Optimal. Leaf size=223 \[ -\frac {131072 a^7 \cos ^7(c+d x)}{969969 d (a \sin (c+d x)+a)^{7/2}}-\frac {32768 a^6 \cos ^7(c+d x)}{138567 d (a \sin (c+d x)+a)^{5/2}}-\frac {12288 a^5 \cos ^7(c+d x)}{46189 d (a \sin (c+d x)+a)^{3/2}}-\frac {1024 a^4 \cos ^7(c+d x)}{4199 d \sqrt {a \sin (c+d x)+a}}-\frac {64 a^3 \cos ^7(c+d x) \sqrt {a \sin (c+d x)+a}}{323 d}-\frac {48 a^2 \cos ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{323 d}-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{19 d} \]
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Rubi [A] time = 0.43, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2674, 2673} \[ -\frac {48 a^2 \cos ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{323 d}-\frac {64 a^3 \cos ^7(c+d x) \sqrt {a \sin (c+d x)+a}}{323 d}-\frac {1024 a^4 \cos ^7(c+d x)}{4199 d \sqrt {a \sin (c+d x)+a}}-\frac {12288 a^5 \cos ^7(c+d x)}{46189 d (a \sin (c+d x)+a)^{3/2}}-\frac {32768 a^6 \cos ^7(c+d x)}{138567 d (a \sin (c+d x)+a)^{5/2}}-\frac {131072 a^7 \cos ^7(c+d x)}{969969 d (a \sin (c+d x)+a)^{7/2}}-\frac {2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{19 d} \]
Antiderivative was successfully verified.
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Rule 2673
Rule 2674
Rubi steps
\begin {align*} \int \cos ^6(c+d x) (a+a \sin (c+d x))^{7/2} \, dx &=-\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*}
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Mathematica [A] time = 0.55, size = 102, normalized size = 0.46 \[ -\frac {2 a^3 \left (51051 \sin ^6(c+d x)+378378 \sin ^5(c+d x)+1222221 \sin ^4(c+d x)+2244396 \sin ^3(c+d x)+2546901 \sin ^2(c+d x)+1778602 \sin (c+d x)+646739\right ) \cos ^7(c+d x) \sqrt {a (\sin (c+d x)+1)}}{969969 d (\sin (c+d x)+1)^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 296, normalized size = 1.33 \[ \frac {2 \, {\left (51051 \, a^{3} \cos \left (d x + c\right )^{10} + 225225 \, a^{3} \cos \left (d x + c\right )^{9} - 270270 \, a^{3} \cos \left (d x + c\right )^{8} - 562716 \, a^{3} \cos \left (d x + c\right )^{7} + 10752 \, a^{3} \cos \left (d x + c\right )^{6} - 14336 \, a^{3} \cos \left (d x + c\right )^{5} + 20480 \, a^{3} \cos \left (d x + c\right )^{4} - 32768 \, a^{3} \cos \left (d x + c\right )^{3} + 65536 \, a^{3} \cos \left (d x + c\right )^{2} - 262144 \, a^{3} \cos \left (d x + c\right ) - 524288 \, a^{3} + {\left (51051 \, a^{3} \cos \left (d x + c\right )^{9} - 174174 \, a^{3} \cos \left (d x + c\right )^{8} - 444444 \, a^{3} \cos \left (d x + c\right )^{7} + 118272 \, a^{3} \cos \left (d x + c\right )^{6} + 129024 \, a^{3} \cos \left (d x + c\right )^{5} + 143360 \, a^{3} \cos \left (d x + c\right )^{4} + 163840 \, a^{3} \cos \left (d x + c\right )^{3} + 196608 \, a^{3} \cos \left (d x + c\right )^{2} + 262144 \, a^{3} \cos \left (d x + c\right ) + 524288 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{969969 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 8.97, size = 636, normalized size = 2.85 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 107, normalized size = 0.48 \[ -\frac {2 \left (1+\sin \left (d x +c \right )\right ) a^{4} \left (\sin \left (d x +c \right )-1\right )^{4} \left (51051 \left (\sin ^{6}\left (d x +c \right )\right )+378378 \left (\sin ^{5}\left (d x +c \right )\right )+1222221 \left (\sin ^{4}\left (d x +c \right )\right )+2244396 \left (\sin ^{3}\left (d x +c \right )\right )+2546901 \left (\sin ^{2}\left (d x +c \right )\right )+1778602 \sin \left (d x +c \right )+646739\right )}{969969 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
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 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \cos \left (d x + c\right )^{6}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\cos \left (c+d\,x\right )}^6\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{7/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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