Integrand size = 21, antiderivative size = 114 \[ \int (a+a \cos (c+d x))^3 \sec ^6(c+d x) \, dx=\frac {13 a^3 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {4 a^3 \tan (c+d x)}{d}+\frac {13 a^3 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {3 a^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {5 a^3 \tan ^3(c+d x)}{3 d}+\frac {a^3 \tan ^5(c+d x)}{5 d} \]
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Time = 0.16 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2836, 3853, 3855, 3852} \[ \int (a+a \cos (c+d x))^3 \sec ^6(c+d x) \, dx=\frac {13 a^3 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^3 \tan ^5(c+d x)}{5 d}+\frac {5 a^3 \tan ^3(c+d x)}{3 d}+\frac {4 a^3 \tan (c+d x)}{d}+\frac {3 a^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {13 a^3 \tan (c+d x) \sec (c+d x)}{8 d} \]
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Rule 2836
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 \sec ^3(c+d x)+3 a^3 \sec ^4(c+d x)+3 a^3 \sec ^5(c+d x)+a^3 \sec ^6(c+d x)\right ) \, dx \\ & = a^3 \int \sec ^3(c+d x) \, dx+a^3 \int \sec ^6(c+d x) \, dx+\left (3 a^3\right ) \int \sec ^4(c+d x) \, dx+\left (3 a^3\right ) \int \sec ^5(c+d x) \, dx \\ & = \frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {3 a^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{2} a^3 \int \sec (c+d x) \, dx+\frac {1}{4} \left (9 a^3\right ) \int \sec ^3(c+d x) \, dx-\frac {a^3 \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d} \\ & = \frac {a^3 \text {arctanh}(\sin (c+d x))}{2 d}+\frac {4 a^3 \tan (c+d x)}{d}+\frac {13 a^3 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {3 a^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {5 a^3 \tan ^3(c+d x)}{3 d}+\frac {a^3 \tan ^5(c+d x)}{5 d}+\frac {1}{8} \left (9 a^3\right ) \int \sec (c+d x) \, dx \\ & = \frac {13 a^3 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {4 a^3 \tan (c+d x)}{d}+\frac {13 a^3 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {3 a^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {5 a^3 \tan ^3(c+d x)}{3 d}+\frac {a^3 \tan ^5(c+d x)}{5 d} \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00 \[ \int (a+a \cos (c+d x))^3 \sec ^6(c+d x) \, dx=\frac {13 a^3 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {4 a^3 \tan (c+d x)}{d}+\frac {13 a^3 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {3 a^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {5 a^3 \tan ^3(c+d x)}{3 d}+\frac {a^3 \tan ^5(c+d x)}{5 d} \]
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Result contains complex when optimal does not.
Time = 4.07 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.27
method | result | size |
risch | \(-\frac {i a^{3} \left (195 \,{\mathrm e}^{9 i \left (d x +c \right )}+750 \,{\mathrm e}^{7 i \left (d x +c \right )}-720 \,{\mathrm e}^{6 i \left (d x +c \right )}-2320 \,{\mathrm e}^{4 i \left (d x +c \right )}-750 \,{\mathrm e}^{3 i \left (d x +c \right )}-1520 \,{\mathrm e}^{2 i \left (d x +c \right )}-195 \,{\mathrm e}^{i \left (d x +c \right )}-304\right )}{60 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}+\frac {13 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}-\frac {13 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}\) | \(145\) |
derivativedivides | \(\frac {a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+3 a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(146\) |
default | \(\frac {a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+3 a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(146\) |
parts | \(-\frac {a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {3 a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {3 a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(154\) |
norman | \(\frac {-\frac {51 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {193 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {31 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 d}-\frac {857 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 d}-\frac {1127 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 d}+\frac {481 a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 d}+\frac {65 a^{3} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {13 a^{3} \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}-\frac {13 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {13 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(224\) |
parallelrisch | \(-\frac {a^{3} \left (975 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (3 d x +3 c \right )-975 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (3 d x +3 c \right )+1950 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (d x +c \right )-1950 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (d x +c \right )+195 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (5 d x +5 c \right )-195 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (5 d x +5 c \right )-1600 \sin \left (d x +c \right )-304 \sin \left (5 d x +5 c \right )-390 \sin \left (4 d x +4 c \right )-1520 \sin \left (3 d x +3 c \right )-1500 \sin \left (2 d x +2 c \right )\right )}{120 d \left (\cos \left (5 d x +5 c \right )+5 \cos \left (3 d x +3 c \right )+10 \cos \left (d x +c \right )\right )}\) | \(225\) |
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Time = 0.25 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.09 \[ \int (a+a \cos (c+d x))^3 \sec ^6(c+d x) \, dx=\frac {195 \, a^{3} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 195 \, a^{3} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (304 \, a^{3} \cos \left (d x + c\right )^{4} + 195 \, a^{3} \cos \left (d x + c\right )^{3} + 152 \, a^{3} \cos \left (d x + c\right )^{2} + 90 \, a^{3} \cos \left (d x + c\right ) + 24 \, a^{3}\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \]
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Timed out. \[ \int (a+a \cos (c+d x))^3 \sec ^6(c+d x) \, dx=\text {Timed out} \]
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Time = 0.24 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.57 \[ \int (a+a \cos (c+d x))^3 \sec ^6(c+d x) \, dx=\frac {16 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} a^{3} + 240 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{3} - 45 \, a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 60 \, a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{240 \, d} \]
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Time = 0.35 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.21 \[ \int (a+a \cos (c+d x))^3 \sec ^6(c+d x) \, dx=\frac {195 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 195 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (195 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 910 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1664 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1330 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 765 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5}}}{120 \, d} \]
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Time = 18.35 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.49 \[ \int (a+a \cos (c+d x))^3 \sec ^6(c+d x) \, dx=\frac {13\,a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{4\,d}-\frac {\frac {13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4}-\frac {91\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{6}+\frac {416\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{15}-\frac {133\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{6}+\frac {51\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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