Integrand size = 43, antiderivative size = 26 \[ \int (b \sec (c+d x)+a \sin (c+d x))^2 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx=\frac {(b \sec (c+d x)+a \sin (c+d x))^3}{3 d} \]
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Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {4470} \[ \int (b \sec (c+d x)+a \sin (c+d x))^2 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx=\frac {(a \sin (c+d x)+b \sec (c+d x))^3}{3 d} \]
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Rule 4470
Rubi steps \begin{align*} \text {integral}& = \frac {(b \sec (c+d x)+a \sin (c+d x))^3}{3 d} \\ \end{align*}
Time = 5.95 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int (b \sec (c+d x)+a \sin (c+d x))^2 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx=\frac {(b \sec (c+d x)+a \sin (c+d x))^3}{3 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(151\) vs. \(2(24)=48\).
Time = 48.38 (sec) , antiderivative size = 152, normalized size of antiderivative = 5.85
method | result | size |
derivativedivides | \(\frac {\frac {\sin \left (d x +c \right )^{3} a^{3}}{3}+a^{2} b \left (\frac {\sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )}+\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )\right )-2 \cos \left (d x +c \right ) a^{2} b +2 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+\frac {b^{3}}{3 \cos \left (d x +c \right )^{3}}}{d}\) | \(152\) |
default | \(\frac {\frac {\sin \left (d x +c \right )^{3} a^{3}}{3}+a^{2} b \left (\frac {\sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )}+\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )\right )-2 \cos \left (d x +c \right ) a^{2} b +2 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+\frac {b^{3}}{3 \cos \left (d x +c \right )^{3}}}{d}\) | \(152\) |
parts | \(\frac {\frac {\sin \left (d x +c \right )^{3} a^{3}}{3}-2 \cos \left (d x +c \right ) a^{2} b +a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {b^{3} \sec \left (d x +c \right )^{3}}{3 d}+\frac {a^{2} b \left (\frac {\sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )}+\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )\right )}{d}+\frac {2 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(162\) |
risch | \(\frac {-3 i a^{3} {\mathrm e}^{5 i \left (d x +c \right )}+3 i a^{3} {\mathrm e}^{i \left (d x +c \right )}-12 \,{\mathrm e}^{-i \left (d x +c \right )} a^{2} b +24 \,{\mathrm e}^{3 i \left (d x +c \right )} a^{2} b -i a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}+64 \,{\mathrm e}^{3 i \left (d x +c \right )} b^{3}+i a^{3} {\mathrm e}^{9 i \left (d x +c \right )}-12 a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}+48 i {\mathrm e}^{i \left (d x +c \right )} a \,b^{2}-48 i {\mathrm e}^{5 i \left (d x +c \right )} a \,b^{2}}{24 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}\) | \(171\) |
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Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (24) = 48\).
Time = 0.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.54 \[ \int (b \sec (c+d x)+a \sin (c+d x))^2 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx=-\frac {3 \, a^{2} b \cos \left (d x + c\right )^{4} - 3 \, a^{2} b \cos \left (d x + c\right )^{2} - b^{3} + {\left (a^{3} \cos \left (d x + c\right )^{5} - a^{3} \cos \left (d x + c\right )^{3} - 3 \, a b^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3 \, d \cos \left (d x + c\right )^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (20) = 40\).
Time = 1.00 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.85 \[ \int (b \sec (c+d x)+a \sin (c+d x))^2 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx=\begin {cases} \frac {a^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {a^{2} b \sin ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{d} + \frac {a b^{2} \sin {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{d} + \frac {b^{3} \sec ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + b \sec {\left (c \right )}\right )^{2} \left (a \cos {\left (c \right )} + b \tan {\left (c \right )} \sec {\left (c \right )}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int (b \sec (c+d x)+a \sin (c+d x))^2 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx=\frac {{\left (b \sec \left (d x + c\right ) + a \sin \left (d x + c\right )\right )}^{3}}{3 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (24) = 48\).
Time = 0.70 (sec) , antiderivative size = 321, normalized size of antiderivative = 12.35 \[ \int (b \sec (c+d x)+a \sin (c+d x))^2 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx=\frac {2 \, {\left (3 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 6 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 3 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 4 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 9 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 9 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 6 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 10 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 4 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b^{3}\right )}}{3 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1\right )}^{3} d} \]
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Time = 26.84 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.85 \[ \int (b \sec (c+d x)+a \sin (c+d x))^2 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx=\frac {a^3\,\sin \left (c+d\,x\right )}{3\,d}+\frac {a^2\,b\,{\cos \left (c+d\,x\right )}^2+\sin \left (c+d\,x\right )\,a\,b^2\,\cos \left (c+d\,x\right )+\frac {b^3}{3}}{d\,{\cos \left (c+d\,x\right )}^3}-\frac {a^3\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d}-\frac {a^2\,b\,\cos \left (c+d\,x\right )}{d} \]
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