Optimal. Leaf size=58 \[ -\frac {\cos ^7(c+d x)}{9 d (a+a \sin (c+d x))^8}-\frac {\cos ^7(c+d x)}{63 a d (a+a \sin (c+d x))^7} \]
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Rubi [A]
time = 0.06, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2751, 2750}
\begin {gather*} -\frac {\cos ^7(c+d x)}{63 a d (a \sin (c+d x)+a)^7}-\frac {\cos ^7(c+d x)}{9 d (a \sin (c+d x)+a)^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 2750
Rule 2751
Rubi steps
\begin {align*} \int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=-\frac {\cos ^7(c+d x)}{9 d (a+a \sin (c+d x))^8}+\frac {\int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^7} \, dx}{9 a}\\ &=-\frac {\cos ^7(c+d x)}{9 d (a+a \sin (c+d x))^8}-\frac {\cos ^7(c+d x)}{63 a d (a+a \sin (c+d x))^7}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 36, normalized size = 0.62 \begin {gather*} -\frac {\cos ^7(c+d x) (8+\sin (c+d x))}{63 a^8 d (1+\sin (c+d x))^8} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(144\) vs.
\(2(54)=108\).
time = 0.22, size = 145, normalized size = 2.50
method | result | size |
risch | \(\frac {2 i \left (-105 i {\mathrm e}^{6 i \left (d x +c \right )}+63 \,{\mathrm e}^{7 i \left (d x +c \right )}+189 i {\mathrm e}^{4 i \left (d x +c \right )}-315 \,{\mathrm e}^{5 i \left (d x +c \right )}-27 i {\mathrm e}^{2 i \left (d x +c \right )}+189 \,{\mathrm e}^{3 i \left (d x +c \right )}-i-9 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{63 d \,a^{8} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{9}}\) | \(107\) |
derivativedivides | \(\frac {-\frac {1856}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {152}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {128}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {172}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {14}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {256}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}-\frac {272}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {992}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}}{d \,a^{8}}\) | \(145\) |
default | \(\frac {-\frac {1856}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {152}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {128}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {172}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {14}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {256}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}-\frac {272}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {992}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}}{d \,a^{8}}\) | \(145\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 375 vs.
\(2 (54) = 108\).
time = 0.32, size = 375, normalized size = 6.47 \begin {gather*} -\frac {2 \, {\left (\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {225 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {189 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {693 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {315 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {483 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {63 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {63 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + 8\right )}}{63 \, {\left (a^{8} + \frac {9 \, a^{8} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {36 \, a^{8} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {84 \, a^{8} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {126 \, a^{8} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {126 \, a^{8} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {84 \, a^{8} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {36 \, a^{8} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {9 \, a^{8} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {a^{8} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}\right )} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 239 vs.
\(2 (54) = 108\).
time = 0.34, size = 239, normalized size = 4.12 \begin {gather*} \frac {\cos \left (d x + c\right )^{5} - 4 \, \cos \left (d x + c\right )^{4} + 19 \, \cos \left (d x + c\right )^{3} + 52 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{3} + 24 \, \cos \left (d x + c\right )^{2} - 28 \, \cos \left (d x + c\right ) - 56\right )} \sin \left (d x + c\right ) - 28 \, \cos \left (d x + c\right ) - 56}{63 \, {\left (a^{8} d \cos \left (d x + c\right )^{5} + 5 \, a^{8} d \cos \left (d x + c\right )^{4} - 8 \, a^{8} d \cos \left (d x + c\right )^{3} - 20 \, a^{8} d \cos \left (d x + c\right )^{2} + 8 \, a^{8} d \cos \left (d x + c\right ) + 16 \, a^{8} d + {\left (a^{8} d \cos \left (d x + c\right )^{4} - 4 \, a^{8} d \cos \left (d x + c\right )^{3} - 12 \, a^{8} d \cos \left (d x + c\right )^{2} + 8 \, a^{8} d \cos \left (d x + c\right ) + 16 \, a^{8} d\right )} \sin \left (d x + c\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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. 125 vs.
\(2 (54) = 108\).
time = 7.32, size = 125, normalized size = 2.16 \begin {gather*} -\frac {2 \, {\left (63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 483 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 315 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 693 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 189 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 225 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8\right )}}{63 \, a^{8} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.63, size = 118, normalized size = 2.03 \begin {gather*} -\frac {\sqrt {2}\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {63\,\sin \left (c+d\,x\right )}{2}-\frac {257\,\cos \left (c+d\,x\right )}{8}-\frac {113\,\cos \left (2\,c+2\,d\,x\right )}{4}+\frac {37\,\cos \left (3\,c+3\,d\,x\right )}{8}+\frac {7\,\cos \left (4\,c+4\,d\,x\right )}{16}-\frac {63\,\sin \left (2\,c+2\,d\,x\right )}{8}-\frac {9\,\sin \left (3\,c+3\,d\,x\right )}{2}+\frac {9\,\sin \left (4\,c+4\,d\,x\right )}{16}+\frac {1013}{16}\right )}{1008\,a^8\,d\,{\cos \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d\,x}{2}\right )}^9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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