Integrand size = 19, antiderivative size = 67 \[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \, dx=3 a^2 b x+\frac {3 a b^2 \text {arctanh}(\sin (c+d x))}{d}+\frac {a \left (a^2-b^2\right ) \sin (c+d x)}{d}+\frac {b^2 (a+b \sec (c+d x)) \sin (c+d x)}{d} \]
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Time = 0.14 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3927, 4132, 8, 4130, 3855} \[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {a \left (a^2-b^2\right ) \sin (c+d x)}{d}+3 a^2 b x+\frac {3 a b^2 \text {arctanh}(\sin (c+d x))}{d}+\frac {b^2 \sin (c+d x) (a+b \sec (c+d x))}{d} \]
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Rule 8
Rule 3855
Rule 3927
Rule 4130
Rule 4132
Rubi steps \begin{align*} \text {integral}& = \frac {b^2 (a+b \sec (c+d x)) \sin (c+d x)}{d}+\int \cos (c+d x) \left (a \left (a^2-b^2\right )+3 a^2 b \sec (c+d x)+3 a b^2 \sec ^2(c+d x)\right ) \, dx \\ & = \frac {b^2 (a+b \sec (c+d x)) \sin (c+d x)}{d}+\left (3 a^2 b\right ) \int 1 \, dx+\int \cos (c+d x) \left (a \left (a^2-b^2\right )+3 a b^2 \sec ^2(c+d x)\right ) \, dx \\ & = 3 a^2 b x+\frac {a \left (a^2-b^2\right ) \sin (c+d x)}{d}+\frac {b^2 (a+b \sec (c+d x)) \sin (c+d x)}{d}+\left (3 a b^2\right ) \int \sec (c+d x) \, dx \\ & = 3 a^2 b x+\frac {3 a b^2 \text {arctanh}(\sin (c+d x))}{d}+\frac {a \left (a^2-b^2\right ) \sin (c+d x)}{d}+\frac {b^2 (a+b \sec (c+d x)) \sin (c+d x)}{d} \\ \end{align*}
Time = 0.86 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.31 \[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {3 a b \left (a c+a d x-b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+a^3 \sin (c+d x)+b^3 \tan (c+d x)}{d} \]
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Time = 0.79 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(\frac {a^{3} \sin \left (d x +c \right )+3 a^{2} b \left (d x +c \right )+3 a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+\tan \left (d x +c \right ) b^{3}}{d}\) | \(57\) |
default | \(\frac {a^{3} \sin \left (d x +c \right )+3 a^{2} b \left (d x +c \right )+3 a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+\tan \left (d x +c \right ) b^{3}}{d}\) | \(57\) |
parallelrisch | \(\frac {6 a^{2} b x d \cos \left (d x +c \right )-6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a \,b^{2} \cos \left (d x +c \right )+6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a \,b^{2} \cos \left (d x +c \right )+2 b^{3} \sin \left (d x +c \right )+a^{3} \sin \left (2 d x +2 c \right )}{2 d \cos \left (d x +c \right )}\) | \(101\) |
risch | \(3 a^{2} b x -\frac {i a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {i a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {2 i b^{3}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b^{2}}{d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b^{2}}{d}\) | \(111\) |
norman | \(\frac {3 a^{2} b x -\frac {4 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}+\frac {2 \left (a^{3}-b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}+\frac {2 \left (a^{3}+b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-3 a^{2} b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-3 a^{2} b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+3 a^{2} b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}-\frac {3 a \,b^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}+\frac {3 a \,b^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(202\) |
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Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.40 \[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {6 \, a^{2} b d x \cos \left (d x + c\right ) + 3 \, a b^{2} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, a b^{2} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (a^{3} \cos \left (d x + c\right ) + b^{3}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
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\[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{3} \cos {\left (c + d x \right )}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.99 \[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {6 \, {\left (d x + c\right )} a^{2} b + 3 \, a b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, a^{3} \sin \left (d x + c\right ) + 2 \, b^{3} \tan \left (d x + c\right )}{2 \, d} \]
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Time = 0.33 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.96 \[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {3 \, {\left (d x + c\right )} a^{2} b + 3 \, a b^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, a b^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1}}{d} \]
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Time = 13.91 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.45 \[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {a^3\,\sin \left (c+d\,x\right )}{d}+\frac {b^3\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {6\,a^2\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {6\,a\,b^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
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