Optimal. Leaf size=119 \[ \frac {256 a^4 \sin (c+d x)}{35 d \sqrt {a \cos (c+d x)+a}}+\frac {64 a^3 \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{35 d}+\frac {24 a^2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac {2 a \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d} \]
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Rubi [A] time = 0.07, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2647, 2646} \[ \frac {256 a^4 \sin (c+d x)}{35 d \sqrt {a \cos (c+d x)+a}}+\frac {64 a^3 \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{35 d}+\frac {24 a^2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac {2 a \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2647
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^{7/2} \, dx &=\frac {2 a (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {1}{7} (12 a) \int (a+a \cos (c+d x))^{5/2} \, dx\\ &=\frac {24 a^2 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac {2 a (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {1}{35} \left (96 a^2\right ) \int (a+a \cos (c+d x))^{3/2} \, dx\\ &=\frac {64 a^3 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{35 d}+\frac {24 a^2 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac {2 a (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {1}{35} \left (128 a^3\right ) \int \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {256 a^4 \sin (c+d x)}{35 d \sqrt {a+a \cos (c+d x)}}+\frac {64 a^3 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{35 d}+\frac {24 a^2 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac {2 a (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 83, normalized size = 0.70 \[ \frac {a^3 \left (1225 \sin \left (\frac {1}{2} (c+d x)\right )+245 \sin \left (\frac {3}{2} (c+d x)\right )+49 \sin \left (\frac {5}{2} (c+d x)\right )+5 \sin \left (\frac {7}{2} (c+d x)\right )\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cos (c+d x)+1)}}{140 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 75, normalized size = 0.63 \[ \frac {2 \, {\left (5 \, a^{3} \cos \left (d x + c\right )^{3} + 27 \, a^{3} \cos \left (d x + c\right )^{2} + 71 \, a^{3} \cos \left (d x + c\right ) + 177 \, a^{3}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{35 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.80, size = 117, normalized size = 0.98 \[ \frac {1}{140} \, \sqrt {2} {\left (\frac {5 \, a^{3} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )}{d} + \frac {49 \, a^{3} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )}{d} + \frac {245 \, a^{3} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )}{d} + \frac {1225 \, a^{3} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 86, normalized size = 0.72 \[ \frac {16 a^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (5 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+16\right ) \sqrt {2}}{35 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 77, normalized size = 0.65 \[ \frac {{\left (5 \, \sqrt {2} a^{3} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 49 \, \sqrt {2} a^{3} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 245 \, \sqrt {2} a^{3} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 1225 \, \sqrt {2} a^{3} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{140 \, d} \]
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 \[ \int {\left (a+a\,\cos \left (c+d\,x\right )\right )}^{7/2} \,d x \]
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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