Optimal. Leaf size=112 \[ \frac {13 \sin (c+d x)}{105 d \left (a^4 \cos (c+d x)+a^4\right )}+\frac {13 \sin (c+d x)}{105 d \left (a^2 \cos (c+d x)+a^2\right )^2}-\frac {11 \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}+\frac {\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
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Rubi [A] time = 0.11, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2758, 2750, 2650, 2648} \[ \frac {13 \sin (c+d x)}{105 d \left (a^4 \cos (c+d x)+a^4\right )}+\frac {13 \sin (c+d x)}{105 d \left (a^2 \cos (c+d x)+a^2\right )^2}-\frac {11 \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}+\frac {\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2650
Rule 2750
Rule 2758
Rubi steps
\begin {align*} \int \frac {\cos ^2(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 {-4 a+7 a \cos (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 {11 \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {13 \int \frac {1}{(a+a \cos (c+d x))^2} \, dx}{35 a^2}\\ &=\frac {\sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {11 \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {13 \sin (c+d x)}{105 d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac {13 \int \frac {1}{a+a \cos (c+d x)} \, dx}{105 a^3}\\ &=\frac {\sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {11 \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {13 \sin (c+d x)}{105 d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac {13 \sin (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 99, normalized size = 0.88 \[ \frac {\sec \left (\frac {c}{2}\right ) \left (-175 \sin \left (c+\frac {d x}{2}\right )+168 \sin \left (c+\frac {3 d x}{2}\right )-105 \sin \left (2 c+\frac {3 d x}{2}\right )+91 \sin \left (2 c+\frac {5 d x}{2}\right )+13 \sin \left (3 c+\frac {7 d x}{2}\right )+280 \sin \left (\frac {d x}{2}\right )\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right )}{6720 a^4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 99, normalized size = 0.88 \[ \frac {{\left (13 \, \cos \left (d x + c\right )^{3} + 52 \, \cos \left (d x + c\right )^{2} + 32 \, \cos \left (d x + c\right ) + 8\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.56, size = 59, normalized size = 0.53 \[ \frac {15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 35 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{840 \, a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 58, normalized size = 0.52 \[ \frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.84, size = 87, normalized size = 0.78 \[ \frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{840 \, a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 58, normalized size = 0.52 \[ -\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (-15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-105\right )}{840\,a^4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.05, size = 87, normalized size = 0.78 \[ \begin {cases} \frac {\tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{4} d} - \frac {\tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{40 a^{4} d} - \frac {\tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{24 a^{4} d} + \frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{2}{\relax (c )}}{\left (a \cos {\relax (c )} + a\right )^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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