Integrand size = 31, antiderivative size = 63 \[ \int (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a A x+\frac {a (2 A+2 B+C) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {a (B+C) \tan (c+d x)}{d}+\frac {a C \sec (c+d x) \tan (c+d x)}{2 d} \]
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Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {4133, 3855, 3852, 8} \[ \int (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a (2 A+2 B+C) \text {arctanh}(\sin (c+d x))}{2 d}+a A x+\frac {a (B+C) \tan (c+d x)}{d}+\frac {a C \tan (c+d x) \sec (c+d x)}{2 d} \]
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Rule 8
Rule 3852
Rule 3855
Rule 4133
Rubi steps \begin{align*} \text {integral}& = \frac {a C \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \int \left (2 a A+a (2 A+2 B+C) \sec (c+d x)+2 a (B+C) \sec ^2(c+d x)\right ) \, dx \\ & = a A x+\frac {a C \sec (c+d x) \tan (c+d x)}{2 d}+(a (B+C)) \int \sec ^2(c+d x) \, dx+\frac {1}{2} (a (2 A+2 B+C)) \int \sec (c+d x) \, dx \\ & = a A x+\frac {a (2 A+2 B+C) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {a C \sec (c+d x) \tan (c+d x)}{2 d}-\frac {(a (B+C)) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d} \\ & = a A x+\frac {a (2 A+2 B+C) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {a (B+C) \tan (c+d x)}{d}+\frac {a C \sec (c+d x) \tan (c+d x)}{2 d} \\ \end{align*}
Time = 1.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a (2 A d x+(2 A+2 B+C) \text {arctanh}(\sin (c+d x))+(2 (B+C)+C \sec (c+d x)) \tan (c+d x))}{2 d} \]
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Time = 0.38 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.35
method | result | size |
parts | \(a A x +\frac {\left (a A +a B \right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (a B +C a \right ) \tan \left (d x +c \right )}{d}+\frac {C a \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(85\) |
derivativedivides | \(\frac {a A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a B \tan \left (d x +c \right )+C a \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a A \left (d x +c \right )+a B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C a \tan \left (d x +c \right )}{d}\) | \(100\) |
default | \(\frac {a A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a B \tan \left (d x +c \right )+C a \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a A \left (d x +c \right )+a B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C a \tan \left (d x +c \right )}{d}\) | \(100\) |
parallelrisch | \(-\frac {\left (\left (1+\cos \left (2 d x +2 c \right )\right ) \left (A +B +\frac {C}{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\left (1+\cos \left (2 d x +2 c \right )\right ) \left (A +B +\frac {C}{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-d x A \cos \left (2 d x +2 c \right )+\left (-B -C \right ) \sin \left (2 d x +2 c \right )-d x A -C \sin \left (d x +c \right )\right ) a}{d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(127\) |
norman | \(\frac {a A x +a A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\frac {a \left (2 B +3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-2 a A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\frac {a \left (2 B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2}}-\frac {a \left (2 A +2 B +C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {a \left (2 A +2 B +C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(149\) |
risch | \(a A x -\frac {i a \left (C \,{\mathrm e}^{3 i \left (d x +c \right )}-2 B \,{\mathrm e}^{2 i \left (d x +c \right )}-2 C \,{\mathrm e}^{2 i \left (d x +c \right )}-C \,{\mathrm e}^{i \left (d x +c \right )}-2 B -2 C \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{d}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{d}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{2 d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{2 d}\) | \(198\) |
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Time = 0.29 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.73 \[ \int (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {4 \, A a d x \cos \left (d x + c\right )^{2} + {\left (2 \, A + 2 \, B + C\right )} a \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (2 \, A + 2 \, B + C\right )} a \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, {\left (B + C\right )} a \cos \left (d x + c\right ) + C a\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
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\[ \int (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a \left (\int A\, dx + \int A \sec {\left (c + d x \right )}\, dx + \int B \sec {\left (c + d x \right )}\, dx + \int B \sec ^{2}{\left (c + d x \right )}\, dx + \int C \sec ^{2}{\left (c + d x \right )}\, dx + \int C \sec ^{3}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.21 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.84 \[ \int (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {4 \, {\left (d x + c\right )} A a - C a {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, A a \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 4 \, B a \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 4 \, B a \tan \left (d x + c\right ) + 4 \, C a \tan \left (d x + c\right )}{4 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (59) = 118\).
Time = 0.32 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.24 \[ \int (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (d x + c\right )} A a + {\left (2 \, A a + 2 \, B a + C a\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (2 \, A a + 2 \, B a + C a\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (2 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \]
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Time = 18.12 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.79 \[ \int (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\frac {C\,a\,\sin \left (c+d\,x\right )}{2}+\frac {B\,a\,\sin \left (2\,c+2\,d\,x\right )}{2}+\frac {C\,a\,\sin \left (2\,c+2\,d\,x\right )}{2}}{d\,\left (\frac {\cos \left (2\,c+2\,d\,x\right )}{2}+\frac {1}{2}\right )}-\frac {2\,\left (-A\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+A\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,1{}\mathrm {i}+B\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,1{}\mathrm {i}+\frac {C\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,1{}\mathrm {i}}{2}\right )}{d} \]
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