Optimal. Leaf size=159 \[ -\frac {4096 a^5 \cos ^7(c+d x)}{45045 d (a+a \sin (c+d x))^{7/2}}-\frac {1024 a^4 \cos ^7(c+d x)}{6435 d (a+a \sin (c+d x))^{5/2}}-\frac {128 a^3 \cos ^7(c+d x)}{715 d (a+a \sin (c+d x))^{3/2}}-\frac {32 a^2 \cos ^7(c+d x)}{195 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)}}{15 d} \]
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Rubi [A]
time = 0.21, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps
used = 5, 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 {4096 a^5 \cos ^7(c+d x)}{45045 d (a \sin (c+d x)+a)^{7/2}}-\frac {1024 a^4 \cos ^7(c+d x)}{6435 d (a \sin (c+d x)+a)^{5/2}}-\frac {128 a^3 \cos ^7(c+d x)}{715 d (a \sin (c+d x)+a)^{3/2}}-\frac {32 a^2 \cos ^7(c+d x)}{195 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos ^7(c+d x) \sqrt {a \sin (c+d x)+a}}{15 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2752
Rule 2753
Rubi steps
\begin {align*} \int \cos ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=-\frac {2 a \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)}}{15 d}+\frac {1}{15} (16 a) \int \cos ^6(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {32 a^2 \cos ^7(c+d x)}{195 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)}}{15 d}+\frac {1}{65} \left (64 a^2\right ) \int \frac {\cos ^6(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=-\frac {128 a^3 \cos ^7(c+d x)}{715 d (a+a \sin (c+d x))^{3/2}}-\frac {32 a^2 \cos ^7(c+d x)}{195 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)}}{15 d}+\frac {1}{715} \left (512 a^3\right ) \int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac {1024 a^4 \cos ^7(c+d x)}{6435 d (a+a \sin (c+d x))^{5/2}}-\frac {128 a^3 \cos ^7(c+d x)}{715 d (a+a \sin (c+d x))^{3/2}}-\frac {32 a^2 \cos ^7(c+d x)}{195 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)}}{15 d}+\frac {\left (2048 a^4\right ) \int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx}{6435}\\ &=-\frac {4096 a^5 \cos ^7(c+d x)}{45045 d (a+a \sin (c+d x))^{7/2}}-\frac {1024 a^4 \cos ^7(c+d x)}{6435 d (a+a \sin (c+d x))^{5/2}}-\frac {128 a^3 \cos ^7(c+d x)}{715 d (a+a \sin (c+d x))^{3/2}}-\frac {32 a^2 \cos ^7(c+d x)}{195 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)}}{15 d}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 79, normalized size = 0.50 \begin {gather*} -\frac {2 \cos ^7(c+d x) (a (1+\sin (c+d x)))^{3/2} \left (16363+34748 \sin (c+d x)+33138 \sin ^2(c+d x)+15708 \sin ^3(c+d x)+3003 \sin ^4(c+d x)\right )}{45045 d (1+\sin (c+d x))^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.49, size = 87, normalized size = 0.55
method | result | size |
default | \(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) a^{2} \left (\sin \left (d x +c \right )-1\right )^{4} \left (3003 \left (\sin ^{4}\left (d x +c \right )\right )+15708 \left (\sin ^{3}\left (d x +c \right )\right )+33138 \left (\sin ^{2}\left (d x +c \right )\right )+34748 \sin \left (d x +c \right )+16363\right )}{45045 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(87\) |
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 = 0.35, size = 210, normalized size = 1.32 \begin {gather*} -\frac {2 \, {\left (3003 \, a \cos \left (d x + c\right )^{8} + 6699 \, a \cos \left (d x + c\right )^{7} - 336 \, a \cos \left (d x + c\right )^{6} + 448 \, a \cos \left (d x + c\right )^{5} - 640 \, a \cos \left (d x + c\right )^{4} + 1024 \, a \cos \left (d x + c\right )^{3} - 2048 \, a \cos \left (d x + c\right )^{2} + 8192 \, a \cos \left (d x + c\right ) + {\left (3003 \, a \cos \left (d x + c\right )^{7} - 3696 \, a \cos \left (d x + c\right )^{6} - 4032 \, a \cos \left (d x + c\right )^{5} - 4480 \, a \cos \left (d x + c\right )^{4} - 5120 \, a \cos \left (d x + c\right )^{3} - 6144 \, a \cos \left (d x + c\right )^{2} - 8192 \, a \cos \left (d x + c\right ) - 16384 \, a\right )} \sin \left (d x + c\right ) + 16384 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{45045 \, {\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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.59, size = 162, normalized size = 1.02 \begin {gather*} \frac {256 \, \sqrt {2} {\left (3003 \, a \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} - 13860 \, a \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} + 24570 \, a \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} - 20020 \, a \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} + 6435 \, a \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}}{45045 \, 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\,{\left (a+a\,\sin \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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