3855, 2686, 8} \[ \int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx=\frac {a \text {arctanh}(\sin (c+d x))}{d}+\frac {b \sec (c+d x)}{d} \]
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Rule 8
Rule 2686
Rule 3169
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int (a \sec (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx \\ & = a \int \sec (c+d x) \, dx+b \int \sec (c+d x) \tan (c+d x) \, dx \\ & = \frac {a \text {arctanh}(\sin (c+d x))}{d}+\frac {b \text {Subst}(\int 1 \, dx,x,\sec (c+d x))}{d} \\ & = \frac {a \text {arctanh}(\sin (c+d x))}{d}+\frac {b \sec (c+d x)}{d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx=\frac {a \text {arctanh}(\sin (c+d x))}{d}+\frac {b \sec (c+d x)}{d} \]
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Time = 0.60 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33
method | result | size |
derivativedivides | \(\frac {a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+\frac {b}{\cos \left (d x +c \right )}}{d}\) | \(32\) |
default | \(\frac {a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+\frac {b}{\cos \left (d x +c \right )}}{d}\) | \(32\) |
parts | \(\frac {a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {b \sec \left (d x +c \right )}{d}\) | \(32\) |
parallelrisch | \(\frac {-\cos \left (d x +c \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a +\cos \left (d x +c \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a +\cos \left (d x +c \right ) b +b}{d \cos \left (d x +c \right )}\) | \(64\) |
risch | \(\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {a \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}\) | \(67\) |
norman | \(\frac {-\frac {2 b}{d}-\frac {2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}+\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(92\) |
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (24) = 48\).
Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.25 \[ \int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx=\frac {a \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - a \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, b}{2 \, d \cos \left (d x + c\right )} \]
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\[ \int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx=\int \left (a \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}\, dx \]
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none
Time = 0.24 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx=\frac {a {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + \frac {2 \, b}{\cos \left (d x + c\right )}}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (24) = 48\).
Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.25 \[ \int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx=\frac {a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, b}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}}{d} \]
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Time = 21.12 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.58 \[ \int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx=\frac {2\,a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {2\,b}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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