Optimal. Leaf size=183 \[ \frac {3 a^2 \cos (c+d x)}{d}+\frac {4 a^2 \cos ^2(c+d x)}{d}-\frac {2 a^2 \cos ^3(c+d x)}{3 d}-\frac {3 a^2 \cos ^4(c+d x)}{d}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}+\frac {4 a^2 \cos ^6(c+d x)}{3 d}+\frac {3 a^2 \cos ^7(c+d x)}{7 d}-\frac {a^2 \cos ^8(c+d x)}{4 d}-\frac {a^2 \cos ^9(c+d x)}{9 d}-\frac {2 a^2 \log (\cos (c+d x))}{d}+\frac {a^2 \sec (c+d x)}{d} \]
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Rubi [A]
time = 0.14, 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 = {3957, 2915, 12,
90} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 90
Rule 2915
Rule 3957
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 {\text {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 {\text {Subst}\left (\int \frac {(-a-x)^4 (-a+x)^6}{x^2} \, dx,x,-a \cos (c+d x)\right )}{a^7 d}\\ &=\frac {\text {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 = 0.58, size = 127, normalized size = 0.69 \begin {gather*} -\frac {a^2 (-714420-361620 \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) (205+3072 \log (\cos (c+d x)))) \sec (c+d x)}{322560 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 181, normalized size = 0.99
method | result | size |
derivativedivides | \(\frac {a^{2} \left (\frac {\sin ^{10}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {128}{35}+\sin ^{8}\left (d x +c \right )+\frac {8 \left (\sin ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\sin ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\sin ^{2}\left (d x +c \right )\right )}{35}\right ) \cos \left (d x +c \right )\right )+2 a^{2} \left (-\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{8}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{2} \left (\frac {128}{35}+\sin ^{8}\left (d x +c \right )+\frac {8 \left (\sin ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\sin ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\sin ^{2}\left (d x +c \right )\right )}{35}\right ) \cos \left (d x +c \right )}{9}}{d}\) | \(181\) |
default | \(\frac {a^{2} \left (\frac {\sin ^{10}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {128}{35}+\sin ^{8}\left (d x +c \right )+\frac {8 \left (\sin ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\sin ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\sin ^{2}\left (d x +c \right )\right )}{35}\right ) \cos \left (d x +c \right )\right )+2 a^{2} \left (-\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{8}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{2} \left (\frac {128}{35}+\sin ^{8}\left (d x +c \right )+\frac {8 \left (\sin ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\sin ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\sin ^{2}\left (d x +c \right )\right )}{35}\right ) \cos \left (d x +c \right )}{9}}{d}\) | \(181\) |
risch | \(\frac {65 a^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{128 d}+2 i a^{2} x +\frac {4 i a^{2} c}{d}-\frac {a^{2} \cos \left (9 d x +9 c \right )}{2304 d}-\frac {2 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}+\frac {2 a^{2} {\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {311 a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{256 d}+\frac {311 a^{2} {\mathrm e}^{i \left (d x +c \right )}}{256 d}+\frac {65 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{128 d}-\frac {a^{2} \cos \left (8 d x +8 c \right )}{512 d}+\frac {5 a^{2} \cos \left (7 d x +7 c \right )}{1792 d}+\frac {5 a^{2} \cos \left (6 d x +6 c \right )}{192 d}+\frac {a^{2} \cos \left (5 d x +5 c \right )}{160 d}-\frac {23 a^{2} \cos \left (4 d x +4 c \right )}{128 d}-\frac {3 a^{2} \cos \left (3 d x +3 c \right )}{16 d}\) | \(256\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 146, normalized size = 0.80 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.45, size = 167, normalized size = 0.91 \begin {gather*} -\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 370 vs.
\(2 (171) = 342\).
time = 0.73, size = 370, normalized size = 2.02 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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