Optimal. Leaf size=127 \[ -\frac {256 a^4 \cos ^3(c+d x)}{315 d (a+a \sin (c+d x))^{3/2}}-\frac {64 a^3 \cos ^3(c+d x)}{105 d \sqrt {a+a \sin (c+d x)}}-\frac {8 a^2 \cos ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{21 d}-\frac {2 a \cos ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{9 d} \]
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Rubi [A]
time = 0.15, 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 ^3(c+d x)}{315 d (a \sin (c+d x)+a)^{3/2}}-\frac {64 a^3 \cos ^3(c+d x)}{105 d \sqrt {a \sin (c+d x)+a}}-\frac {8 a^2 \cos ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{21 d}-\frac {2 a \cos ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{9 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2752
Rule 2753
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+a \sin (c+d x))^{5/2} \, dx &=-\frac {2 a \cos ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{9 d}+\frac {1}{3} (4 a) \int \cos ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\\ &=-\frac {8 a^2 \cos ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{21 d}-\frac {2 a \cos ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{9 d}+\frac {1}{21} \left (32 a^2\right ) \int \cos ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {64 a^3 \cos ^3(c+d x)}{105 d \sqrt {a+a \sin (c+d x)}}-\frac {8 a^2 \cos ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{21 d}-\frac {2 a \cos ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{9 d}+\frac {1}{105} \left (128 a^3\right ) \int \frac {\cos ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=-\frac {256 a^4 \cos ^3(c+d x)}{315 d (a+a \sin (c+d x))^{3/2}}-\frac {64 a^3 \cos ^3(c+d x)}{105 d \sqrt {a+a \sin (c+d x)}}-\frac {8 a^2 \cos ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{21 d}-\frac {2 a \cos ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{9 d}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 69, normalized size = 0.54 \begin {gather*} -\frac {2 \cos ^3(c+d x) (a (1+\sin (c+d x)))^{5/2} \left (319+321 \sin (c+d x)+165 \sin ^2(c+d x)+35 \sin ^3(c+d x)\right )}{315 d (1+\sin (c+d x))^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 77, normalized size = 0.61
method | result | size |
default | \(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) a^{3} \left (\sin \left (d x +c \right )-1\right )^{2} \left (35 \left (\sin ^{3}\left (d x +c \right )\right )+165 \left (\sin ^{2}\left (d x +c \right )\right )+321 \sin \left (d x +c \right )+319\right )}{315 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(77\) |
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 = 167, normalized size = 1.31 \begin {gather*} \frac {2 \, {\left (35 \, a^{2} \cos \left (d x + c\right )^{5} - 95 \, a^{2} \cos \left (d x + c\right )^{4} - 226 \, a^{2} \cos \left (d x + c\right )^{3} + 32 \, a^{2} \cos \left (d x + c\right )^{2} - 128 \, a^{2} \cos \left (d x + c\right ) - 256 \, a^{2} - {\left (35 \, a^{2} \cos \left (d x + c\right )^{4} + 130 \, a^{2} \cos \left (d x + c\right )^{3} - 96 \, a^{2} \cos \left (d x + c\right )^{2} - 128 \, a^{2} \cos \left (d x + c\right ) - 256 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{315 \, {\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 \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}} \cos ^{2}{\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 = 4.91, size = 140, normalized size = 1.10 \begin {gather*} -\frac {32 \, \sqrt {2} {\left (35 \, a^{2} \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} - 135 \, a^{2} \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} + 189 \, a^{2} \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 )^{5} - 105 \, a^{2} \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 )^{3}\right )} \sqrt {a}}{315 \, 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 )}^2\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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