Optimal. Leaf size=114 \[ \frac {4 (a-a \cos (c+d x))^5}{5 a^7 d}-\frac {2 (a-a \cos (c+d x))^6}{a^8 d}+\frac {13 (a-a \cos (c+d x))^7}{7 a^9 d}-\frac {3 (a-a \cos (c+d x))^8}{4 a^{10} d}+\frac {(a-a \cos (c+d x))^9}{9 a^{11} d} \]
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Rubi [A]
time = 0.13, antiderivative size = 114, 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-a \cos (c+d x))^9}{9 a^{11} d}-\frac {3 (a-a \cos (c+d x))^8}{4 a^{10} d}+\frac {13 (a-a \cos (c+d x))^7}{7 a^9 d}-\frac {2 (a-a \cos (c+d x))^6}{a^8 d}+\frac {4 (a-a \cos (c+d x))^5}{5 a^7 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 90
Rule 2915
Rule 3957
Rubi steps
\begin {align*} \int \frac {\sin ^9(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=\int \frac {\cos ^2(c+d x) \sin ^9(c+d x)}{(-a-a \cos (c+d x))^2} \, dx\\ &=\frac {\text {Subst}\left (\int \frac {(-a-x)^4 x^2 (-a+x)^2}{a^2} \, dx,x,-a \cos (c+d x)\right )}{a^9 d}\\ &=\frac {\text {Subst}\left (\int (-a-x)^4 x^2 (-a+x)^2 \, dx,x,-a \cos (c+d x)\right )}{a^{11} d}\\ &=\frac {\text {Subst}\left (\int \left (4 a^4 (-a-x)^4+12 a^3 (-a-x)^5+13 a^2 (-a-x)^6+6 a (-a-x)^7+(-a-x)^8\right ) \, dx,x,-a \cos (c+d x)\right )}{a^{11} d}\\ &=\frac {4 (a-a \cos (c+d x))^5}{5 a^7 d}-\frac {2 (a-a \cos (c+d x))^6}{a^8 d}+\frac {13 (a-a \cos (c+d x))^7}{7 a^9 d}-\frac {3 (a-a \cos (c+d x))^8}{4 a^{10} d}+\frac {(a-a \cos (c+d x))^9}{9 a^{11} d}\\ \end {align*}
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Mathematica [A]
time = 2.28, size = 62, normalized size = 0.54 \begin {gather*} \frac {2 (992+1615 \cos (c+d x)+970 \cos (2 (c+d x))+385 \cos (3 (c+d x))+70 \cos (4 (c+d x))) \sin ^{10}\left (\frac {1}{2} (c+d x)\right )}{315 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 79, normalized size = 0.69
method | result | size |
derivativedivides | \(\frac {\frac {1}{2 \sec \left (d x +c \right )^{4}}+\frac {1}{4 \sec \left (d x +c \right )^{8}}-\frac {1}{3 \sec \left (d x +c \right )^{3}}-\frac {1}{9 \sec \left (d x +c \right )^{9}}-\frac {2}{3 \sec \left (d x +c \right )^{6}}+\frac {1}{7 \sec \left (d x +c \right )^{7}}+\frac {1}{5 \sec \left (d x +c \right )^{5}}}{d \,a^{2}}\) | \(79\) |
default | \(\frac {\frac {1}{2 \sec \left (d x +c \right )^{4}}+\frac {1}{4 \sec \left (d x +c \right )^{8}}-\frac {1}{3 \sec \left (d x +c \right )^{3}}-\frac {1}{9 \sec \left (d x +c \right )^{9}}-\frac {2}{3 \sec \left (d x +c \right )^{6}}+\frac {1}{7 \sec \left (d x +c \right )^{7}}+\frac {1}{5 \sec \left (d x +c \right )^{5}}}{d \,a^{2}}\) | \(79\) |
norman | \(\frac {-\frac {64}{315 a d}-\frac {128 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {64 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 a d}-\frac {256 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 a d}-\frac {256 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 a d}-\frac {128 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9} a}\) | \(124\) |
risch | \(-\frac {13 \cos \left (d x +c \right )}{128 a^{2} d}-\frac {\cos \left (9 d x +9 c \right )}{2304 d \,a^{2}}+\frac {\cos \left (8 d x +8 c \right )}{512 d \,a^{2}}-\frac {3 \cos \left (7 d x +7 c \right )}{1792 d \,a^{2}}-\frac {\cos \left (6 d x +6 c \right )}{192 d \,a^{2}}+\frac {\cos \left (5 d x +5 c \right )}{80 d \,a^{2}}-\frac {\cos \left (4 d x +4 c \right )}{128 d \,a^{2}}-\frac {\cos \left (3 d x +3 c \right )}{96 d \,a^{2}}+\frac {3 \cos \left (2 d x +2 c \right )}{64 d \,a^{2}}\) | \(152\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 79, normalized size = 0.69 \begin {gather*} -\frac {140 \, \cos \left (d x + c\right )^{9} - 315 \, \cos \left (d x + c\right )^{8} - 180 \, \cos \left (d x + c\right )^{7} + 840 \, \cos \left (d x + c\right )^{6} - 252 \, \cos \left (d x + c\right )^{5} - 630 \, \cos \left (d x + c\right )^{4} + 420 \, \cos \left (d x + c\right )^{3}}{1260 \, a^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.35, size = 79, normalized size = 0.69 \begin {gather*} -\frac {140 \, \cos \left (d x + c\right )^{9} - 315 \, \cos \left (d x + c\right )^{8} - 180 \, \cos \left (d x + c\right )^{7} + 840 \, \cos \left (d x + c\right )^{6} - 252 \, \cos \left (d x + c\right )^{5} - 630 \, \cos \left (d x + c\right )^{4} + 420 \, \cos \left (d x + c\right )^{3}}{1260 \, a^{2} d} \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 [A]
time = 0.50, size = 141, normalized size = 1.24 \begin {gather*} -\frac {64 \, {\left (\frac {9 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {36 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {84 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {126 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {210 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 1\right )}}{315 \, a^{2} d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.91, size = 96, normalized size = 0.84 \begin {gather*} \frac {\frac {{\cos \left (c+d\,x\right )}^4}{2\,a^2}-\frac {{\cos \left (c+d\,x\right )}^3}{3\,a^2}+\frac {{\cos \left (c+d\,x\right )}^5}{5\,a^2}-\frac {2\,{\cos \left (c+d\,x\right )}^6}{3\,a^2}+\frac {{\cos \left (c+d\,x\right )}^7}{7\,a^2}+\frac {{\cos \left (c+d\,x\right )}^8}{4\,a^2}-\frac {{\cos \left (c+d\,x\right )}^9}{9\,a^2}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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