Optimal. Leaf size=103 \[ \frac {7 x}{8 a^3}+\frac {7 \cos ^5(c+d x)}{15 a^3 d}+\frac {7 \cos (c+d x) \sin (c+d x)}{8 a^3 d}+\frac {7 \cos ^3(c+d x) \sin (c+d x)}{12 a^3 d}+\frac {2 \cos ^7(c+d x)}{3 a d (a+a \sin (c+d x))^2} \]
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Rubi [A]
time = 0.08, antiderivative size = 103, 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 = {2759, 2761,
2715, 8} \begin {gather*} \frac {7 \cos ^5(c+d x)}{15 a^3 d}+\frac {7 \sin (c+d x) \cos ^3(c+d x)}{12 a^3 d}+\frac {7 \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac {7 x}{8 a^3}+\frac {2 \cos ^7(c+d x)}{3 a d (a \sin (c+d x)+a)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 2759
Rule 2761
Rubi steps
\begin {align*} \int \frac {\cos ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {2 \cos ^7(c+d x)}{3 a d (a+a \sin (c+d x))^2}+\frac {7 \int \frac {\cos ^6(c+d x)}{a+a \sin (c+d x)} \, dx}{3 a^2}\\ &=\frac {7 \cos ^5(c+d x)}{15 a^3 d}+\frac {2 \cos ^7(c+d x)}{3 a d (a+a \sin (c+d x))^2}+\frac {7 \int \cos ^4(c+d x) \, dx}{3 a^3}\\ &=\frac {7 \cos ^5(c+d x)}{15 a^3 d}+\frac {7 \cos ^3(c+d x) \sin (c+d x)}{12 a^3 d}+\frac {2 \cos ^7(c+d x)}{3 a d (a+a \sin (c+d x))^2}+\frac {7 \int \cos ^2(c+d x) \, dx}{4 a^3}\\ &=\frac {7 \cos ^5(c+d x)}{15 a^3 d}+\frac {7 \cos (c+d x) \sin (c+d x)}{8 a^3 d}+\frac {7 \cos ^3(c+d x) \sin (c+d x)}{12 a^3 d}+\frac {2 \cos ^7(c+d x)}{3 a d (a+a \sin (c+d x))^2}+\frac {7 \int 1 \, dx}{8 a^3}\\ &=\frac {7 x}{8 a^3}+\frac {7 \cos ^5(c+d x)}{15 a^3 d}+\frac {7 \cos (c+d x) \sin (c+d x)}{8 a^3 d}+\frac {7 \cos ^3(c+d x) \sin (c+d x)}{12 a^3 d}+\frac {2 \cos ^7(c+d x)}{3 a d (a+a \sin (c+d x))^2}\\ \end {align*}
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Mathematica [A]
time = 0.67, size = 141, normalized size = 1.37 \begin {gather*} -\frac {\cos ^9(c+d x) \left (-210 \sin ^{-1}\left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right ) \sqrt {1-\sin (c+d x)}+\sqrt {1+\sin (c+d x)} \left (136-121 \sin (c+d x)-127 \sin ^2(c+d x)+202 \sin ^3(c+d x)-114 \sin ^4(c+d x)+24 \sin ^5(c+d x)\right )\right )}{120 a^3 d (-1+\sin (c+d x))^5 (1+\sin (c+d x))^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 142, normalized size = 1.38
method | result | size |
risch | \(\frac {7 x}{8 a^{3}}+\frac {7 \cos \left (d x +c \right )}{8 a^{3} d}-\frac {\cos \left (5 d x +5 c \right )}{80 a^{3} d}-\frac {3 \sin \left (4 d x +4 c \right )}{32 a^{3} d}+\frac {13 \cos \left (3 d x +3 c \right )}{48 a^{3} d}+\frac {\sin \left (2 d x +2 c \right )}{4 a^{3} d}\) | \(90\) |
derivativedivides | \(\frac {\frac {2 \left (-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+3 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {13 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+8 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {10 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {8 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8}+\frac {17}{15}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {7 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{a^{3} d}\) | \(142\) |
default | \(\frac {\frac {2 \left (-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+3 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {13 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+8 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {10 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {8 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8}+\frac {17}{15}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {7 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{a^{3} d}\) | \(142\) |
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. 310 vs.
\(2 (93) = 186\).
time = 0.54, size = 310, normalized size = 3.01 \begin {gather*} \frac {\frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {320 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {390 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {400 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {960 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {390 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {360 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {15 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + 136}{a^{3} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} + \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{60 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 60, normalized size = 0.58 \begin {gather*} -\frac {24 \, \cos \left (d x + c\right )^{5} - 160 \, \cos \left (d x + c\right )^{3} - 105 \, d x + 15 \, {\left (6 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, a^{3} d} \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. 1717 vs.
\(2 (97) = 194\).
time = 120.35, size = 1717, normalized size = 16.67 \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 = 6.94, size = 140, normalized size = 1.36 \begin {gather*} \frac {\frac {105 \, {\left (d x + c\right )}}{a^{3}} - \frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 360 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 390 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 400 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 390 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 320 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 136\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5} a^{3}}}{120 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.72, size = 81, normalized size = 0.79 \begin {gather*} \frac {7\,x}{8\,a^3}+\frac {4\,{\cos \left (c+d\,x\right )}^3}{3\,a^3\,d}-\frac {{\cos \left (c+d\,x\right )}^5}{5\,a^3\,d}-\frac {3\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{4\,a^3\,d}+\frac {7\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{8\,a^3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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