Optimal. Leaf size=103 \[ \frac {24 \sin (c+d x)}{5 a^3 d}-\frac {3 \sin (c+d x)}{d \left (a^3 \sec (c+d x)+a^3\right )}-\frac {3 x}{a^3}-\frac {3 \sin (c+d x)}{5 a d (a \sec (c+d x)+a)^2}-\frac {\sin (c+d x)}{5 d (a \sec (c+d x)+a)^3} \]
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Rubi [A] time = 0.22, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ \frac {24 \sin (c+d x)}{5 a^3 d}-\frac {3 \sin (c+d x)}{d \left (a^3 \sec (c+d x)+a^3\right )}-\frac {3 x}{a^3}-\frac {3 \sin (c+d x)}{5 a d (a \sec (c+d x)+a)^2}-\frac {\sin (c+d x)}{5 d (a \sec (c+d x)+a)^3} \]
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))^3} \, dx &=-\frac {\sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {\int \frac {\cos (c+d x) (-6 a+3 a \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac {\sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {3 \sin (c+d x)}{5 a d (a+a \sec (c+d x))^2}-\frac {\int \frac {\cos (c+d x) \left (-27 a^2+18 a^2 \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{15 a^4}\\ &=-\frac {\sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {3 \sin (c+d x)}{5 a d (a+a \sec (c+d x))^2}-\frac {3 \sin (c+d x)}{d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {\int \cos (c+d x) \left (-72 a^3+45 a^3 \sec (c+d x)\right ) \, dx}{15 a^6}\\ &=-\frac {\sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {3 \sin (c+d x)}{5 a d (a+a \sec (c+d x))^2}-\frac {3 \sin (c+d x)}{d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {3 \int 1 \, dx}{a^3}+\frac {24 \int \cos (c+d x) \, dx}{5 a^3}\\ &=-\frac {3 x}{a^3}+\frac {24 \sin (c+d x)}{5 a^3 d}-\frac {\sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {3 \sin (c+d x)}{5 a d (a+a \sec (c+d x))^2}-\frac {3 \sin (c+d x)}{d \left (a^3+a^3 \sec (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.58, size = 169, normalized size = 1.64 \[ \frac {2 \cos \left (\frac {1}{2} (c+d x)\right ) \sec ^3(c+d x) \left (20 (\sin (c+d x)-3 d x) \cos ^5\left (\frac {1}{2} (c+d x)\right )-12 \tan \left (\frac {c}{2}\right ) \cos ^3\left (\frac {1}{2} (c+d x)\right )+\tan \left (\frac {c}{2}\right ) \cos \left (\frac {1}{2} (c+d x)\right )+\sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )+96 \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) \cos ^4\left (\frac {1}{2} (c+d x)\right )-12 \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) \cos ^2\left (\frac {1}{2} (c+d x)\right )\right )}{5 a^3 d (\sec (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.30, size = 126, normalized size = 1.22 \[ -\frac {15 \, d x \cos \left (d x + c\right )^{3} + 45 \, d x \cos \left (d x + c\right )^{2} + 45 \, d x \cos \left (d x + c\right ) + 15 \, d x - {\left (5 \, \cos \left (d x + c\right )^{3} + 39 \, \cos \left (d x + c\right )^{2} + 57 \, \cos \left (d x + c\right ) + 24\right )} \sin \left (d x + c\right )}{5 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 4.36, size = 96, normalized size = 0.93 \[ -\frac {\frac {60 \, {\left (d x + c\right )}}{a^{3}} - \frac {40 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{3}} - \frac {a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 10 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 85 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{20 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.61, size = 107, normalized size = 1.04 \[ \frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{20 d \,a^{3}}-\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{3}}+\frac {17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \,a^{3}}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {6 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 137, normalized size = 1.33 \[ \frac {\frac {40 \, \sin \left (d x + c\right )}{{\left (a^{3} + \frac {a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {85 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{20 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.73, size = 113, normalized size = 1.10 \[ \frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-12\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+96\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+40\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-60\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (c+d\,x\right )}{20\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \]
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 \[ \frac {\int \frac {\cos {\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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