Optimal. Leaf size=155 \[ -\frac {\cos ^3(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {11 \cos ^2(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}+\frac {67 \sin (c+d x)}{315 a^2 d (a+a \cos (c+d x))^3}-\frac {142 \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}+\frac {83 \sin (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )} \]
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Rubi [A]
time = 0.19, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2844, 3056,
3047, 3098, 2829, 2727} \begin {gather*} \frac {83 \sin (c+d x)}{315 d \left (a^5 \cos (c+d x)+a^5\right )}-\frac {142 \sin (c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}+\frac {67 \sin (c+d x)}{315 a^2 d (a \cos (c+d x)+a)^3}-\frac {\sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac {11 \sin (c+d x) \cos ^2(c+d x)}{63 a d (a \cos (c+d x)+a)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 2727
Rule 2829
Rule 2844
Rule 3047
Rule 3056
Rule 3098
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^5} \, dx &=-\frac {\cos ^3(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {\int \frac {\cos ^2(c+d x) (3 a-8 a \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx}{9 a^2}\\ &=-\frac {\cos ^3(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {11 \cos ^2(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {\int \frac {\cos (c+d x) \left (22 a^2-45 a^2 \cos (c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx}{63 a^4}\\ &=-\frac {\cos ^3(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {11 \cos ^2(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {\int \frac {22 a^2 \cos (c+d x)-45 a^2 \cos ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx}{63 a^4}\\ &=-\frac {\cos ^3(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {11 \cos ^2(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}+\frac {67 \sin (c+d x)}{315 a^2 d (a+a \cos (c+d x))^3}+\frac {\int \frac {-201 a^3+225 a^3 \cos (c+d x)}{(a+a \cos (c+d x))^2} \, dx}{315 a^6}\\ &=-\frac {\cos ^3(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {11 \cos ^2(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}+\frac {67 \sin (c+d x)}{315 a^2 d (a+a \cos (c+d x))^3}-\frac {142 \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}+\frac {83 \int \frac {1}{a+a \cos (c+d x)} \, dx}{315 a^4}\\ &=-\frac {\cos ^3(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {11 \cos ^2(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}+\frac {67 \sin (c+d x)}{315 a^2 d (a+a \cos (c+d x))^3}-\frac {142 \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}+\frac {83 \sin (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 138, normalized size = 0.89 \begin {gather*} \frac {\sec \left (\frac {c}{2}\right ) \sec ^9\left (\frac {1}{2} (c+d x)\right ) \left (5418 \sin \left (\frac {d x}{2}\right )-5040 \sin \left (c+\frac {d x}{2}\right )+3612 \sin \left (c+\frac {3 d x}{2}\right )-3360 \sin \left (2 c+\frac {3 d x}{2}\right )+1728 \sin \left (2 c+\frac {5 d x}{2}\right )-1260 \sin \left (3 c+\frac {5 d x}{2}\right )+432 \sin \left (3 c+\frac {7 d x}{2}\right )-315 \sin \left (4 c+\frac {7 d x}{2}\right )+83 \sin \left (4 c+\frac {9 d x}{2}\right )\right )}{80640 a^5 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 71, normalized size = 0.46
method | result | size |
derivativedivides | \(\frac {\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {4 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{5}}\) | \(71\) |
default | \(\frac {\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {4 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{5}}\) | \(71\) |
risch | \(\frac {2 i \left (315 \,{\mathrm e}^{8 i \left (d x +c \right )}+1260 \,{\mathrm e}^{7 i \left (d x +c \right )}+3360 \,{\mathrm e}^{6 i \left (d x +c \right )}+5040 \,{\mathrm e}^{5 i \left (d x +c \right )}+5418 \,{\mathrm e}^{4 i \left (d x +c \right )}+3612 \,{\mathrm e}^{3 i \left (d x +c \right )}+1728 \,{\mathrm e}^{2 i \left (d x +c \right )}+432 \,{\mathrm e}^{i \left (d x +c \right )}+83\right )}{315 d \,a^{5} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{9}}\) | \(113\) |
norman | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a d}+\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{6 a d}+\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 a d}+\frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{70 a d}+\frac {109 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2520 a d}+\frac {19 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{630 a d}-\frac {11 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{420 a d}-\frac {\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )}{126 a d}+\frac {\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )}{144 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} a^{4}}\) | \(190\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 107, normalized size = 0.69 \begin {gather*} \frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {420 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {378 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {180 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{5040 \, a^{5} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 123, normalized size = 0.79 \begin {gather*} \frac {{\left (83 \, \cos \left (d x + c\right )^{4} + 100 \, \cos \left (d x + c\right )^{3} + 84 \, \cos \left (d x + c\right )^{2} + 40 \, \cos \left (d x + c\right ) + 8\right )} \sin \left (d x + c\right )}{315 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 5.74, size = 107, normalized size = 0.69 \begin {gather*} \begin {cases} \frac {\tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{144 a^{5} d} - \frac {\tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{28 a^{5} d} + \frac {3 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{40 a^{5} d} - \frac {\tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{12 a^{5} d} + \frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16 a^{5} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{4}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{5}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 72, normalized size = 0.46 \begin {gather*} \frac {35 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 180 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 378 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 420 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 315 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{5040 \, a^{5} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.43, size = 127, normalized size = 0.82 \begin {gather*} \frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (315\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-420\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+378\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-180\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+35\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\right )}{5040\,a^5\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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