Optimal. Leaf size=71 \[ \frac {2 (a \sin (c+d x)+a)^{5/2}}{5 a^5 d}-\frac {8 (a \sin (c+d x)+a)^{3/2}}{3 a^4 d}+\frac {8 \sqrt {a \sin (c+d x)+a}}{a^3 d} \]
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Rubi [A] time = 0.07, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2667, 43} \[ \frac {2 (a \sin (c+d x)+a)^{5/2}}{5 a^5 d}-\frac {8 (a \sin (c+d x)+a)^{3/2}}{3 a^4 d}+\frac {8 \sqrt {a \sin (c+d x)+a}}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^2}{\sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {4 a^2}{\sqrt {a+x}}-4 a \sqrt {a+x}+(a+x)^{3/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {8 \sqrt {a+a \sin (c+d x)}}{a^3 d}-\frac {8 (a+a \sin (c+d x))^{3/2}}{3 a^4 d}+\frac {2 (a+a \sin (c+d x))^{5/2}}{5 a^5 d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 44, normalized size = 0.62 \[ \frac {2 \left (3 \sin ^2(c+d x)-14 \sin (c+d x)+43\right ) \sqrt {a (\sin (c+d x)+1)}}{15 a^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 40, normalized size = 0.56 \[ -\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} + 14 \, \sin \left (d x + c\right ) - 46\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{15 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 5.24, size = 172, normalized size = 2.42 \[ \frac {2 \, {\left ({\left ({\left ({\left ({\left (\frac {43 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + \frac {15}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {70}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {70}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {15}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {43}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}\right )}}{15 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {5}{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 41, normalized size = 0.58 \[ -\frac {2 \sqrt {a +a \sin \left (d x +c \right )}\, \left (3 \left (\cos ^{2}\left (d x +c \right )\right )+14 \sin \left (d x +c \right )-46\right )}{15 a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 55, normalized size = 0.77 \[ \frac {2 \, {\left (3 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} - 20 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a + 60 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{2}\right )}}{15 \, a^{5} d} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^5}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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