Optimal. Leaf size=118 \[ -\frac {\cos ^5(c+d x)}{11 d (a+a \sin (c+d x))^8}-\frac {\cos ^5(c+d x)}{33 a d (a+a \sin (c+d x))^7}-\frac {2 \cos ^5(c+d x)}{231 a^2 d (a+a \sin (c+d x))^6}-\frac {2 \cos ^5(c+d x)}{1155 a^3 d (a+a \sin (c+d x))^5} \]
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Rubi [A]
time = 0.12, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps
used = 4, 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 {2 \cos ^5(c+d x)}{1155 a^3 d (a \sin (c+d x)+a)^5}-\frac {2 \cos ^5(c+d x)}{231 a^2 d (a \sin (c+d x)+a)^6}-\frac {\cos ^5(c+d x)}{33 a d (a \sin (c+d x)+a)^7}-\frac {\cos ^5(c+d x)}{11 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 ^4(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=-\frac {\cos ^5(c+d x)}{11 d (a+a \sin (c+d x))^8}+\frac {3 \int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^7} \, dx}{11 a}\\ &=-\frac {\cos ^5(c+d x)}{11 d (a+a \sin (c+d x))^8}-\frac {\cos ^5(c+d x)}{33 a d (a+a \sin (c+d x))^7}+\frac {2 \int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^6} \, dx}{33 a^2}\\ &=-\frac {\cos ^5(c+d x)}{11 d (a+a \sin (c+d x))^8}-\frac {\cos ^5(c+d x)}{33 a d (a+a \sin (c+d x))^7}-\frac {2 \cos ^5(c+d x)}{231 a^2 d (a+a \sin (c+d x))^6}+\frac {2 \int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^5} \, dx}{231 a^3}\\ &=-\frac {\cos ^5(c+d x)}{11 d (a+a \sin (c+d x))^8}-\frac {\cos ^5(c+d x)}{33 a d (a+a \sin (c+d x))^7}-\frac {2 \cos ^5(c+d x)}{231 a^2 d (a+a \sin (c+d x))^6}-\frac {2 \cos ^5(c+d x)}{1155 a^3 d (a+a \sin (c+d x))^5}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 58, normalized size = 0.49 \begin {gather*} -\frac {\cos ^5(c+d x) \left (152+61 \sin (c+d x)+16 \sin ^2(c+d x)+2 \sin ^3(c+d x)\right )}{1155 a^8 d (1+\sin (c+d x))^8} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 175, normalized size = 1.48
method | result | size |
risch | \(\frac {4 i \left (-2079 i {\mathrm e}^{6 i \left (d x +c \right )}+1155 \,{\mathrm e}^{7 i \left (d x +c \right )}+825 i {\mathrm e}^{4 i \left (d x +c \right )}-2541 \,{\mathrm e}^{5 i \left (d x +c \right )}+55 i {\mathrm e}^{2 i \left (d x +c \right )}+165 \,{\mathrm e}^{3 i \left (d x +c \right )}-i-11 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{1155 d \,a^{8} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{11}}\) | \(107\) |
derivativedivides | \(\frac {-\frac {4752}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {14}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {256}{11 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{11}}+\frac {584}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {128}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{10}}-\frac {60}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {576}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}+\frac {176}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {1024}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}-\frac {1864}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{d \,a^{8}}\) | \(175\) |
default | \(\frac {-\frac {4752}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {14}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {256}{11 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{11}}+\frac {584}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {128}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{10}}-\frac {60}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {576}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}+\frac {176}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {1024}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}-\frac {1864}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{d \,a^{8}}\) | \(175\) |
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. 461 vs.
\(2 (110) = 220\).
time = 0.33, size = 461, normalized size = 3.91 \begin {gather*} -\frac {2 \, {\left (\frac {517 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {4895 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {11220 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {27060 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {32802 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {37422 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {23100 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {13860 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {3465 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {1155 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + 152\right )}}{1155 \, {\left (a^{8} + \frac {11 \, a^{8} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {55 \, a^{8} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {165 \, a^{8} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {330 \, a^{8} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {462 \, a^{8} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {462 \, a^{8} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {330 \, a^{8} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {165 \, a^{8} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {55 \, a^{8} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {11 \, a^{8} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a^{8} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}\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. 291 vs.
\(2 (110) = 220\).
time = 0.38, size = 291, normalized size = 2.47 \begin {gather*} \frac {2 \, \cos \left (d x + c\right )^{6} + 12 \, \cos \left (d x + c\right )^{5} - 25 \, \cos \left (d x + c\right )^{4} - 70 \, \cos \left (d x + c\right )^{3} - 245 \, \cos \left (d x + c\right )^{2} + {\left (2 \, \cos \left (d x + c\right )^{5} - 10 \, \cos \left (d x + c\right )^{4} - 35 \, \cos \left (d x + c\right )^{3} + 35 \, \cos \left (d x + c\right )^{2} - 210 \, \cos \left (d x + c\right ) - 420\right )} \sin \left (d x + c\right ) + 210 \, \cos \left (d x + c\right ) + 420}{1155 \, {\left (a^{8} d \cos \left (d x + c\right )^{6} - 5 \, a^{8} d \cos \left (d x + c\right )^{5} - 18 \, a^{8} d \cos \left (d x + c\right )^{4} + 20 \, a^{8} d \cos \left (d x + c\right )^{3} + 48 \, a^{8} d \cos \left (d x + c\right )^{2} - 16 \, a^{8} d \cos \left (d x + c\right ) - 32 \, a^{8} d - {\left (a^{8} d \cos \left (d x + c\right )^{5} + 6 \, a^{8} d \cos \left (d x + c\right )^{4} - 12 \, a^{8} d \cos \left (d x + c\right )^{3} - 32 \, a^{8} d \cos \left (d x + c\right )^{2} + 16 \, a^{8} d \cos \left (d x + c\right ) + 32 \, 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 [B] Leaf count of result is larger than twice the leaf count of optimal. 2470 vs.
\(2 (107) = 214\).
time = 201.18, size = 2470, normalized size = 20.93 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.20, size = 151, normalized size = 1.28 \begin {gather*} -\frac {2 \, {\left (1155 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 3465 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 13860 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 23100 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 37422 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 32802 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 27060 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 11220 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4895 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 517 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 152\right )}}{1155 \, a^{8} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.15, size = 140, normalized size = 1.19 \begin {gather*} -\frac {\sqrt {2}\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {7623\,\sin \left (c+d\,x\right )}{4}-697\,\cos \left (c+d\,x\right )-\frac {3977\,\cos \left (2\,c+2\,d\,x\right )}{4}+\frac {3203\,\cos \left (3\,c+3\,d\,x\right )}{16}+\frac {461\,\cos \left (4\,c+4\,d\,x\right )}{8}-\frac {75\,\cos \left (5\,c+5\,d\,x\right )}{16}-462\,\sin \left (2\,c+2\,d\,x\right )-\frac {4983\,\sin \left (3\,c+3\,d\,x\right )}{16}+\frac {187\,\sin \left (4\,c+4\,d\,x\right )}{4}+\frac {77\,\sin \left (5\,c+5\,d\,x\right )}{16}+\frac {12721}{8}\right )}{36960\,a^8\,d\,{\cos \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d\,x}{2}\right )}^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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