Optimal. Leaf size=120 \[ \frac {\tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac {32 \sin (c+d x)}{21 a^4 d (\cos (c+d x)+1)}-\frac {11 \sin (c+d x)}{21 a^4 d (\cos (c+d x)+1)^2}-\frac {2 \sin (c+d x)}{7 a d (a \cos (c+d x)+a)^3}-\frac {\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
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Rubi [A] time = 0.29, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2766, 2978, 12, 3770} \[ \frac {\tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac {32 \sin (c+d x)}{21 a^4 d (\cos (c+d x)+1)}-\frac {11 \sin (c+d x)}{21 a^4 d (\cos (c+d x)+1)^2}-\frac {2 \sin (c+d x)}{7 a d (a \cos (c+d x)+a)^3}-\frac {\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2766
Rule 2978
Rule 3770
Rubi steps
\begin {align*} \int \frac {\sec (c+d x)}{(a+a \cos (c+d x))^4} \, dx &=-\frac {\sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {\int \frac {(7 a-3 a \cos (c+d x)) \sec (c+d x)}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac {\sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {2 \sin (c+d x)}{7 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {\left (35 a^2-20 a^2 \cos (c+d x)\right ) \sec (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 {\sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {2 \sin (c+d x)}{7 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {\left (105 a^3-55 a^3 \cos (c+d x)\right ) \sec (c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=-\frac {11 \sin (c+d x)}{21 a^4 d (1+\cos (c+d x))^2}-\frac {\sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {2 \sin (c+d x)}{7 a d (a+a \cos (c+d x))^3}-\frac {32 \sin (c+d x)}{21 d \left (a^4+a^4 \cos (c+d x)\right )}+\frac {\int 105 a^4 \sec (c+d x) \, dx}{105 a^8}\\ &=-\frac {11 \sin (c+d x)}{21 a^4 d (1+\cos (c+d x))^2}-\frac {\sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {2 \sin (c+d x)}{7 a d (a+a \cos (c+d x))^3}-\frac {32 \sin (c+d x)}{21 d \left (a^4+a^4 \cos (c+d x)\right )}+\frac {\int \sec (c+d x) \, dx}{a^4}\\ &=\frac {\tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac {11 \sin (c+d x)}{21 a^4 d (1+\cos (c+d x))^2}-\frac {\sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {2 \sin (c+d x)}{7 a d (a+a \cos (c+d x))^3}-\frac {32 \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 = 0.86, size = 185, normalized size = 1.54 \[ \frac {\sec \left (\frac {c}{2}\right ) \left (434 \sin \left (c+\frac {d x}{2}\right )-525 \sin \left (c+\frac {3 d x}{2}\right )+147 \sin \left (2 c+\frac {3 d x}{2}\right )-203 \sin \left (2 c+\frac {5 d x}{2}\right )+21 \sin \left (3 c+\frac {5 d x}{2}\right )-32 \sin \left (3 c+\frac {7 d x}{2}\right )-686 \sin \left (\frac {d x}{2}\right )\right ) \cos \left (\frac {1}{2} (c+d x)\right )-1344 \cos ^8\left (\frac {1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{84 a^4 d (\cos (c+d x)+1)^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 202, normalized size = 1.68 \[ \frac {21 \, {\left (\cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right )^{2} + 4 \, \cos \left (d x + c\right ) + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 21 \, {\left (\cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right )^{2} + 4 \, \cos \left (d x + c\right ) + 1\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (32 \, \cos \left (d x + c\right )^{3} + 107 \, \cos \left (d x + c\right )^{2} + 124 \, \cos \left (d x + c\right ) + 52\right )} \sin \left (d x + c\right )}{42 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.69, size = 110, normalized size = 0.92 \[ \frac {\frac {168 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {168 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} - \frac {3 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 21 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 77 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 315 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{168 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 115, normalized size = 0.96 \[ -\frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{56 d \,a^{4}}-\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}-\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d \,a^{4}}-\frac {15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d \,a^{4}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \,a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.48, size = 139, normalized size = 1.16 \[ -\frac {\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}}{168 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.37, size = 83, normalized size = 0.69 \[ -\frac {\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a^4}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{8\,a^4}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{56\,a^4}-\frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^4}+\frac {15\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a^4}}{d} \]
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 \[ \frac {\int \frac {\sec {\left (c + d x \right )}}{\cos ^{4}{\left (c + d x \right )} + 4 \cos ^{3}{\left (c + d x \right )} + 6 \cos ^{2}{\left (c + d x \right )} + 4 \cos {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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