Integrand size = 23, antiderivative size = 58 \[ \int (a+b \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=a A x+\frac {b (2 A+C) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {a C \tan (c+d x)}{d}+\frac {b C \sec (c+d x) \tan (c+d x)}{2 d} \]
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Time = 0.07 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {4134, 3855, 3852, 8} \[ \int (a+b \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=a A x+\frac {a C \tan (c+d x)}{d}+\frac {b (2 A+C) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {b C \tan (c+d x) \sec (c+d x)}{2 d} \]
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Rule 8
Rule 3852
Rule 3855
Rule 4134
Rubi steps \begin{align*} \text {integral}& = \frac {b C \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \int \left (2 a A+b (2 A+C) \sec (c+d x)+2 a C \sec ^2(c+d x)\right ) \, dx \\ & = a A x+\frac {b C \sec (c+d x) \tan (c+d x)}{2 d}+(a C) \int \sec ^2(c+d x) \, dx+\frac {1}{2} (b (2 A+C)) \int \sec (c+d x) \, dx \\ & = a A x+\frac {b (2 A+C) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {b C \sec (c+d x) \tan (c+d x)}{2 d}-\frac {(a C) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d} \\ & = a A x+\frac {b (2 A+C) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {a C \tan (c+d x)}{d}+\frac {b C \sec (c+d x) \tan (c+d x)}{2 d} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.16 \[ \int (a+b \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=a A x+\frac {A b \text {arctanh}(\sin (c+d x))}{d}+\frac {b C \text {arctanh}(\sin (c+d x))}{2 d}+\frac {a C \tan (c+d x)}{d}+\frac {b C \sec (c+d x) \tan (c+d x)}{2 d} \]
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Time = 0.44 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.28
method | result | size |
derivativedivides | \(\frac {a A \left (d x +c \right )+C a \tan \left (d x +c \right )+A b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(74\) |
default | \(\frac {a A \left (d x +c \right )+C a \tan \left (d x +c \right )+A b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(74\) |
parts | \(a A x +\frac {A b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a C \tan \left (d x +c \right )}{d}+\frac {C b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(75\) |
parallelrisch | \(\frac {-b \left (1+\cos \left (2 d x +2 c \right )\right ) \left (A +\frac {C}{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+b \left (1+\cos \left (2 d x +2 c \right )\right ) \left (A +\frac {C}{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+a A x d \cos \left (2 d x +2 c \right )+a A x d +C a \sin \left (2 d x +2 c \right )+C \sin \left (d x +c \right ) b}{d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(120\) |
norman | \(\frac {a A x +a A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\frac {\left (2 a +b \right ) C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {C \left (2 a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}-2 a A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2}}-\frac {b \left (2 A +C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {b \left (2 A +C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(143\) |
risch | \(a A x -\frac {i C \left (b \,{\mathrm e}^{3 i \left (d x +c \right )}-2 a \,{\mathrm e}^{2 i \left (d x +c \right )}-b \,{\mathrm e}^{i \left (d x +c \right )}-2 a \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{d}-\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{2 d}+\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{d}+\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{2 d}\) | \(144\) |
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Time = 0.27 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.74 \[ \int (a+b \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {4 \, A a d x \cos \left (d x + c\right )^{2} + {\left (2 \, A + C\right )} b \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (2 \, A + C\right )} b \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, C a \cos \left (d x + c\right ) + C b\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
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\[ \int (a+b \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )\, dx \]
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Time = 0.22 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.52 \[ \int (a+b \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {4 \, {\left (d x + c\right )} A a - C b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, A b \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 4 \, C a \tan \left (d x + c\right )}{4 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (54) = 108\).
Time = 0.32 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.31 \[ \int (a+b \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (d x + c\right )} A a + {\left (2 \, A b + C b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (2 \, A b + C b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (2 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \]
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Time = 15.57 (sec) , antiderivative size = 135, normalized size of antiderivative = 2.33 \[ \int (a+b \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2\,A\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {C\,a\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {C\,b\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2}-\frac {A\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,2{}\mathrm {i}}{d}-\frac {C\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,1{}\mathrm {i}}{d} \]
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