Optimal. Leaf size=127 \[ -\frac {256 a^4 \cos ^5(c+d x)}{1155 d (a+a \sin (c+d x))^{5/2}}-\frac {64 a^3 \cos ^5(c+d x)}{231 d (a+a \sin (c+d x))^{3/2}}-\frac {8 a^2 \cos ^5(c+d x)}{33 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d} \]
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Rubi [A]
time = 0.16, 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 ^5(c+d x)}{1155 d (a \sin (c+d x)+a)^{5/2}}-\frac {64 a^3 \cos ^5(c+d x)}{231 d (a \sin (c+d x)+a)^{3/2}}-\frac {8 a^2 \cos ^5(c+d x)}{33 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{11 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2752
Rule 2753
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=-\frac {2 a \cos ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}+\frac {1}{11} (12 a) \int \cos ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {8 a^2 \cos ^5(c+d x)}{33 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}+\frac {1}{33} \left (32 a^2\right ) \int \frac {\cos ^4(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=-\frac {64 a^3 \cos ^5(c+d x)}{231 d (a+a \sin (c+d x))^{3/2}}-\frac {8 a^2 \cos ^5(c+d x)}{33 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}+\frac {1}{231} \left (128 a^3\right ) \int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac {256 a^4 \cos ^5(c+d x)}{1155 d (a+a \sin (c+d x))^{5/2}}-\frac {64 a^3 \cos ^5(c+d x)}{231 d (a+a \sin (c+d x))^{3/2}}-\frac {8 a^2 \cos ^5(c+d x)}{33 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 69, normalized size = 0.54 \begin {gather*} -\frac {2 \cos ^5(c+d x) (a (1+\sin (c+d x)))^{3/2} \left (533+755 \sin (c+d x)+455 \sin ^2(c+d x)+105 \sin ^3(c+d x)\right )}{1155 d (1+\sin (c+d x))^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 77, normalized size = 0.61
method | result | size |
default | \(\frac {2 \left (1+\sin \left (d x +c \right )\right ) a^{2} \left (\sin \left (d x +c \right )-1\right )^{3} \left (105 \left (\sin ^{3}\left (d x +c \right )\right )+455 \left (\sin ^{2}\left (d x +c \right )\right )+755 \sin \left (d x +c \right )+533\right )}{1155 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(77\) |
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 = 0.36, size = 166, normalized size = 1.31 \begin {gather*} -\frac {2 \, {\left (105 \, a \cos \left (d x + c\right )^{6} + 245 \, a \cos \left (d x + c\right )^{5} - 20 \, a \cos \left (d x + c\right )^{4} + 32 \, a \cos \left (d x + c\right )^{3} - 64 \, a \cos \left (d x + c\right )^{2} + 256 \, a \cos \left (d x + c\right ) + {\left (105 \, a \cos \left (d x + c\right )^{5} - 140 \, a \cos \left (d x + c\right )^{4} - 160 \, a \cos \left (d x + c\right )^{3} - 192 \, a \cos \left (d x + c\right )^{2} - 256 \, a \cos \left (d x + c\right ) - 512 \, a\right )} \sin \left (d x + c\right ) + 512 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{1155 \, {\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 {3}{2}} \cos ^{4}{\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 = 5.42, size = 132, normalized size = 1.04 \begin {gather*} -\frac {64 \, \sqrt {2} {\left (105 \, 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} - 385 \, 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} + 495 \, 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} - 231 \, 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 )^{5}\right )} \sqrt {a}}{1155 \, 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 )}^4\,{\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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