Optimal. Leaf size=183 \[ -\frac {a^2 \cos ^9(c+d x)}{9 d}-\frac {a^2 \cos ^8(c+d x)}{4 d}+\frac {3 a^2 \cos ^7(c+d x)}{7 d}+\frac {4 a^2 \cos ^6(c+d x)}{3 d}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}-\frac {3 a^2 \cos ^4(c+d x)}{d}-\frac {2 a^2 \cos ^3(c+d x)}{3 d}+\frac {4 a^2 \cos ^2(c+d x)}{d}+\frac {3 a^2 \cos (c+d x)}{d}+\frac {a^2 \sec (c+d x)}{d}-\frac {2 a^2 \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.19, antiderivative size = 183, 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} \[ -\frac {a^2 \cos ^9(c+d x)}{9 d}-\frac {a^2 \cos ^8(c+d x)}{4 d}+\frac {3 a^2 \cos ^7(c+d x)}{7 d}+\frac {4 a^2 \cos ^6(c+d x)}{3 d}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}-\frac {3 a^2 \cos ^4(c+d x)}{d}-\frac {2 a^2 \cos ^3(c+d x)}{3 d}+\frac {4 a^2 \cos ^2(c+d x)}{d}+\frac {3 a^2 \cos (c+d x)}{d}+\frac {a^2 \sec (c+d x)}{d}-\frac {2 a^2 \log (\cos (c+d x))}{d} \]
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))^2 \sin ^9(c+d x) \, dx &=\int (-a-a \cos (c+d x))^2 \sin ^7(c+d x) \tan ^2(c+d x) \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {a^2 (-a-x)^4 (-a+x)^6}{x^2} \, dx,x,-a \cos (c+d x)\right )}{a^9 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(-a-x)^4 (-a+x)^6}{x^2} \, dx,x,-a \cos (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-3 a^8+\frac {a^{10}}{x^2}-\frac {2 a^9}{x}+8 a^7 x+2 a^6 x^2-12 a^5 x^3+2 a^4 x^4+8 a^3 x^5-3 a^2 x^6-2 a x^7+x^8\right ) \, dx,x,-a \cos (c+d x)\right )}{a^7 d}\\ &=\frac {3 a^2 \cos (c+d x)}{d}+\frac {4 a^2 \cos ^2(c+d x)}{d}-\frac {2 a^2 \cos ^3(c+d x)}{3 d}-\frac {3 a^2 \cos ^4(c+d x)}{d}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}+\frac {4 a^2 \cos ^6(c+d x)}{3 d}+\frac {3 a^2 \cos ^7(c+d x)}{7 d}-\frac {a^2 \cos ^8(c+d x)}{4 d}-\frac {a^2 \cos ^9(c+d x)}{9 d}-\frac {2 a^2 \log (\cos (c+d x))}{d}+\frac {a^2 \sec (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.94, size = 127, normalized size = 0.69 \[ -\frac {a^2 \sec (c+d x) (-361620 \cos (2 (c+d x))-134820 \cos (3 (c+d x))+29232 \cos (4 (c+d x))+24780 \cos (5 (c+d x))-1458 \cos (6 (c+d x))-3885 \cos (7 (c+d x))-380 \cos (8 (c+d x))+315 \cos (9 (c+d x))+70 \cos (10 (c+d x))+210 \cos (c+d x) (3072 \log (\cos (c+d x))+205)-714420)}{322560 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 167, normalized size = 0.91 \[ -\frac {17920 \, a^{2} \cos \left (d x + c\right )^{10} + 40320 \, a^{2} \cos \left (d x + c\right )^{9} - 69120 \, a^{2} \cos \left (d x + c\right )^{8} - 215040 \, a^{2} \cos \left (d x + c\right )^{7} + 64512 \, a^{2} \cos \left (d x + c\right )^{6} + 483840 \, a^{2} \cos \left (d x + c\right )^{5} + 107520 \, a^{2} \cos \left (d x + c\right )^{4} - 645120 \, a^{2} \cos \left (d x + c\right )^{3} - 483840 \, a^{2} \cos \left (d x + c\right )^{2} + 322560 \, a^{2} \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) + 197295 \, a^{2} \cos \left (d x + c\right ) - 161280 \, a^{2}}{161280 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.93, size = 370, normalized size = 2.02 \[ \frac {2520 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 2520 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {2520 \, {\left (2 \, a^{2} + \frac {a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}{\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1} + \frac {1457 \, a^{2} - \frac {20673 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {123012 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {421428 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {949662 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {1009134 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {666036 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {276804 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {66681 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {7129 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{9}}}{1260 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.74, size = 206, normalized size = 1.13 \[ \frac {1024 a^{2} \cos \left (d x +c \right )}{315 d}+\frac {8 a^{2} \left (\sin ^{8}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{9 d}+\frac {64 a^{2} \cos \left (d x +c \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{63 d}+\frac {128 a^{2} \cos \left (d x +c \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{105 d}+\frac {512 a^{2} \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{315 d}-\frac {a^{2} \left (\sin ^{8}\left (d x +c \right )\right )}{4 d}-\frac {a^{2} \left (\sin ^{6}\left (d x +c \right )\right )}{3 d}-\frac {a^{2} \left (\sin ^{4}\left (d x +c \right )\right )}{2 d}-\frac {a^{2} \left (\sin ^{2}\left (d x +c \right )\right )}{d}-\frac {2 a^{2} \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {a^{2} \left (\sin ^{10}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 146, normalized size = 0.80 \[ -\frac {140 \, a^{2} \cos \left (d x + c\right )^{9} + 315 \, a^{2} \cos \left (d x + c\right )^{8} - 540 \, a^{2} \cos \left (d x + c\right )^{7} - 1680 \, a^{2} \cos \left (d x + c\right )^{6} + 504 \, a^{2} \cos \left (d x + c\right )^{5} + 3780 \, a^{2} \cos \left (d x + c\right )^{4} + 840 \, a^{2} \cos \left (d x + c\right )^{3} - 5040 \, a^{2} \cos \left (d x + c\right )^{2} - 3780 \, a^{2} \cos \left (d x + c\right ) + 2520 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) - \frac {1260 \, a^{2}}{\cos \left (d x + c\right )}}{1260 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.99, size = 146, normalized size = 0.80 \[ -\frac {\frac {2\,a^2\,{\cos \left (c+d\,x\right )}^3}{3}-\frac {a^2}{\cos \left (c+d\,x\right )}-4\,a^2\,{\cos \left (c+d\,x\right )}^2-3\,a^2\,\cos \left (c+d\,x\right )+3\,a^2\,{\cos \left (c+d\,x\right )}^4+\frac {2\,a^2\,{\cos \left (c+d\,x\right )}^5}{5}-\frac {4\,a^2\,{\cos \left (c+d\,x\right )}^6}{3}-\frac {3\,a^2\,{\cos \left (c+d\,x\right )}^7}{7}+\frac {a^2\,{\cos \left (c+d\,x\right )}^8}{4}+\frac {a^2\,{\cos \left (c+d\,x\right )}^9}{9}+2\,a^2\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]
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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