Optimal. Leaf size=159 \[ \frac {496 \sin (c+d x)}{63 a^5 d}-\frac {5 \sin (c+d x)}{d \left (a^5 \sec (c+d x)+a^5\right )}-\frac {5 x}{a^5}-\frac {67 \sin (c+d x)}{63 a^3 d (a \sec (c+d x)+a)^2}-\frac {29 \sin (c+d x)}{63 a^2 d (a \sec (c+d x)+a)^3}-\frac {5 \sin (c+d x)}{21 a d (a \sec (c+d x)+a)^4}-\frac {\sin (c+d x)}{9 d (a \sec (c+d x)+a)^5} \]
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Rubi [A] time = 0.40, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3817, 4020, 3787, 2637, 8} \[ \frac {496 \sin (c+d x)}{63 a^5 d}-\frac {5 \sin (c+d x)}{d \left (a^5 \sec (c+d x)+a^5\right )}-\frac {67 \sin (c+d x)}{63 a^3 d (a \sec (c+d x)+a)^2}-\frac {29 \sin (c+d x)}{63 a^2 d (a \sec (c+d x)+a)^3}-\frac {5 x}{a^5}-\frac {5 \sin (c+d x)}{21 a d (a \sec (c+d x)+a)^4}-\frac {\sin (c+d x)}{9 d (a \sec (c+d x)+a)^5} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2637
Rule 3787
Rule 3817
Rule 4020
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{(a+a \sec (c+d x))^5} \, dx &=-\frac {\sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {\int \frac {\cos (c+d x) (-10 a+5 a \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx}{9 a^2}\\ &=-\frac {\sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {5 \sin (c+d x)}{21 a d (a+a \sec (c+d x))^4}-\frac {\int \frac {\cos (c+d x) \left (-85 a^2+60 a^2 \sec (c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx}{63 a^4}\\ &=-\frac {\sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {5 \sin (c+d x)}{21 a d (a+a \sec (c+d x))^4}-\frac {29 \sin (c+d x)}{63 a^2 d (a+a \sec (c+d x))^3}-\frac {\int \frac {\cos (c+d x) \left (-570 a^3+435 a^3 \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{315 a^6}\\ &=-\frac {\sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {5 \sin (c+d x)}{21 a d (a+a \sec (c+d x))^4}-\frac {29 \sin (c+d x)}{63 a^2 d (a+a \sec (c+d x))^3}-\frac {67 \sin (c+d x)}{63 a^3 d (a+a \sec (c+d x))^2}-\frac {\int \frac {\cos (c+d x) \left (-2715 a^4+2010 a^4 \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{945 a^8}\\ &=-\frac {\sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {5 \sin (c+d x)}{21 a d (a+a \sec (c+d x))^4}-\frac {29 \sin (c+d x)}{63 a^2 d (a+a \sec (c+d x))^3}-\frac {67 \sin (c+d x)}{63 a^3 d (a+a \sec (c+d x))^2}-\frac {5 \sin (c+d x)}{d \left (a^5+a^5 \sec (c+d x)\right )}-\frac {\int \cos (c+d x) \left (-7440 a^5+4725 a^5 \sec (c+d x)\right ) \, dx}{945 a^{10}}\\ &=-\frac {\sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {5 \sin (c+d x)}{21 a d (a+a \sec (c+d x))^4}-\frac {29 \sin (c+d x)}{63 a^2 d (a+a \sec (c+d x))^3}-\frac {67 \sin (c+d x)}{63 a^3 d (a+a \sec (c+d x))^2}-\frac {5 \sin (c+d x)}{d \left (a^5+a^5 \sec (c+d x)\right )}-\frac {5 \int 1 \, dx}{a^5}+\frac {496 \int \cos (c+d x) \, dx}{63 a^5}\\ &=-\frac {5 x}{a^5}+\frac {496 \sin (c+d x)}{63 a^5 d}-\frac {\sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {5 \sin (c+d x)}{21 a d (a+a \sec (c+d x))^4}-\frac {29 \sin (c+d x)}{63 a^2 d (a+a \sec (c+d x))^3}-\frac {67 \sin (c+d x)}{63 a^3 d (a+a \sec (c+d x))^2}-\frac {5 \sin (c+d x)}{d \left (a^5+a^5 \sec (c+d x)\right )}\\ \end {align*}
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Mathematica [B] time = 0.70, size = 319, normalized size = 2.01 \[ -\frac {\sec \left (\frac {c}{2}\right ) \sec ^9\left (\frac {1}{2} (c+d x)\right ) \left (143010 \sin \left (c+\frac {d x}{2}\right )-138726 \sin \left (c+\frac {3 d x}{2}\right )+73290 \sin \left (2 c+\frac {3 d x}{2}\right )-70389 \sin \left (2 c+\frac {5 d x}{2}\right )+20475 \sin \left (3 c+\frac {5 d x}{2}\right )-21141 \sin \left (3 c+\frac {7 d x}{2}\right )+1575 \sin \left (4 c+\frac {7 d x}{2}\right )-3091 \sin \left (4 c+\frac {9 d x}{2}\right )-567 \sin \left (5 c+\frac {9 d x}{2}\right )-63 \sin \left (5 c+\frac {11 d x}{2}\right )-63 \sin \left (6 c+\frac {11 d x}{2}\right )+79380 d x \cos \left (c+\frac {d x}{2}\right )+52920 d x \cos \left (c+\frac {3 d x}{2}\right )+52920 d x \cos \left (2 c+\frac {3 d x}{2}\right )+22680 d x \cos \left (2 c+\frac {5 d x}{2}\right )+22680 d x \cos \left (3 c+\frac {5 d x}{2}\right )+5670 d x \cos \left (3 c+\frac {7 d x}{2}\right )+5670 d x \cos \left (4 c+\frac {7 d x}{2}\right )+630 d x \cos \left (4 c+\frac {9 d x}{2}\right )+630 d x \cos \left (5 c+\frac {9 d x}{2}\right )-175014 \sin \left (\frac {d x}{2}\right )+79380 d x \cos \left (\frac {d x}{2}\right )\right )}{64512 a^5 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 198, normalized size = 1.25 \[ -\frac {315 \, d x \cos \left (d x + c\right )^{5} + 1575 \, d x \cos \left (d x + c\right )^{4} + 3150 \, d x \cos \left (d x + c\right )^{3} + 3150 \, d x \cos \left (d x + c\right )^{2} + 1575 \, d x \cos \left (d x + c\right ) + 315 \, d x - {\left (63 \, \cos \left (d x + c\right )^{5} + 946 \, \cos \left (d x + c\right )^{4} + 2840 \, \cos \left (d x + c\right )^{3} + 3633 \, \cos \left (d x + c\right )^{2} + 2165 \, \cos \left (d x + c\right ) + 496\right )} \sin \left (d x + c\right )}{63 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.12, size = 129, normalized size = 0.81 \[ -\frac {\frac {5040 \, {\left (d x + c\right )}}{a^{5}} - \frac {2016 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{5}} - \frac {7 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 72 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 378 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1512 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8127 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{45}}}{1008 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.65, size = 145, normalized size = 0.91 \[ \frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{144 d \,a^{5}}-\frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{14 d \,a^{5}}+\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{5}}-\frac {3 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{5}}+\frac {129 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{5}}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{5} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {10 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 178, normalized size = 1.12 \[ \frac {\frac {2016 \, \sin \left (d x + c\right )}{{\left (a^{5} + \frac {a^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {8127 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1512 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {378 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {72 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {7 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{5}} - \frac {10080 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{5}}}{1008 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.83, size = 159, normalized size = 1.00 \[ \frac {7\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-100\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+636\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-2512\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+10096\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+2016\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-5040\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (c+d\,x\right )}{1008\,a^5\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cos {\left (c + d x \right )}}{\sec ^{5}{\left (c + d x \right )} + 5 \sec ^{4}{\left (c + d x \right )} + 10 \sec ^{3}{\left (c + d x \right )} + 10 \sec ^{2}{\left (c + d x \right )} + 5 \sec {\left (c + d x \right )} + 1}\, dx}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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