Optimal. Leaf size=30 \[ -\frac {2 a \cos ^5(c+d x)}{5 d (a+a \sin (c+d x))^{5/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2752}
\begin {gather*} -\frac {2 a \cos ^5(c+d x)}{5 d (a \sin (c+d x)+a)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2752
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=-\frac {2 a \cos ^5(c+d x)}{5 d (a+a \sin (c+d x))^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 42, normalized size = 1.40 \begin {gather*} -\frac {2 \cos ^5(c+d x) \sqrt {a (1+\sin (c+d x))}}{5 a^2 d (1+\sin (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 47, normalized size = 1.57
method | result | size |
default | \(\frac {2 \left (1+\sin \left (d x +c \right )\right ) \left (\sin \left (d x +c \right )-1\right )^{3}}{5 a \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(47\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 98 vs.
\(2 (26) = 52\).
time = 0.34, size = 98, normalized size = 3.27 \begin {gather*} \frac {2 \, {\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) - 4\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 4\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{5 \, {\left (a^{2} d \cos \left (d x + c\right ) + a^{2} d \sin \left (d x + c\right ) + a^{2} 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 \frac {\cos ^{4}{\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.32, size = 40, normalized size = 1.33 \begin {gather*} \frac {8 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}{5 \, a^{\frac {3}{2}} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \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 = -0.03 \begin {gather*} \int \frac {{\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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