Optimal. Leaf size=127 \[ \frac {x}{a^4}+\frac {11 \sin (c+d x)}{21 a^4 d (1+\cos (c+d x))^2}-\frac {43 \sin (c+d x)}{21 a^4 d (1+\cos (c+d x))}-\frac {\cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {2 \cos ^2(c+d x) \sin (c+d x)}{7 a d (a+a \cos (c+d x))^3} \]
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Rubi [A]
time = 0.19, antiderivative size = 127, 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, 2814, 2727} \begin {gather*} -\frac {43 \sin (c+d x)}{21 a^4 d (\cos (c+d x)+1)}+\frac {11 \sin (c+d x)}{21 a^4 d (\cos (c+d x)+1)^2}+\frac {x}{a^4}-\frac {\sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac {2 \sin (c+d x) \cos ^2(c+d x)}{7 a d (a \cos (c+d x)+a)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2727
Rule 2814
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))^4} \, dx &=-\frac {\cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {\int \frac {\cos ^2(c+d x) (3 a-7 a \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac {\cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {2 \cos ^2(c+d x) \sin (c+d x)}{7 a d (a+a \cos (c+d x))^3}-\frac {\int \frac {\cos (c+d x) \left (20 a^2-35 a^2 \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac {\cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {2 \cos ^2(c+d x) \sin (c+d x)}{7 a d (a+a \cos (c+d x))^3}-\frac {\int \frac {20 a^2 \cos (c+d x)-35 a^2 \cos ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=\frac {11 \sin (c+d x)}{21 a^4 d (1+\cos (c+d x))^2}-\frac {\cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {2 \cos ^2(c+d x) \sin (c+d x)}{7 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {-110 a^3+105 a^3 \cos (c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=\frac {x}{a^4}+\frac {11 \sin (c+d x)}{21 a^4 d (1+\cos (c+d x))^2}-\frac {\cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {2 \cos ^2(c+d x) \sin (c+d x)}{7 a d (a+a \cos (c+d x))^3}-\frac {43 \int \frac {1}{a+a \cos (c+d x)} \, dx}{21 a^3}\\ &=\frac {x}{a^4}+\frac {11 \sin (c+d x)}{21 a^4 d (1+\cos (c+d x))^2}-\frac {\cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {2 \cos ^2(c+d x) \sin (c+d x)}{7 a d (a+a \cos (c+d x))^3}-\frac {43 \sin (c+d x)}{21 d \left (a^4+a^4 \cos (c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.36, size = 224, normalized size = 1.76 \begin {gather*} \frac {\sec \left (\frac {c}{2}\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right ) \left (735 d x \cos \left (\frac {d x}{2}\right )+735 d x \cos \left (c+\frac {d x}{2}\right )+441 d x \cos \left (c+\frac {3 d x}{2}\right )+441 d x \cos \left (2 c+\frac {3 d x}{2}\right )+147 d x \cos \left (2 c+\frac {5 d x}{2}\right )+147 d x \cos \left (3 c+\frac {5 d x}{2}\right )+21 d x \cos \left (3 c+\frac {7 d x}{2}\right )+21 d x \cos \left (4 c+\frac {7 d x}{2}\right )-1988 \sin \left (\frac {d x}{2}\right )+1652 \sin \left (c+\frac {d x}{2}\right )-1428 \sin \left (c+\frac {3 d x}{2}\right )+756 \sin \left (2 c+\frac {3 d x}{2}\right )-560 \sin \left (2 c+\frac {5 d x}{2}\right )+168 \sin \left (3 c+\frac {5 d x}{2}\right )-104 \sin \left (3 c+\frac {7 d x}{2}\right )\right )}{2688 a^4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 72, normalized size = 0.57
method | result | size |
derivativedivides | \(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+16 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(72\) |
default | \(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+16 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(72\) |
risch | \(\frac {x}{a^{4}}-\frac {4 i \left (42 \,{\mathrm e}^{6 i \left (d x +c \right )}+189 \,{\mathrm e}^{5 i \left (d x +c \right )}+413 \,{\mathrm e}^{4 i \left (d x +c \right )}+497 \,{\mathrm e}^{3 i \left (d x +c \right )}+357 \,{\mathrm e}^{2 i \left (d x +c \right )}+140 \,{\mathrm e}^{i \left (d x +c \right )}+26\right )}{21 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(97\) |
norman | \(\frac {\frac {x}{a}+\frac {x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}-\frac {169 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 a d}-\frac {229 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 a d}-\frac {293 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 a d}-\frac {121 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{168 a d}+\frac {11 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{168 a d}-\frac {3 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 a d}+\frac {\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )}{56 a d}+\frac {4 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {6 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {4 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} a^{3}}\) | \(243\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 112, normalized size = 0.88 \begin {gather*} -\frac {\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {336 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}}{168 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 152, normalized size = 1.20 \begin {gather*} \frac {21 \, d x \cos \left (d x + c\right )^{4} + 84 \, d x \cos \left (d x + c\right )^{3} + 126 \, d x \cos \left (d x + c\right )^{2} + 84 \, d x \cos \left (d x + c\right ) + 21 \, d x - {\left (52 \, \cos \left (d x + c\right )^{3} + 124 \, \cos \left (d x + c\right )^{2} + 107 \, \cos \left (d x + c\right ) + 32\right )} \sin \left (d x + c\right )}{21 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.50, size = 95, normalized size = 0.75 \begin {gather*} \begin {cases} \frac {x}{a^{4}} + \frac {\tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{4} d} - \frac {\tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} + \frac {11 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{24 a^{4} d} - \frac {15 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{4}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{4}} & \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 = 83, normalized size = 0.65 \begin {gather*} \frac {\frac {168 \, {\left (d x + c\right )}}{a^{4}} + \frac {3 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 21 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 77 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 315 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{168 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.43, size = 102, normalized size = 0.80 \begin {gather*} \frac {x}{a^4}+\frac {-\frac {52\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{21}+\frac {16\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{21}-\frac {5\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{28}+\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{56}}{a^4\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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