Integrand size = 29, antiderivative size = 60 \[ \int \cos (c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=a (2 A b+a B) x+\frac {b (A b+2 a B) \text {arctanh}(\sin (c+d x))}{d}+\frac {a^2 A \sin (c+d x)}{d}+\frac {b^2 B \tan (c+d x)}{d} \]
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Time = 0.11 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {4109, 3855, 3852, 8} \[ \int \cos (c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {a^2 A \sin (c+d x)}{d}+\frac {b (2 a B+A b) \text {arctanh}(\sin (c+d x))}{d}+a x (a B+2 A b)+\frac {b^2 B \tan (c+d x)}{d} \]
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Rule 8
Rule 3852
Rule 3855
Rule 4109
Rubi steps \begin{align*} \text {integral}& = \frac {a^2 A \sin (c+d x)}{d}-\int \left (-a (2 A b+a B)+\left (-A b^2-2 a b B\right ) \sec (c+d x)-b^2 B \sec ^2(c+d x)\right ) \, dx \\ & = a (2 A b+a B) x+\frac {a^2 A \sin (c+d x)}{d}+\left (b^2 B\right ) \int \sec ^2(c+d x) \, dx+(b (A b+2 a B)) \int \sec (c+d x) \, dx \\ & = a (2 A b+a B) x+\frac {b (A b+2 a B) \text {arctanh}(\sin (c+d x))}{d}+\frac {a^2 A \sin (c+d x)}{d}-\frac {\left (b^2 B\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d} \\ & = a (2 A b+a B) x+\frac {b (A b+2 a B) \text {arctanh}(\sin (c+d x))}{d}+\frac {a^2 A \sin (c+d x)}{d}+\frac {b^2 B \tan (c+d x)}{d} \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.18 \[ \int \cos (c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=2 a A b x+a^2 B x+\frac {A b^2 \text {arctanh}(\sin (c+d x))}{d}+\frac {2 a b B \text {arctanh}(\sin (c+d x))}{d}+\frac {a^2 A \sin (c+d x)}{d}+\frac {b^2 B \tan (c+d x)}{d} \]
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Time = 2.00 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.43
method | result | size |
derivativedivides | \(\frac {A \,a^{2} \sin \left (d x +c \right )+B \,a^{2} \left (d x +c \right )+2 A a b \left (d x +c \right )+2 B a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+A \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \tan \left (d x +c \right ) b^{2}}{d}\) | \(86\) |
default | \(\frac {A \,a^{2} \sin \left (d x +c \right )+B \,a^{2} \left (d x +c \right )+2 A a b \left (d x +c \right )+2 B a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+A \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \tan \left (d x +c \right ) b^{2}}{d}\) | \(86\) |
parallelrisch | \(\frac {-2 b \cos \left (d x +c \right ) \left (A b +2 B a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+2 b \cos \left (d x +c \right ) \left (A b +2 B a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+A \,a^{2} \sin \left (2 d x +2 c \right )+4 \left (A b +\frac {B a}{2}\right ) d x a \cos \left (d x +c \right )+2 B \sin \left (d x +c \right ) b^{2}}{2 d \cos \left (d x +c \right )}\) | \(118\) |
risch | \(2 A a b x +B \,a^{2} x -\frac {i A \,a^{2} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {i A \,a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {2 i b^{2} B}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{2}}{d}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B a b}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{2}}{d}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B a b}{d}\) | \(160\) |
norman | \(\frac {\left (2 A a b +B \,a^{2}\right ) x +\left (-2 A a b -B \,a^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-2 A a b -B \,a^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (2 A a b +B \,a^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {2 \left (A \,a^{2}-b^{2} B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}+\frac {2 \left (A \,a^{2}+b^{2} B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {4 A \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {b \left (A b +2 B a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {b \left (A b +2 B a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(245\) |
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Time = 0.32 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.95 \[ \int \cos (c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {2 \, {\left (B a^{2} + 2 \, A a b\right )} d x \cos \left (d x + c\right ) + {\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (A a^{2} \cos \left (d x + c\right ) + B b^{2}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
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\[ \int \cos (c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\int \left (A + B \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{2} \cos {\left (c + d x \right )}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.72 \[ \int \cos (c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {2 \, {\left (d x + c\right )} B a^{2} + 4 \, {\left (d x + c\right )} A a b + 2 \, B a b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + A b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, A a^{2} \sin \left (d x + c\right ) + 2 \, B b^{2} \tan \left (d x + c\right )}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (60) = 120\).
Time = 0.33 (sec) , antiderivative size = 154, normalized size of antiderivative = 2.57 \[ \int \cos (c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {{\left (B a^{2} + 2 \, A a b\right )} {\left (d x + c\right )} + {\left (2 \, B a b + A b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (2 \, B a b + A b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1}}{d} \]
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Time = 15.25 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.72 \[ \int \cos (c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {B\,b^2\,\mathrm {tan}\left (c+d\,x\right )}{d}+\frac {2\,B\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,A\,b^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {A\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d\,\cos \left (c+d\,x\right )}+\frac {4\,A\,a\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {4\,B\,a\,b\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
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