Optimal. Leaf size=147 \[ -\frac {a^4 \sin ^7(c+d x)}{7 d}+\frac {9 a^4 \sin ^5(c+d x)}{5 d}-\frac {16 a^4 \sin ^3(c+d x)}{3 d}+\frac {8 a^4 \sin (c+d x)}{d}+\frac {2 a^4 \sin (c+d x) \cos ^5(c+d x)}{3 d}+\frac {11 a^4 \sin (c+d x) \cos ^3(c+d x)}{6 d}+\frac {11 a^4 \sin (c+d x) \cos (c+d x)}{4 d}+\frac {11 a^4 x}{4} \]
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Rubi [A] time = 0.15, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3791, 2633, 2635, 8} \[ -\frac {a^4 \sin ^7(c+d x)}{7 d}+\frac {9 a^4 \sin ^5(c+d x)}{5 d}-\frac {16 a^4 \sin ^3(c+d x)}{3 d}+\frac {8 a^4 \sin (c+d x)}{d}+\frac {2 a^4 \sin (c+d x) \cos ^5(c+d x)}{3 d}+\frac {11 a^4 \sin (c+d x) \cos ^3(c+d x)}{6 d}+\frac {11 a^4 \sin (c+d x) \cos (c+d x)}{4 d}+\frac {11 a^4 x}{4} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 3791
Rubi steps
\begin {align*} \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 \, dx &=\int \left (a^4 \cos ^3(c+d x)+4 a^4 \cos ^4(c+d x)+6 a^4 \cos ^5(c+d x)+4 a^4 \cos ^6(c+d x)+a^4 \cos ^7(c+d x)\right ) \, dx\\ &=a^4 \int \cos ^3(c+d x) \, dx+a^4 \int \cos ^7(c+d x) \, dx+\left (4 a^4\right ) \int \cos ^4(c+d x) \, dx+\left (4 a^4\right ) \int \cos ^6(c+d x) \, dx+\left (6 a^4\right ) \int \cos ^5(c+d x) \, dx\\ &=\frac {a^4 \cos ^3(c+d x) \sin (c+d x)}{d}+\frac {2 a^4 \cos ^5(c+d x) \sin (c+d x)}{3 d}+\left (3 a^4\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{3} \left (10 a^4\right ) \int \cos ^4(c+d x) \, dx-\frac {a^4 \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}-\frac {a^4 \operatorname {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (c+d x)\right )}{d}-\frac {\left (6 a^4\right ) \operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac {8 a^4 \sin (c+d x)}{d}+\frac {3 a^4 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {11 a^4 \cos ^3(c+d x) \sin (c+d x)}{6 d}+\frac {2 a^4 \cos ^5(c+d x) \sin (c+d x)}{3 d}-\frac {16 a^4 \sin ^3(c+d x)}{3 d}+\frac {9 a^4 \sin ^5(c+d x)}{5 d}-\frac {a^4 \sin ^7(c+d x)}{7 d}+\frac {1}{2} \left (3 a^4\right ) \int 1 \, dx+\frac {1}{2} \left (5 a^4\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac {3 a^4 x}{2}+\frac {8 a^4 \sin (c+d x)}{d}+\frac {11 a^4 \cos (c+d x) \sin (c+d x)}{4 d}+\frac {11 a^4 \cos ^3(c+d x) \sin (c+d x)}{6 d}+\frac {2 a^4 \cos ^5(c+d x) \sin (c+d x)}{3 d}-\frac {16 a^4 \sin ^3(c+d x)}{3 d}+\frac {9 a^4 \sin ^5(c+d x)}{5 d}-\frac {a^4 \sin ^7(c+d x)}{7 d}+\frac {1}{4} \left (5 a^4\right ) \int 1 \, dx\\ &=\frac {11 a^4 x}{4}+\frac {8 a^4 \sin (c+d x)}{d}+\frac {11 a^4 \cos (c+d x) \sin (c+d x)}{4 d}+\frac {11 a^4 \cos ^3(c+d x) \sin (c+d x)}{6 d}+\frac {2 a^4 \cos ^5(c+d x) \sin (c+d x)}{3 d}-\frac {16 a^4 \sin ^3(c+d x)}{3 d}+\frac {9 a^4 \sin ^5(c+d x)}{5 d}-\frac {a^4 \sin ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 83, normalized size = 0.56 \[ \frac {a^4 (33915 \sin (c+d x)+13020 \sin (2 (c+d x))+5495 \sin (3 (c+d x))+2100 \sin (4 (c+d x))+651 \sin (5 (c+d x))+140 \sin (6 (c+d x))+15 \sin (7 (c+d x))+18480 d x)}{6720 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 102, normalized size = 0.69 \[ \frac {1155 \, a^{4} d x + {\left (60 \, a^{4} \cos \left (d x + c\right )^{6} + 280 \, a^{4} \cos \left (d x + c\right )^{5} + 576 \, a^{4} \cos \left (d x + c\right )^{4} + 770 \, a^{4} \cos \left (d x + c\right )^{3} + 908 \, a^{4} \cos \left (d x + c\right )^{2} + 1155 \, a^{4} \cos \left (d x + c\right ) + 1816 \, a^{4}\right )} \sin \left (d x + c\right )}{420 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 144, normalized size = 0.98 \[ \frac {1155 \, {\left (d x + c\right )} a^{4} + \frac {2 \, {\left (1155 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 7700 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 21791 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 33792 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 31521 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 14700 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5565 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{7}}}{420 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.58, size = 185, normalized size = 1.26 \[ \frac {\frac {a^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+4 a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {6 a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+4 a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.72, size = 187, normalized size = 1.27 \[ -\frac {48 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} a^{4} - 672 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a^{4} + 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} + 560 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{4} - 210 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4}}{1680 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.64, size = 137, normalized size = 0.93 \[ \frac {11\,a^4\,x}{4}+\frac {\frac {11\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{2}+\frac {110\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{3}+\frac {3113\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{30}+\frac {5632\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{35}+\frac {1501\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{10}+70\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {53\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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