Optimal. Leaf size=138 \[ \frac {2 (3 A+4 B) \sin (c+d x)}{105 d \left (a^4 \cos (c+d x)+a^4\right )}+\frac {2 (3 A+4 B) \sin (c+d x)}{105 d \left (a^2 \cos (c+d x)+a^2\right )^2}+\frac {(3 A+4 B) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}+\frac {(A-B) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
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Rubi [A] time = 0.14, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2750, 2650, 2648} \[ \frac {2 (3 A+4 B) \sin (c+d x)}{105 d \left (a^4 \cos (c+d x)+a^4\right )}+\frac {2 (3 A+4 B) \sin (c+d x)}{105 d \left (a^2 \cos (c+d x)+a^2\right )^2}+\frac {(3 A+4 B) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}+\frac {(A-B) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2650
Rule 2750
Rubi steps
\begin {align*} \int \frac {A+B \cos (c+d x)}{(a+a \cos (c+d x))^4} \, dx &=\frac {(A-B) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(3 A+4 B) \int \frac {1}{(a+a \cos (c+d x))^3} \, dx}{7 a}\\ &=\frac {(A-B) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(3 A+4 B) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {(2 (3 A+4 B)) \int \frac {1}{(a+a \cos (c+d x))^2} \, dx}{35 a^2}\\ &=\frac {(A-B) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(3 A+4 B) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {2 (3 A+4 B) \sin (c+d x)}{105 d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac {(2 (3 A+4 B)) \int \frac {1}{a+a \cos (c+d x)} \, dx}{105 a^3}\\ &=\frac {(A-B) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(3 A+4 B) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {2 (3 A+4 B) \sin (c+d x)}{105 d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac {2 (3 A+4 B) \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 = 0.37, size = 109, normalized size = 0.79 \[ \frac {\sec \left (\frac {c}{2}\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \left ((3 A+4 B) \left (21 \sin \left (c+\frac {3 d x}{2}\right )+7 \sin \left (2 c+\frac {5 d x}{2}\right )+\sin \left (3 c+\frac {7 d x}{2}\right )\right )+35 (3 A+2 B) \sin \left (\frac {d x}{2}\right )-70 B \sin \left (c+\frac {d x}{2}\right )\right )}{210 a^4 d (\cos (c+d x)+1)^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 125, normalized size = 0.91 \[ \frac {{\left (2 \, {\left (3 \, A + 4 \, B\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left (3 \, A + 4 \, B\right )} \cos \left (d x + c\right )^{2} + 13 \, {\left (3 \, A + 4 \, B\right )} \cos \left (d x + c\right ) + 36 \, A + 13 \, B\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.93, size = 117, normalized size = 0.85 \[ \frac {15 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 63 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 21 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 105 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 35 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 105 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{840 \, a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 88, normalized size = 0.64 \[ \frac {\frac {\left (A -B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {\left (3 A -B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {\left (3 A +B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 175, normalized size = 1.27 \[ \frac {\frac {B {\left (\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}} + \frac {3 \, A {\left (\frac {35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.24, size = 87, normalized size = 0.63 \[ \frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (3\,A+B\right )}{24\,a^4}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A-B\right )}{56\,a^4}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A+B\right )}{8\,a^4}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (3\,A-B\right )}{40\,a^4}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.87, size = 177, normalized size = 1.28 \[ \begin {cases} \frac {A \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{4} d} + \frac {3 A \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{40 a^{4} d} + \frac {A \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} + \frac {A \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} - \frac {B \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{4} d} - \frac {B \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{40 a^{4} d} + \frac {B \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{24 a^{4} d} + \frac {B \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \left (A + B \cos {\relax (c )}\right )}{\left (a \cos {\relax (c )} + a\right )^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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