Optimal. Leaf size=30 \[ -\frac {2 a \cos ^7(c+d x)}{7 d (a+a \sin (c+d x))^{7/2}} \]
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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 ^7(c+d x)}{7 d (a \sin (c+d x)+a)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2752
Rubi steps
\begin {align*} \int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=-\frac {2 a \cos ^7(c+d x)}{7 d (a+a \sin (c+d x))^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 42, normalized size = 1.40 \begin {gather*} -\frac {2 \cos ^7(c+d x) \sqrt {a (1+\sin (c+d x))}}{7 a^3 d (1+\sin (c+d x))^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 47, normalized size = 1.57
method | result | size |
default | \(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) \left (\sin \left (d x +c \right )-1\right )^{4}}{7 a^{2} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(47\) |
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. 117 vs.
\(2 (26) = 52\).
time = 0.36, size = 117, normalized size = 3.90 \begin {gather*} -\frac {2 \, {\left (\cos \left (d x + c\right )^{4} - 3 \, \cos \left (d x + c\right )^{3} - 8 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{3} + 4 \, \cos \left (d x + c\right )^{2} - 4 \, \cos \left (d x + c\right ) - 8\right )} \sin \left (d x + c\right ) + 4 \, \cos \left (d x + c\right ) + 8\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{7 \, {\left (a^{3} d \cos \left (d x + c\right ) + a^{3} d \sin \left (d x + c\right ) + a^{3} 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 = 4.79, size = 40, normalized size = 1.33 \begin {gather*} \frac {16 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}{7 \, a^{\frac {5}{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 )}^6}{{\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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