Optimal. Leaf size=70 \[ \frac {B x}{a^2}+\frac {(2 A-5 B) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {(A-B) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2} \]
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Rubi [A]
time = 0.10, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {3047, 3098,
2814, 2727} \begin {gather*} \frac {(2 A-5 B) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac {B x}{a^2}-\frac {(A-B) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2727
Rule 2814
Rule 3047
Rule 3098
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx &=\int \frac {A \cos (c+d x)+B \cos ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx\\ &=-\frac {(A-B) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac {\int \frac {-2 a (A-B)-3 a B \cos (c+d x)}{a+a \cos (c+d x)} \, dx}{3 a^2}\\ &=\frac {B x}{a^2}-\frac {(A-B) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {(2 A-5 B) \int \frac {1}{a+a \cos (c+d x)} \, dx}{3 a}\\ &=\frac {B x}{a^2}-\frac {(A-B) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {(2 A-5 B) \sin (c+d x)}{3 d \left (a^2+a^2 \cos (c+d x)\right )}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(153\) vs. \(2(70)=140\).
time = 0.38, size = 153, normalized size = 2.19 \begin {gather*} \frac {\sec \left (\frac {c}{2}\right ) \sec ^3\left (\frac {1}{2} (c+d x)\right ) \left (9 B d x \cos \left (\frac {d x}{2}\right )+9 B d x \cos \left (c+\frac {d x}{2}\right )+3 B d x \cos \left (c+\frac {3 d x}{2}\right )+3 B d x \cos \left (2 c+\frac {3 d x}{2}\right )+6 A \sin \left (\frac {d x}{2}\right )-18 B \sin \left (\frac {d x}{2}\right )-6 A \sin \left (c+\frac {d x}{2}\right )+12 B \sin \left (c+\frac {d x}{2}\right )+4 A \sin \left (c+\frac {3 d x}{2}\right )-10 B \sin \left (c+\frac {3 d x}{2}\right )\right )}{24 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 74, normalized size = 1.06
method | result | size |
derivativedivides | \(\frac {-\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{3}+\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{3}+A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+4 B \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{2}}\) | \(74\) |
default | \(\frac {-\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{3}+\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{3}+A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+4 B \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{2}}\) | \(74\) |
risch | \(\frac {B x}{a^{2}}+\frac {2 i \left (3 A \,{\mathrm e}^{2 i \left (d x +c \right )}-6 B \,{\mathrm e}^{2 i \left (d x +c \right )}+3 A \,{\mathrm e}^{i \left (d x +c \right )}-9 B \,{\mathrm e}^{i \left (d x +c \right )}+2 A -5 B \right )}{3 d \,a^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3}}\) | \(85\) |
norman | \(\frac {\frac {B x}{a}+\frac {B x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {2 B x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {\left (A -7 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}+\frac {\left (A -3 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}-\frac {\left (A -B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}+\frac {\left (5 A -17 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} a}\) | \(158\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 120, normalized size = 1.71 \begin {gather*} -\frac {B {\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 91, normalized size = 1.30 \begin {gather*} \frac {3 \, B d x \cos \left (d x + c\right )^{2} + 6 \, B d x \cos \left (d x + c\right ) + 3 \, B d x + {\left ({\left (2 \, A - 5 \, B\right )} \cos \left (d x + c\right ) + A - 4 \, B\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.76, size = 105, normalized size = 1.50 \begin {gather*} \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 {B x}{a^{2}} + \frac {B \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{2} d} - \frac {3 B \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a^{2} d} & \text {for}\: d \neq 0 \\\frac {x \left (A + B \cos {\left (c \right )}\right ) \cos {\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 86, normalized size = 1.23 \begin {gather*} \frac {\frac {6 \, {\left (d x + c\right )} B}{a^{2}} - \frac {A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - B 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 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.22, size = 65, normalized size = 0.93 \begin {gather*} \frac {3\,A\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-9\,B\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-A\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+B\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+6\,B\,d\,x}{6\,a^2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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