Optimal. Leaf size=203 \[ -\frac {a^3 \cos ^9(c+d x)}{9 d}-\frac {3 a^3 \cos ^8(c+d x)}{8 d}+\frac {a^3 \cos ^7(c+d x)}{7 d}+\frac {11 a^3 \cos ^6(c+d x)}{6 d}+\frac {6 a^3 \cos ^5(c+d x)}{5 d}-\frac {7 a^3 \cos ^4(c+d x)}{2 d}-\frac {14 a^3 \cos ^3(c+d x)}{3 d}+\frac {3 a^3 \cos ^2(c+d x)}{d}+\frac {11 a^3 \cos (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.20, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3872, 2836, 12, 88} \[ -\frac {a^3 \cos ^9(c+d x)}{9 d}-\frac {3 a^3 \cos ^8(c+d x)}{8 d}+\frac {a^3 \cos ^7(c+d x)}{7 d}+\frac {11 a^3 \cos ^6(c+d x)}{6 d}+\frac {6 a^3 \cos ^5(c+d x)}{5 d}-\frac {7 a^3 \cos ^4(c+d x)}{2 d}-\frac {14 a^3 \cos ^3(c+d x)}{3 d}+\frac {3 a^3 \cos ^2(c+d x)}{d}+\frac {11 a^3 \cos (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rule 3872
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^3 \sin ^9(c+d x) \, dx &=-\int (-a-a \cos (c+d x))^3 \sin ^6(c+d x) \tan ^3(c+d x) \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {a^3 (-a-x)^4 (-a+x)^7}{x^3} \, dx,x,-a \cos (c+d x)\right )}{a^9 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(-a-x)^4 (-a+x)^7}{x^3} \, dx,x,-a \cos (c+d x)\right )}{a^6 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-11 a^8-\frac {a^{11}}{x^3}+\frac {3 a^{10}}{x^2}+\frac {a^9}{x}+6 a^7 x+14 a^6 x^2-14 a^5 x^3-6 a^4 x^4+11 a^3 x^5-a^2 x^6-3 a x^7+x^8\right ) \, dx,x,-a \cos (c+d x)\right )}{a^6 d}\\ &=\frac {11 a^3 \cos (c+d x)}{d}+\frac {3 a^3 \cos ^2(c+d x)}{d}-\frac {14 a^3 \cos ^3(c+d x)}{3 d}-\frac {7 a^3 \cos ^4(c+d x)}{2 d}+\frac {6 a^3 \cos ^5(c+d x)}{5 d}+\frac {11 a^3 \cos ^6(c+d x)}{6 d}+\frac {a^3 \cos ^7(c+d x)}{7 d}-\frac {3 a^3 \cos ^8(c+d x)}{8 d}-\frac {a^3 \cos ^9(c+d x)}{9 d}+\frac {a^3 \log (\cos (c+d x))}{d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 1.79, size = 148, normalized size = 0.73 \[ \frac {a^3 \sec ^2(c+d x) (11624760 \cos (c+d x)+2188872 \cos (3 (c+d x))+41160 \cos (4 (c+d x))-204156 \cos (5 (c+d x))-35805 \cos (6 (c+d x))+22972 \cos (7 (c+d x))+9030 \cos (8 (c+d x))-820 \cos (9 (c+d x))-945 \cos (10 (c+d x))-140 \cos (11 (c+d x))+645120 \log (\cos (c+d x))+210 \cos (2 (c+d x)) (3072 \log (\cos (c+d x))-413)+471450)}{1290240 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 182, normalized size = 0.90 \[ -\frac {35840 \, a^{3} \cos \left (d x + c\right )^{11} + 120960 \, a^{3} \cos \left (d x + c\right )^{10} - 46080 \, a^{3} \cos \left (d x + c\right )^{9} - 591360 \, a^{3} \cos \left (d x + c\right )^{8} - 387072 \, a^{3} \cos \left (d x + c\right )^{7} + 1128960 \, a^{3} \cos \left (d x + c\right )^{6} + 1505280 \, a^{3} \cos \left (d x + c\right )^{5} - 967680 \, a^{3} \cos \left (d x + c\right )^{4} - 3548160 \, a^{3} \cos \left (d x + c\right )^{3} - 322560 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) + 212205 \, a^{3} \cos \left (d x + c\right )^{2} - 967680 \, a^{3} \cos \left (d x + c\right ) - 161280 \, a^{3}}{322560 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.55, size = 396, normalized size = 1.95 \[ -\frac {2520 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 2520 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) - \frac {1260 \, {\left (9 \, a^{3} + \frac {2 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {3 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}} + \frac {45257 \, a^{3} - \frac {392193 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {1467972 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {3001908 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3232782 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {2359854 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {1190196 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {397764 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {79281 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {7129 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{9}}}{2520 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.77, size = 230, normalized size = 1.13 \[ \frac {3328 a^{3} \cos \left (d x +c \right )}{315 d}+\frac {26 a^{3} \left (\sin ^{8}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{9 d}+\frac {208 a^{3} \cos \left (d x +c \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{63 d}+\frac {416 a^{3} \cos \left (d x +c \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{105 d}+\frac {1664 a^{3} \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{315 d}+\frac {a^{3} \left (\sin ^{8}\left (d x +c \right )\right )}{8 d}+\frac {a^{3} \left (\sin ^{6}\left (d x +c \right )\right )}{6 d}+\frac {a^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{4 d}+\frac {a^{3} \left (\sin ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a^{3} \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {3 a^{3} \left (\sin ^{10}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )}+\frac {a^{3} \left (\sin ^{10}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 158, normalized size = 0.78 \[ -\frac {280 \, a^{3} \cos \left (d x + c\right )^{9} + 945 \, a^{3} \cos \left (d x + c\right )^{8} - 360 \, a^{3} \cos \left (d x + c\right )^{7} - 4620 \, a^{3} \cos \left (d x + c\right )^{6} - 3024 \, a^{3} \cos \left (d x + c\right )^{5} + 8820 \, a^{3} \cos \left (d x + c\right )^{4} + 11760 \, a^{3} \cos \left (d x + c\right )^{3} - 7560 \, a^{3} \cos \left (d x + c\right )^{2} - 27720 \, a^{3} \cos \left (d x + c\right ) - 2520 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) - \frac {1260 \, {\left (6 \, a^{3} \cos \left (d x + c\right ) + a^{3}\right )}}{\cos \left (d x + c\right )^{2}}}{2520 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.96, size = 157, normalized size = 0.77 \[ \frac {\frac {3\,a^3\,\cos \left (c+d\,x\right )+\frac {a^3}{2}}{{\cos \left (c+d\,x\right )}^2}+11\,a^3\,\cos \left (c+d\,x\right )+3\,a^3\,{\cos \left (c+d\,x\right )}^2-\frac {14\,a^3\,{\cos \left (c+d\,x\right )}^3}{3}-\frac {7\,a^3\,{\cos \left (c+d\,x\right )}^4}{2}+\frac {6\,a^3\,{\cos \left (c+d\,x\right )}^5}{5}+\frac {11\,a^3\,{\cos \left (c+d\,x\right )}^6}{6}+\frac {a^3\,{\cos \left (c+d\,x\right )}^7}{7}-\frac {3\,a^3\,{\cos \left (c+d\,x\right )}^8}{8}-\frac {a^3\,{\cos \left (c+d\,x\right )}^9}{9}+a^3\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]
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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