Optimal. Leaf size=152 \[ -\frac {a \cos ^9(c+d x)}{9 d}-\frac {a \cos ^8(c+d x)}{8 d}+\frac {4 a \cos ^7(c+d x)}{7 d}+\frac {2 a \cos ^6(c+d x)}{3 d}-\frac {6 a \cos ^5(c+d x)}{5 d}-\frac {3 a \cos ^4(c+d x)}{2 d}+\frac {4 a \cos ^3(c+d x)}{3 d}+\frac {2 a \cos ^2(c+d x)}{d}-\frac {a \cos (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.11, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3872, 2836, 12, 88} \[ -\frac {a \cos ^9(c+d x)}{9 d}-\frac {a \cos ^8(c+d x)}{8 d}+\frac {4 a \cos ^7(c+d x)}{7 d}+\frac {2 a \cos ^6(c+d x)}{3 d}-\frac {6 a \cos ^5(c+d x)}{5 d}-\frac {3 a \cos ^4(c+d x)}{2 d}+\frac {4 a \cos ^3(c+d x)}{3 d}+\frac {2 a \cos ^2(c+d x)}{d}-\frac {a \cos (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d} \]
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)) \sin ^9(c+d x) \, dx &=-\int (-a-a \cos (c+d x)) \sin ^8(c+d x) \tan (c+d x) \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {a (-a-x)^4 (-a+x)^5}{x} \, dx,x,-a \cos (c+d x)\right )}{a^9 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(-a-x)^4 (-a+x)^5}{x} \, dx,x,-a \cos (c+d x)\right )}{a^8 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^8-\frac {a^9}{x}+4 a^7 x-4 a^6 x^2-6 a^5 x^3+6 a^4 x^4+4 a^3 x^5-4 a^2 x^6-a x^7+x^8\right ) \, dx,x,-a \cos (c+d x)\right )}{a^8 d}\\ &=-\frac {a \cos (c+d x)}{d}+\frac {2 a \cos ^2(c+d x)}{d}+\frac {4 a \cos ^3(c+d x)}{3 d}-\frac {3 a \cos ^4(c+d x)}{2 d}-\frac {6 a \cos ^5(c+d x)}{5 d}+\frac {2 a \cos ^6(c+d x)}{3 d}+\frac {4 a \cos ^7(c+d x)}{7 d}-\frac {a \cos ^8(c+d x)}{8 d}-\frac {a \cos ^9(c+d x)}{9 d}-\frac {a \log (\cos (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 106, normalized size = 0.70 \[ -\frac {a \left (10080 \cos ^8(c+d x)-53760 \cos ^6(c+d x)+120960 \cos ^4(c+d x)-161280 \cos ^2(c+d x)+39690 \cos (c+d x)-8820 \cos (3 (c+d x))+2268 \cos (5 (c+d x))-405 \cos (7 (c+d x))+35 \cos (9 (c+d x))+80640 \log (\cos (c+d x))\right )}{80640 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.71, size = 115, normalized size = 0.76 \[ -\frac {280 \, a \cos \left (d x + c\right )^{9} + 315 \, a \cos \left (d x + c\right )^{8} - 1440 \, a \cos \left (d x + c\right )^{7} - 1680 \, a \cos \left (d x + c\right )^{6} + 3024 \, a \cos \left (d x + c\right )^{5} + 3780 \, a \cos \left (d x + c\right )^{4} - 3360 \, a \cos \left (d x + c\right )^{3} - 5040 \, a \cos \left (d x + c\right )^{2} + 2520 \, a \cos \left (d x + c\right ) + 2520 \, a \log \left (-\cos \left (d x + c\right )\right )}{2520 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 293, normalized size = 1.93 \[ \frac {2520 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 2520 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {9177 \, a - \frac {87633 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {375732 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {953988 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1594782 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {1336734 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {781956 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {302004 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {69201 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {7129 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{9}}}{2520 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.67, size = 163, normalized size = 1.07 \[ -\frac {128 a \cos \left (d x +c \right )}{315 d}-\frac {\left (\sin ^{8}\left (d x +c \right )\right ) \cos \left (d x +c \right ) a}{9 d}-\frac {8 a \cos \left (d x +c \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{63 d}-\frac {16 a \cos \left (d x +c \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{105 d}-\frac {64 a \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{315 d}-\frac {a \left (\sin ^{8}\left (d x +c \right )\right )}{8 d}-\frac {a \left (\sin ^{6}\left (d x +c \right )\right )}{6 d}-\frac {a \left (\sin ^{4}\left (d x +c \right )\right )}{4 d}-\frac {a \left (\sin ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a \ln \left (\cos \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 113, normalized size = 0.74 \[ -\frac {280 \, a \cos \left (d x + c\right )^{9} + 315 \, a \cos \left (d x + c\right )^{8} - 1440 \, a \cos \left (d x + c\right )^{7} - 1680 \, a \cos \left (d x + c\right )^{6} + 3024 \, a \cos \left (d x + c\right )^{5} + 3780 \, a \cos \left (d x + c\right )^{4} - 3360 \, a \cos \left (d x + c\right )^{3} - 5040 \, a \cos \left (d x + c\right )^{2} + 2520 \, a \cos \left (d x + c\right ) + 2520 \, a \log \left (\cos \left (d x + c\right )\right )}{2520 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 111, normalized size = 0.73 \[ -\frac {a\,\cos \left (c+d\,x\right )-2\,a\,{\cos \left (c+d\,x\right )}^2-\frac {4\,a\,{\cos \left (c+d\,x\right )}^3}{3}+\frac {3\,a\,{\cos \left (c+d\,x\right )}^4}{2}+\frac {6\,a\,{\cos \left (c+d\,x\right )}^5}{5}-\frac {2\,a\,{\cos \left (c+d\,x\right )}^6}{3}-\frac {4\,a\,{\cos \left (c+d\,x\right )}^7}{7}+\frac {a\,{\cos \left (c+d\,x\right )}^8}{8}+\frac {a\,{\cos \left (c+d\,x\right )}^9}{9}+a\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]
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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