Optimal. Leaf size=143 \[ \frac {2 \sin (c+d x)}{63 d \left (a^5 \cos (c+d x)+a^5\right )}+\frac {2 \sin (c+d x)}{63 a d \left (a^2 \cos (c+d x)+a^2\right )^2}+\frac {\sin (c+d x)}{21 a^2 d (a \cos (c+d x)+a)^3}+\frac {5 \sin (c+d x)}{63 a d (a \cos (c+d x)+a)^4}-\frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5} \]
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Rubi [A] time = 0.11, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2750, 2650, 2648} \[ \frac {2 \sin (c+d x)}{63 d \left (a^5 \cos (c+d x)+a^5\right )}+\frac {2 \sin (c+d x)}{63 a d \left (a^2 \cos (c+d x)+a^2\right )^2}+\frac {\sin (c+d x)}{21 a^2 d (a \cos (c+d x)+a)^3}+\frac {5 \sin (c+d x)}{63 a d (a \cos (c+d x)+a)^4}-\frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2650
Rule 2750
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^5} \, dx &=-\frac {\sin (c+d x)}{9 d (a+a \cos (c+d x))^5}+\frac {5 \int \frac {1}{(a+a \cos (c+d x))^4} \, dx}{9 a}\\ &=-\frac {\sin (c+d x)}{9 d (a+a \cos (c+d x))^5}+\frac {5 \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}+\frac {5 \int \frac {1}{(a+a \cos (c+d x))^3} \, dx}{21 a^2}\\ &=-\frac {\sin (c+d x)}{9 d (a+a \cos (c+d x))^5}+\frac {5 \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}+\frac {\sin (c+d x)}{21 a^2 d (a+a \cos (c+d x))^3}+\frac {2 \int \frac {1}{(a+a \cos (c+d x))^2} \, dx}{21 a^3}\\ &=-\frac {\sin (c+d x)}{9 d (a+a \cos (c+d x))^5}+\frac {5 \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}+\frac {\sin (c+d x)}{21 a^2 d (a+a \cos (c+d x))^3}+\frac {2 \sin (c+d x)}{63 a^3 d (a+a \cos (c+d x))^2}+\frac {2 \int \frac {1}{a+a \cos (c+d x)} \, dx}{63 a^4}\\ &=-\frac {\sin (c+d x)}{9 d (a+a \cos (c+d x))^5}+\frac {5 \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}+\frac {\sin (c+d x)}{21 a^2 d (a+a \cos (c+d x))^3}+\frac {2 \sin (c+d x)}{63 a^3 d (a+a \cos (c+d x))^2}+\frac {2 \sin (c+d x)}{63 d \left (a^5+a^5 \cos (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 97, normalized size = 0.68 \[ \frac {\sec \left (\frac {c}{2}\right ) \left (-63 \sin \left (c+\frac {d x}{2}\right )+84 \sin \left (c+\frac {3 d x}{2}\right )+36 \sin \left (2 c+\frac {5 d x}{2}\right )+9 \sin \left (3 c+\frac {7 d x}{2}\right )+\sin \left (4 c+\frac {9 d x}{2}\right )+63 \sin \left (\frac {d x}{2}\right )\right ) \sec ^9\left (\frac {1}{2} (c+d x)\right )}{8064 a^5 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 123, normalized size = 0.86 \[ \frac {{\left (2 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3} + 21 \, \cos \left (d x + c\right )^{2} + 25 \, \cos \left (d x + c\right ) + 5\right )} \sin \left (d x + c\right )}{63 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.57, size = 59, normalized size = 0.41 \[ -\frac {7 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 18 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 42 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{1008 \, a^{5} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 58, normalized size = 0.41 \[ \frac {-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {2 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 87, normalized size = 0.61 \[ \frac {\frac {63 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {42 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {18 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {7 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{1008 \, a^{5} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 58, normalized size = 0.41 \[ \frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-18\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+42\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+63\right )}{1008\,a^5\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.18, size = 85, normalized size = 0.59 \[ \begin {cases} - \frac {\tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{144 a^{5} d} - \frac {\tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{5} d} + \frac {\tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{24 a^{5} d} + \frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16 a^{5} d} & \text {for}\: d \neq 0 \\\frac {x \cos {\relax (c )}}{\left (a \cos {\relax (c )} + a\right )^{5}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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