Optimal. Leaf size=66 \[ \frac {(A-5 C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac {C x}{a^2}+\frac {(A+C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.13, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3020, 2735, 2648} \[ \frac {(A-5 C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac {C x}{a^2}+\frac {(A+C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2735
Rule 3020
Rubi steps
\begin {align*} \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx &=\frac {(A+C) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac {\int \frac {-a (A-2 C)-3 a C \cos (c+d x)}{a+a \cos (c+d x)} \, dx}{3 a^2}\\ &=\frac {C x}{a^2}+\frac {(A+C) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {(A-5 C) \int \frac {1}{a+a \cos (c+d x)} \, dx}{3 a}\\ &=\frac {C x}{a^2}+\frac {(A+C) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {(A-5 C) \sin (c+d x)}{3 d \left (a^2+a^2 \cos (c+d x)\right )}\\ \end {align*}
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Mathematica [B] time = 0.34, size = 141, normalized size = 2.14 \[ \frac {\sec \left (\frac {c}{2}\right ) \sec ^3\left (\frac {1}{2} (c+d x)\right ) \left (2 A \sin \left (c+\frac {3 d x}{2}\right )+6 A \sin \left (\frac {d x}{2}\right )+12 C \sin \left (c+\frac {d x}{2}\right )-10 C \sin \left (c+\frac {3 d x}{2}\right )+9 C d x \cos \left (c+\frac {d x}{2}\right )+3 C d x \cos \left (c+\frac {3 d x}{2}\right )+3 C d x \cos \left (2 c+\frac {3 d x}{2}\right )-18 C \sin \left (\frac {d x}{2}\right )+9 C d x \cos \left (\frac {d x}{2}\right )\right )}{24 a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.67, size = 91, normalized size = 1.38 \[ \frac {3 \, C d x \cos \left (d x + c\right )^{2} + 6 \, C d x \cos \left (d x + c\right ) + 3 \, C d x + {\left ({\left (A - 5 \, C\right )} \cos \left (d x + c\right ) + 2 \, A - 4 \, C\right )} \sin \left (d x + c\right )}{3 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 84, normalized size = 1.27 \[ \frac {\frac {6 \, {\left (d x + c\right )} C}{a^{2}} + \frac {A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 9 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 97, normalized size = 1.47 \[ \frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{6 d \,a^{2}}+\frac {C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d \,a^{2}}+\frac {A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{2}}-\frac {3 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{2}}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 119, normalized size = 1.80 \[ -\frac {C {\left (\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {12 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )} - \frac {A {\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{2}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.89, size = 64, normalized size = 0.97 \[ \frac {3\,A\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-9\,C\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+A\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+C\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+6\,C\,d\,x}{6\,a^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.26, size = 104, normalized size = 1.58 \[ \begin {cases} \frac {A \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{2} d} + \frac {A \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a^{2} d} + \frac {C x}{a^{2}} + \frac {C \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{2} d} - \frac {3 C \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a^{2} d} & \text {for}\: d \neq 0 \\\frac {x \left (A + C \cos ^{2}{\relax (c )}\right )}{\left (a \cos {\relax (c )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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