Integrand size = 28, antiderivative size = 63 \[ \int \sec ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {B \text {arctanh}(\sin (c+d x))}{2 d}+\frac {C \tan (c+d x)}{d}+\frac {B \sec (c+d x) \tan (c+d x)}{2 d}+\frac {C \tan ^3(c+d x)}{3 d} \]
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Time = 0.06 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {4132, 3853, 3855, 12, 3852} \[ \int \sec ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {B \text {arctanh}(\sin (c+d x))}{2 d}+\frac {B \tan (c+d x) \sec (c+d x)}{2 d}+\frac {C \tan ^3(c+d x)}{3 d}+\frac {C \tan (c+d x)}{d} \]
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Rule 12
Rule 3852
Rule 3853
Rule 3855
Rule 4132
Rubi steps \begin{align*} \text {integral}& = B \int \sec ^3(c+d x) \, dx+\int C \sec ^4(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+C \int \sec ^4(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}-\frac {C \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d} \\ & = \frac {B \text {arctanh}(\sin (c+d x))}{2 d}+\frac {C \tan (c+d x)}{d}+\frac {B \sec (c+d x) \tan (c+d x)}{2 d}+\frac {C \tan ^3(c+d x)}{3 d} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.95 \[ \int \sec ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {B \text {arctanh}(\sin (c+d x))}{2 d}+\frac {B \sec (c+d x) \tan (c+d x)}{2 d}+\frac {C \left (\tan (c+d x)+\frac {1}{3} \tan ^3(c+d x)\right )}{d} \]
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Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {B \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(60\) |
default | \(\frac {B \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(60\) |
parts | \(\frac {B \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(62\) |
risch | \(-\frac {i \left (3 B \,{\mathrm e}^{5 i \left (d x +c \right )}-12 C \,{\mathrm e}^{2 i \left (d x +c \right )}-3 B \,{\mathrm e}^{i \left (d x +c \right )}-4 C \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {B \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}-\frac {B \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}\) | \(99\) |
norman | \(\frac {\frac {\left (B -2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}+\frac {4 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}-\frac {\left (B +2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{3}}-\frac {B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(111\) |
parallelrisch | \(\frac {-9 \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+9 \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+6 B \sin \left (2 d x +2 c \right )+4 C \left (\sin \left (3 d x +3 c \right )+3 \sin \left (d x +c \right )\right )}{6 d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(126\) |
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Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.40 \[ \int \sec ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, B \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, B \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (4 \, C \cos \left (d x + c\right )^{2} + 3 \, B \cos \left (d x + c\right ) + 2 \, C\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]
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\[ \int \sec ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (B + C \sec {\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.11 \[ \int \sec ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C - 3 \, B {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{12 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (57) = 114\).
Time = 0.30 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.94 \[ \int \sec ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, B \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, B \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (3 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
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Time = 17.12 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.73 \[ \int \sec ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {B\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {\left (B-2\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {4\,C\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\left (-B-2\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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