Optimal. Leaf size=127 \[ -\frac {256 a^4 \cos ^7(c+d x)}{3003 d (a+a \sin (c+d x))^{7/2}}-\frac {64 a^3 \cos ^7(c+d x)}{429 d (a+a \sin (c+d x))^{5/2}}-\frac {24 a^2 \cos ^7(c+d x)}{143 d (a+a \sin (c+d x))^{3/2}}-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a+a \sin (c+d x)}} \]
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Rubi [A]
time = 0.17, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2753, 2752}
\begin {gather*} -\frac {256 a^4 \cos ^7(c+d x)}{3003 d (a \sin (c+d x)+a)^{7/2}}-\frac {64 a^3 \cos ^7(c+d x)}{429 d (a \sin (c+d x)+a)^{5/2}}-\frac {24 a^2 \cos ^7(c+d x)}{143 d (a \sin (c+d x)+a)^{3/2}}-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a \sin (c+d x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2752
Rule 2753
Rubi steps
\begin {align*} \int \cos ^6(c+d x) \sqrt {a+a \sin (c+d x)} \, dx &=-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a+a \sin (c+d x)}}+\frac {1}{13} (12 a) \int \frac {\cos ^6(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=-\frac {24 a^2 \cos ^7(c+d x)}{143 d (a+a \sin (c+d x))^{3/2}}-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a+a \sin (c+d x)}}+\frac {1}{143} \left (96 a^2\right ) \int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac {64 a^3 \cos ^7(c+d x)}{429 d (a+a \sin (c+d x))^{5/2}}-\frac {24 a^2 \cos ^7(c+d x)}{143 d (a+a \sin (c+d x))^{3/2}}-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a+a \sin (c+d x)}}+\frac {1}{429} \left (128 a^3\right ) \int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx\\ &=-\frac {256 a^4 \cos ^7(c+d x)}{3003 d (a+a \sin (c+d x))^{7/2}}-\frac {64 a^3 \cos ^7(c+d x)}{429 d (a+a \sin (c+d x))^{5/2}}-\frac {24 a^2 \cos ^7(c+d x)}{143 d (a+a \sin (c+d x))^{3/2}}-\frac {2 a \cos ^7(c+d x)}{13 d \sqrt {a+a \sin (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 2.37, size = 99, normalized size = 0.78 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^7 \sqrt {a (1+\sin (c+d x))} (-5230+1890 \cos (2 (c+d x))-6377 \sin (c+d x)+231 \sin (3 (c+d x)))}{6006 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 75, normalized size = 0.59
method | result | size |
default | \(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) a \left (\sin \left (d x +c \right )-1\right )^{4} \left (231 \left (\sin ^{3}\left (d x +c \right )\right )+945 \left (\sin ^{2}\left (d x +c \right )\right )+1421 \sin \left (d x +c \right )+835\right )}{3003 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(75\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 172, normalized size = 1.35 \begin {gather*} -\frac {2 \, {\left (231 \, \cos \left (d x + c\right )^{7} - 21 \, \cos \left (d x + c\right )^{6} + 28 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{4} + 64 \, \cos \left (d x + c\right )^{3} - 128 \, \cos \left (d x + c\right )^{2} - {\left (231 \, \cos \left (d x + c\right )^{6} + 252 \, \cos \left (d x + c\right )^{5} + 280 \, \cos \left (d x + c\right )^{4} + 320 \, \cos \left (d x + c\right )^{3} + 384 \, \cos \left (d x + c\right )^{2} + 512 \, \cos \left (d x + c\right ) + 1024\right )} \sin \left (d x + c\right ) + 512 \, \cos \left (d x + c\right ) + 1024\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{3003 \, {\left (d \cos \left (d x + c\right ) + d \sin \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 \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \cos ^{6}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.71, size = 201, normalized size = 1.58 \begin {gather*} \frac {\sqrt {2} {\left (60060 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15015 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right ) - 9009 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 2574 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 2002 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, d x + \frac {9}{2} \, c\right ) - 273 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {11}{4} \, \pi + \frac {11}{2} \, d x + \frac {11}{2} \, c\right ) - 231 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {13}{4} \, \pi + \frac {13}{2} \, d x + \frac {13}{2} \, c\right )\right )} \sqrt {a}}{96096 \, 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 )}^6\,\sqrt {a+a\,\sin \left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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