Integrand size = 40, antiderivative size = 131 \[ \int \frac {\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx=\frac {3 (B-C) \text {arctanh}(\sin (c+d x))}{2 a d}-\frac {(3 B-4 C) \tan (c+d x)}{a d}+\frac {3 (B-C) \sec (c+d x) \tan (c+d x)}{2 a d}+\frac {(B-C) \sec ^3(c+d x) \tan (c+d x)}{d (a+a \sec (c+d x))}-\frac {(3 B-4 C) \tan ^3(c+d x)}{3 a d} \]
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Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4157, 4104, 3872, 3853, 3855, 3852} \[ \int \frac {\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx=\frac {3 (B-C) \text {arctanh}(\sin (c+d x))}{2 a d}-\frac {(3 B-4 C) \tan ^3(c+d x)}{3 a d}-\frac {(3 B-4 C) \tan (c+d x)}{a d}+\frac {(B-C) \tan (c+d x) \sec ^3(c+d x)}{d (a \sec (c+d x)+a)}+\frac {3 (B-C) \tan (c+d x) \sec (c+d x)}{2 a d} \]
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Rule 3852
Rule 3853
Rule 3855
Rule 3872
Rule 4104
Rule 4157
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sec ^4(c+d x) (B+C \sec (c+d x))}{a+a \sec (c+d x)} \, dx \\ & = \frac {(B-C) \sec ^3(c+d x) \tan (c+d x)}{d (a+a \sec (c+d x))}+\frac {\int \sec ^3(c+d x) (3 a (B-C)-a (3 B-4 C) \sec (c+d x)) \, dx}{a^2} \\ & = \frac {(B-C) \sec ^3(c+d x) \tan (c+d x)}{d (a+a \sec (c+d x))}-\frac {(3 B-4 C) \int \sec ^4(c+d x) \, dx}{a}+\frac {(3 (B-C)) \int \sec ^3(c+d x) \, dx}{a} \\ & = \frac {3 (B-C) \sec (c+d x) \tan (c+d x)}{2 a d}+\frac {(B-C) \sec ^3(c+d x) \tan (c+d x)}{d (a+a \sec (c+d x))}+\frac {(3 (B-C)) \int \sec (c+d x) \, dx}{2 a}+\frac {(3 B-4 C) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{a d} \\ & = \frac {3 (B-C) \text {arctanh}(\sin (c+d x))}{2 a d}-\frac {(3 B-4 C) \tan (c+d x)}{a d}+\frac {3 (B-C) \sec (c+d x) \tan (c+d x)}{2 a d}+\frac {(B-C) \sec ^3(c+d x) \tan (c+d x)}{d (a+a \sec (c+d x))}-\frac {(3 B-4 C) \tan ^3(c+d x)}{3 a d} \\ \end{align*}
Time = 0.74 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.76 \[ \int \frac {\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx=\frac {\frac {(B-C) \left (3 \text {arctanh}(\sin (c+d x)) (1+\sec (c+d x))-(1+\cos (c+d x)+2 \cos (2 (c+d x))) \sec ^2(c+d x) \tan (c+d x)\right )}{1+\sec (c+d x)}+\frac {2}{3} C \tan (c+d x) \left (3+\tan ^2(c+d x)\right )}{2 a d} \]
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Time = 0.29 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.27
method | result | size |
parallelrisch | \(\frac {-9 \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \left (B -C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+9 \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \left (B -C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-8 \left (\left (\frac {B}{2}-\frac {2 C}{3}\right ) \cos \left (3 d x +3 c \right )+\frac {\left (B -\frac {7 C}{3}\right ) \cos \left (2 d x +2 c \right )}{4}+\left (B -\frac {11 C}{6}\right ) \cos \left (d x +c \right )+\frac {B}{4}-\frac {11 C}{12}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(167\) |
derivativedivides | \(\frac {-\frac {C}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {B -2 C}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\left (-\frac {3 C}{2}+\frac {3 B}{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {\frac {5 C}{2}-\frac {3 B}{2}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {C}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {2 C -B}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\left (\frac {3 C}{2}-\frac {3 B}{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {\frac {5 C}{2}-\frac {3 B}{2}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}}{d a}\) | \(190\) |
default | \(\frac {-\frac {C}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {B -2 C}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\left (-\frac {3 C}{2}+\frac {3 B}{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {\frac {5 C}{2}-\frac {3 B}{2}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {C}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {2 C -B}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\left (\frac {3 C}{2}-\frac {3 B}{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {\frac {5 C}{2}-\frac {3 B}{2}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}}{d a}\) | \(190\) |
norman | \(\frac {\frac {\left (-9 C +7 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{a d}-\frac {2 \left (B -2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {\left (B -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{a d}+\frac {\left (27 B -37 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a d}-\frac {\left (-49 C +39 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 a d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4}}-\frac {3 \left (B -C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a d}+\frac {3 \left (B -C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a d}\) | \(192\) |
risch | \(-\frac {i \left (9 B \,{\mathrm e}^{6 i \left (d x +c \right )}-9 C \,{\mathrm e}^{6 i \left (d x +c \right )}+9 B \,{\mathrm e}^{5 i \left (d x +c \right )}-9 C \,{\mathrm e}^{5 i \left (d x +c \right )}+24 B \,{\mathrm e}^{4 i \left (d x +c \right )}-24 C \,{\mathrm e}^{4 i \left (d x +c \right )}+12 B \,{\mathrm e}^{3 i \left (d x +c \right )}-24 C \,{\mathrm e}^{3 i \left (d x +c \right )}+27 B \,{\mathrm e}^{2 i \left (d x +c \right )}-39 C \,{\mathrm e}^{2 i \left (d x +c \right )}+3 B \,{\mathrm e}^{i \left (d x +c \right )}-7 C \,{\mathrm e}^{i \left (d x +c \right )}+12 B -16 C \right )}{3 a d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{2 a d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{2 a d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{2 a d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{2 a d}\) | \(276\) |
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Time = 0.27 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.30 \[ \int \frac {\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx=\frac {9 \, {\left ({\left (B - C\right )} \cos \left (d x + c\right )^{4} + {\left (B - C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 9 \, {\left ({\left (B - C\right )} \cos \left (d x + c\right )^{4} + {\left (B - C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (4 \, {\left (3 \, B - 4 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (3 \, B - 7 \, C\right )} \cos \left (d x + c\right )^{2} - {\left (3 \, B - C\right )} \cos \left (d x + c\right ) - 2 \, C\right )} \sin \left (d x + c\right )}{12 \, {\left (a d \cos \left (d x + c\right )^{4} + a d \cos \left (d x + c\right )^{3}\right )}} \]
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\[ \int \frac {\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {B \sec ^{4}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \sec ^{5}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (125) = 250\).
Time = 0.24 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.81 \[ \int \frac {\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx=\frac {C {\left (\frac {2 \, {\left (\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {16 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a - \frac {3 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac {9 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} + \frac {9 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a} + \frac {6 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - 3 \, B {\left (\frac {2 \, {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a - \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac {3 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} + \frac {3 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a} + \frac {2 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{6 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.39 \[ \int \frac {\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx=\frac {\frac {9 \, {\left (B - C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a} - \frac {9 \, {\left (B - C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a} - \frac {6 \, {\left (B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a} + \frac {2 \, {\left (9 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 16 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 9 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} a}}{6 \, d} \]
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Time = 16.29 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.16 \[ \int \frac {\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx=\frac {3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (B-C\right )}{a\,d}-\frac {\left (3\,B-5\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {16\,C}{3}-4\,B\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (B-3\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (B-C\right )}{a\,d} \]
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