Optimal. Leaf size=162 \[ \frac {68 a^2 \sin (c+d x)}{45 d \sqrt {a+a \cos (c+d x)}}+\frac {34 a^2 \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}-\frac {136 a \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac {68 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 d} \]
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Rubi [A]
time = 0.17, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2842, 21, 2849,
2838, 2830, 2725} \begin {gather*} \frac {2 a^2 \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}+\frac {34 a^2 \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt {a \cos (c+d x)+a}}+\frac {68 a^2 \sin (c+d x)}{45 d \sqrt {a \cos (c+d x)+a}}+\frac {68 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}-\frac {136 a \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{315 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 2725
Rule 2830
Rule 2838
Rule 2842
Rule 2849
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \, dx &=\frac {2 a^2 \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}+\frac {2}{9} \int \frac {\cos ^3(c+d x) \left (\frac {17 a^2}{2}+\frac {17}{2} a^2 \cos (c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx\\ &=\frac {2 a^2 \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}+\frac {1}{9} (17 a) \int \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {34 a^2 \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}+\frac {1}{21} (34 a) \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {34 a^2 \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}+\frac {68 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 d}+\frac {68}{105} \int \left (\frac {3 a}{2}-a \cos (c+d x)\right ) \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {34 a^2 \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}-\frac {136 a \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac {68 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 d}+\frac {1}{45} (34 a) \int \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {68 a^2 \sin (c+d x)}{45 d \sqrt {a+a \cos (c+d x)}}+\frac {34 a^2 \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}-\frac {136 a \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac {68 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 d}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 93, normalized size = 0.57 \begin {gather*} \frac {a \sqrt {a (1+\cos (c+d x))} \sec \left (\frac {1}{2} (c+d x)\right ) \left (3780 \sin \left (\frac {1}{2} (c+d x)\right )+1050 \sin \left (\frac {3}{2} (c+d x)\right )+378 \sin \left (\frac {5}{2} (c+d x)\right )+135 \sin \left (\frac {7}{2} (c+d x)\right )+35 \sin \left (\frac {9}{2} (c+d x)\right )\right )}{2520 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 99, normalized size = 0.61
method | result | size |
default | \(\frac {4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (280 \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-220 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+114 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+47 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+94\right ) \sqrt {2}}{315 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(99\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.57, size = 84, normalized size = 0.52 \begin {gather*} \frac {{\left (35 \, \sqrt {2} a \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 135 \, \sqrt {2} a \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 378 \, \sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 1050 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3780 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{2520 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 78, normalized size = 0.48 \begin {gather*} \frac {2 \, {\left (35 \, a \cos \left (d x + c\right )^{4} + 85 \, a \cos \left (d x + c\right )^{3} + 102 \, a \cos \left (d x + c\right )^{2} + 136 \, a \cos \left (d x + c\right ) + 272 \, a\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{315 \, {\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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.82, size = 122, normalized size = 0.75 \begin {gather*} \frac {\sqrt {2} {\left (35 \, a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 135 \, a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 378 \, 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 ) + 1050 \, a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3780 \, a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{2520 \, 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 {\cos \left (c+d\,x\right )}^3\,{\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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