Integrand size = 31, antiderivative size = 108 \[ \int \frac {\sec ^3(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx=-\frac {(2 A-3 B) \text {arctanh}(\sin (c+d x))}{2 a d}+\frac {2 (A-B) \tan (c+d x)}{a d}-\frac {(2 A-3 B) \sec (c+d x) \tan (c+d x)}{2 a d}+\frac {(A-B) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \sec (c+d x))} \]
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Time = 0.19 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4104, 3872, 3852, 8, 3853, 3855} \[ \int \frac {\sec ^3(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx=-\frac {(2 A-3 B) \text {arctanh}(\sin (c+d x))}{2 a d}+\frac {2 (A-B) \tan (c+d x)}{a d}+\frac {(A-B) \tan (c+d x) \sec ^2(c+d x)}{d (a \sec (c+d x)+a)}-\frac {(2 A-3 B) \tan (c+d x) \sec (c+d x)}{2 a d} \]
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Rule 8
Rule 3852
Rule 3853
Rule 3855
Rule 3872
Rule 4104
Rubi steps \begin{align*} \text {integral}& = \frac {(A-B) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \sec (c+d x))}+\frac {\int \sec ^2(c+d x) (2 a (A-B)-a (2 A-3 B) \sec (c+d x)) \, dx}{a^2} \\ & = \frac {(A-B) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \sec (c+d x))}-\frac {(2 A-3 B) \int \sec ^3(c+d x) \, dx}{a}+\frac {(2 (A-B)) \int \sec ^2(c+d x) \, dx}{a} \\ & = -\frac {(2 A-3 B) \sec (c+d x) \tan (c+d x)}{2 a d}+\frac {(A-B) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \sec (c+d x))}-\frac {(2 A-3 B) \int \sec (c+d x) \, dx}{2 a}-\frac {(2 (A-B)) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{a d} \\ & = -\frac {(2 A-3 B) \text {arctanh}(\sin (c+d x))}{2 a d}+\frac {2 (A-B) \tan (c+d x)}{a d}-\frac {(2 A-3 B) \sec (c+d x) \tan (c+d x)}{2 a d}+\frac {(A-B) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \sec (c+d x))} \\ \end{align*}
Time = 0.56 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.78 \[ \int \frac {\sec ^3(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx=\frac {-((2 A-3 B) \text {arctanh}(\sin (c+d x)) (1+\sec (c+d x)))+\left (4 (A-B)+(2 A-B) \sec (c+d x)+B \sec ^2(c+d x)\right ) \tan (c+d x)}{2 a d (1+\sec (c+d x))} \]
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Time = 0.85 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.13
method | result | size |
parallelrisch | \(\frac {\left (-\frac {3 B}{2}+A \right ) \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\left (-\frac {3 B}{2}+A \right ) \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+2 \left (\left (A -B \right ) \cos \left (2 d x +2 c \right )+\left (A -\frac {B}{2}\right ) \left (\cos \left (d x +c \right )+1\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(122\) |
derivativedivides | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\frac {B}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-\frac {3 B}{2}+A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (-\frac {3 B}{2}+A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {B}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\left (\frac {3 B}{2}-A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {-\frac {3 B}{2}+A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{d a}\) | \(142\) |
default | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\frac {B}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-\frac {3 B}{2}+A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (-\frac {3 B}{2}+A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {B}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\left (\frac {3 B}{2}-A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {-\frac {3 B}{2}+A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{d a}\) | \(142\) |
norman | \(\frac {\frac {\left (A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{a d}+\frac {7 \left (A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{a d}-\frac {\left (3 A -2 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {\left (5 A -6 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{a d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}+\frac {\left (2 A -3 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a d}-\frac {\left (2 A -3 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a d}\) | \(170\) |
risch | \(\frac {i \left (2 A \,{\mathrm e}^{4 i \left (d x +c \right )}-3 B \,{\mathrm e}^{4 i \left (d x +c \right )}+2 A \,{\mathrm e}^{3 i \left (d x +c \right )}-3 B \,{\mathrm e}^{3 i \left (d x +c \right )}+6 A \,{\mathrm e}^{2 i \left (d x +c \right )}-5 B \,{\mathrm e}^{2 i \left (d x +c \right )}+2 \,{\mathrm e}^{i \left (d x +c \right )} A -B \,{\mathrm e}^{i \left (d x +c \right )}+4 A -4 B \right )}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{a d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{2 a d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{a d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{2 a d}\) | \(227\) |
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Time = 0.29 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.44 \[ \int \frac {\sec ^3(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx=-\frac {{\left ({\left (2 \, A - 3 \, B\right )} \cos \left (d x + c\right )^{3} + {\left (2 \, A - 3 \, B\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left ({\left (2 \, A - 3 \, B\right )} \cos \left (d x + c\right )^{3} + {\left (2 \, A - 3 \, B\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (4 \, {\left (A - B\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, A - B\right )} \cos \left (d x + c\right ) + B\right )} \sin \left (d x + c\right )}{4 \, {\left (a d \cos \left (d x + c\right )^{3} + a d \cos \left (d x + c\right )^{2}\right )}} \]
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\[ \int \frac {\sec ^3(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {A \sec ^{3}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \sec ^{4}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (104) = 208\).
Time = 0.22 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.61 \[ \int \frac {\sec ^3(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx=-\frac {B {\left (\frac {2 \, {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a - \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac {3 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} + \frac {3 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a} + \frac {2 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} + 2 \, A {\left (\frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} - \frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a} - \frac {2 \, \sin \left (d x + c\right )}{{\left (a - \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} - \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{2 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.44 \[ \int \frac {\sec ^3(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx=-\frac {\frac {{\left (2 \, A - 3 \, B\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a} - \frac {{\left (2 \, A - 3 \, B\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a} - \frac {2 \, {\left (A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a} + \frac {2 \, {\left (2 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a}}{2 \, d} \]
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Time = 13.58 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.10 \[ \int \frac {\sec ^3(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A-B\right )}{a\,d}-\frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (A-\frac {3\,B}{2}\right )}{a\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (2\,A-3\,B\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,A-B\right )}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )} \]
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