Optimal. Leaf size=101 \[ \frac {8 a^2 (5 A+3 B) \sin (c+d x)}{15 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a (5 A+3 B) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{15 d}+\frac {2 B (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d} \]
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Rubi [A]
time = 0.06, antiderivative size = 101, 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 = {2830, 2726,
2725} \begin {gather*} \frac {8 a^2 (5 A+3 B) \sin (c+d x)}{15 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a (5 A+3 B) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{15 d}+\frac {2 B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2725
Rule 2726
Rule 2830
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx &=\frac {2 B (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {1}{5} (5 A+3 B) \int (a+a \cos (c+d x))^{3/2} \, dx\\ &=\frac {2 a (5 A+3 B) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{15 d}+\frac {2 B (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {1}{15} (4 a (5 A+3 B)) \int \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {8 a^2 (5 A+3 B) \sin (c+d x)}{15 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a (5 A+3 B) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{15 d}+\frac {2 B (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 65, normalized size = 0.64 \begin {gather*} \frac {a \sqrt {a (1+\cos (c+d x))} (50 A+39 B+2 (5 A+9 B) \cos (c+d x)+3 B \cos (2 (c+d x))) \tan \left (\frac {1}{2} (c+d x)\right )}{15 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 85, normalized size = 0.84
method | result | size |
default | \(\frac {4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (6 B \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-5 A -15 B \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 A +15 B \right ) \sqrt {2}}{15 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 93, normalized size = 0.92 \begin {gather*} \frac {10 \, {\left (\sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 9 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} A \sqrt {a} + 3 \, {\left (\sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 5 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 20 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} B \sqrt {a}}{30 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 69, normalized size = 0.68 \begin {gather*} \frac {2 \, {\left (3 \, B a \cos \left (d x + c\right )^{2} + {\left (5 \, A + 9 \, B\right )} a \cos \left (d x + c\right ) + {\left (25 \, A + 18 \, B\right )} a\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{15 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \end {gather*}
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 \begin {gather*} \int \left (a \left (\cos {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}} \left (A + B \cos {\left (c + d x \right )}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.50, size = 115, normalized size = 1.14 \begin {gather*} \frac {\sqrt {2} {\left (3 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 5 \, {\left (2 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 3 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 30 \, {\left (3 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 2 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{30 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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