Integrand size = 21, antiderivative size = 59 \[ \int (a+b \cos (c+d x))^2 \sec ^3(c+d x) \, dx=\frac {\left (a^2+2 b^2\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {2 a b \tan (c+d x)}{d}+\frac {a^2 \sec (c+d x) \tan (c+d x)}{2 d} \]
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Time = 0.09 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2868, 3852, 8, 3091, 3855} \[ \int (a+b \cos (c+d x))^2 \sec ^3(c+d x) \, dx=\frac {\left (a^2+2 b^2\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {a^2 \tan (c+d x) \sec (c+d x)}{2 d}+\frac {2 a b \tan (c+d x)}{d} \]
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Rule 8
Rule 2868
Rule 3091
Rule 3852
Rule 3855
Rubi steps \begin{align*} \text {integral}& = (2 a b) \int \sec ^2(c+d x) \, dx+\int \left (a^2+b^2 \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx \\ & = \frac {a^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \left (a^2+2 b^2\right ) \int \sec (c+d x) \, dx-\frac {(2 a b) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d} \\ & = \frac {\left (a^2+2 b^2\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {2 a b \tan (c+d x)}{d}+\frac {a^2 \sec (c+d x) \tan (c+d x)}{2 d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.14 \[ \int (a+b \cos (c+d x))^2 \sec ^3(c+d x) \, dx=\frac {a^2 \text {arctanh}(\sin (c+d x))}{2 d}+\frac {b^2 \text {arctanh}(\sin (c+d x))}{d}+\frac {2 a b \tan (c+d x)}{d}+\frac {a^2 \sec (c+d x) \tan (c+d x)}{2 d} \]
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Time = 3.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.17
method | result | size |
derivativedivides | \(\frac {a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+2 a b \tan \left (d x +c \right )+b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(69\) |
default | \(\frac {a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+2 a b \tan \left (d x +c \right )+b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(69\) |
parts | \(\frac {a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {2 a b \tan \left (d x +c \right )}{d}\) | \(74\) |
parallelrisch | \(\frac {-\left (a^{2}+2 b^{2}\right ) \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (a^{2}+2 b^{2}\right ) \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+2 a^{2} \sin \left (d x +c \right )+4 a b \sin \left (2 d x +2 c \right )}{2 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(111\) |
risch | \(-\frac {i a \left (a \,{\mathrm e}^{3 i \left (d x +c \right )}-4 b \,{\mathrm e}^{2 i \left (d x +c \right )}-a \,{\mathrm e}^{i \left (d x +c \right )}-4 b \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b^{2}}{d}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b^{2}}{d}\) | \(144\) |
norman | \(\frac {\frac {a \left (a -4 b \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a \left (a +4 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {a \left (3 a -4 b \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a \left (3 a +4 b \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {\left (a^{2}+2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {\left (a^{2}+2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(172\) |
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Time = 0.26 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.58 \[ \int (a+b \cos (c+d x))^2 \sec ^3(c+d x) \, dx=\frac {{\left (a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (4 \, a b \cos \left (d x + c\right ) + a^{2}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
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\[ \int (a+b \cos (c+d x))^2 \sec ^3(c+d x) \, dx=\int \left (a + b \cos {\left (c + d x \right )}\right )^{2} \sec ^{3}{\left (c + d x \right )}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.47 \[ \int (a+b \cos (c+d x))^2 \sec ^3(c+d x) \, dx=-\frac {a^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 2 \, b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 8 \, a b \tan \left (d x + c\right )}{4 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (55) = 110\).
Time = 0.30 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.15 \[ \int (a+b \cos (c+d x))^2 \sec ^3(c+d x) \, dx=\frac {{\left (a^{2} + 2 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (a^{2} + 2 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \]
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Time = 15.29 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.68 \[ \int (a+b \cos (c+d x))^2 \sec ^3(c+d x) \, dx=\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a^2+2\,b^2\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (4\,a\,b-a^2\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a^2+4\,b\,a\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
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