Integrand size = 23, antiderivative size = 223 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=-\frac {131072 a^7 \cos ^7(c+d x)}{969969 d (a+a \sin (c+d x))^{7/2}}-\frac {32768 a^6 \cos ^7(c+d x)}{138567 d (a+a \sin (c+d x))^{5/2}}-\frac {12288 a^5 \cos ^7(c+d x)}{46189 d (a+a \sin (c+d x))^{3/2}}-\frac {1024 a^4 \cos ^7(c+d x)}{4199 d \sqrt {a+a \sin (c+d x)}}-\frac {64 a^3 \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)}}{323 d}-\frac {48 a^2 \cos ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{323 d}-\frac {2 a \cos ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{19 d} \]
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Time = 0.31 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2753, 2752} \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=-\frac {131072 a^7 \cos ^7(c+d x)}{969969 d (a \sin (c+d x)+a)^{7/2}}-\frac {32768 a^6 \cos ^7(c+d x)}{138567 d (a \sin (c+d x)+a)^{5/2}}-\frac {12288 a^5 \cos ^7(c+d x)}{46189 d (a \sin (c+d x)+a)^{3/2}}-\frac {1024 a^4 \cos ^7(c+d x)}{4199 d \sqrt {a \sin (c+d x)+a}}-\frac {64 a^3 \cos ^7(c+d x) \sqrt {a \sin (c+d x)+a}}{323 d}-\frac {48 a^2 \cos ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{323 d}-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{19 d} \]
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Rule 2752
Rule 2753
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a \cos ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{19 d}+\frac {1}{19} (24 a) \int \cos ^6(c+d x) (a+a \sin (c+d x))^{5/2} \, dx \\ & = -\frac {48 a^2 \cos ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{323 d}-\frac {2 a \cos ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{19 d}+\frac {1}{323} \left (480 a^2\right ) \int \cos ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx \\ & = -\frac {64 a^3 \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)}}{323 d}-\frac {48 a^2 \cos ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{323 d}-\frac {2 a \cos ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{19 d}+\frac {1}{323} \left (512 a^3\right ) \int \cos ^6(c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {1024 a^4 \cos ^7(c+d x)}{4199 d \sqrt {a+a \sin (c+d x)}}-\frac {64 a^3 \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)}}{323 d}-\frac {48 a^2 \cos ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{323 d}-\frac {2 a \cos ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{19 d}+\frac {\left (6144 a^4\right ) \int \frac {\cos ^6(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{4199} \\ & = -\frac {12288 a^5 \cos ^7(c+d x)}{46189 d (a+a \sin (c+d x))^{3/2}}-\frac {1024 a^4 \cos ^7(c+d x)}{4199 d \sqrt {a+a \sin (c+d x)}}-\frac {64 a^3 \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)}}{323 d}-\frac {48 a^2 \cos ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{323 d}-\frac {2 a \cos ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{19 d}+\frac {\left (49152 a^5\right ) \int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx}{46189} \\ & = -\frac {32768 a^6 \cos ^7(c+d x)}{138567 d (a+a \sin (c+d x))^{5/2}}-\frac {12288 a^5 \cos ^7(c+d x)}{46189 d (a+a \sin (c+d x))^{3/2}}-\frac {1024 a^4 \cos ^7(c+d x)}{4199 d \sqrt {a+a \sin (c+d x)}}-\frac {64 a^3 \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)}}{323 d}-\frac {48 a^2 \cos ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{323 d}-\frac {2 a \cos ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{19 d}+\frac {\left (65536 a^6\right ) \int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx}{138567} \\ & = -\frac {131072 a^7 \cos ^7(c+d x)}{969969 d (a+a \sin (c+d x))^{7/2}}-\frac {32768 a^6 \cos ^7(c+d x)}{138567 d (a+a \sin (c+d x))^{5/2}}-\frac {12288 a^5 \cos ^7(c+d x)}{46189 d (a+a \sin (c+d x))^{3/2}}-\frac {1024 a^4 \cos ^7(c+d x)}{4199 d \sqrt {a+a \sin (c+d x)}}-\frac {64 a^3 \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)}}{323 d}-\frac {48 a^2 \cos ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{323 d}-\frac {2 a \cos ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{19 d} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.46 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=-\frac {2 a^3 \cos ^7(c+d x) \sqrt {a (1+\sin (c+d x))} \left (646739+1778602 \sin (c+d x)+2546901 \sin ^2(c+d x)+2244396 \sin ^3(c+d x)+1222221 \sin ^4(c+d x)+378378 \sin ^5(c+d x)+51051 \sin ^6(c+d x)\right )}{969969 d (1+\sin (c+d x))^4} \]
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Time = 153.25 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.48
method | result | size |
default | \(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) a^{4} \left (\sin \left (d x +c \right )-1\right )^{4} \left (51051 \left (\sin ^{6}\left (d x +c \right )\right )+378378 \left (\sin ^{5}\left (d x +c \right )\right )+1222221 \left (\sin ^{4}\left (d x +c \right )\right )+2244396 \left (\sin ^{3}\left (d x +c \right )\right )+2546901 \left (\sin ^{2}\left (d x +c \right )\right )+1778602 \sin \left (d x +c \right )+646739\right )}{969969 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(107\) |
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Time = 0.28 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.33 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {2 \, {\left (51051 \, a^{3} \cos \left (d x + c\right )^{10} + 225225 \, a^{3} \cos \left (d x + c\right )^{9} - 270270 \, a^{3} \cos \left (d x + c\right )^{8} - 562716 \, a^{3} \cos \left (d x + c\right )^{7} + 10752 \, a^{3} \cos \left (d x + c\right )^{6} - 14336 \, a^{3} \cos \left (d x + c\right )^{5} + 20480 \, a^{3} \cos \left (d x + c\right )^{4} - 32768 \, a^{3} \cos \left (d x + c\right )^{3} + 65536 \, a^{3} \cos \left (d x + c\right )^{2} - 262144 \, a^{3} \cos \left (d x + c\right ) - 524288 \, a^{3} + {\left (51051 \, a^{3} \cos \left (d x + c\right )^{9} - 174174 \, a^{3} \cos \left (d x + c\right )^{8} - 444444 \, a^{3} \cos \left (d x + c\right )^{7} + 118272 \, a^{3} \cos \left (d x + c\right )^{6} + 129024 \, a^{3} \cos \left (d x + c\right )^{5} + 143360 \, a^{3} \cos \left (d x + c\right )^{4} + 163840 \, a^{3} \cos \left (d x + c\right )^{3} + 196608 \, a^{3} \cos \left (d x + c\right )^{2} + 262144 \, a^{3} \cos \left (d x + c\right ) + 524288 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{969969 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]
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Timed out. \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\text {Timed out} \]
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\[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \cos \left (d x + c\right )^{6} \,d x } \]
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Time = 0.33 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.06 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {1024 \, \sqrt {2} {\left (51051 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{19} - 342342 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} + 969969 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} - 1492260 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 1322685 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 646646 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 138567 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}\right )} \sqrt {a}}{969969 \, d} \]
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Timed out. \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\int {\cos \left (c+d\,x\right )}^6\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{7/2} \,d x \]
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