Optimal. Leaf size=116 \[ \frac {152 a^2 \sin (c+d x)}{105 d \sqrt {a \cos (c+d x)+a}}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 a d}-\frac {4 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac {38 a \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{105 d} \]
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Rubi [A] time = 0.14, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2759, 2751, 2647, 2646} \[ \frac {152 a^2 \sin (c+d x)}{105 d \sqrt {a \cos (c+d x)+a}}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 a d}-\frac {4 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac {38 a \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{105 d} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2647
Rule 2751
Rule 2759
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \, dx &=\frac {2 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 a d}+\frac {2 \int \left (\frac {5 a}{2}-a \cos (c+d x)\right ) (a+a \cos (c+d x))^{3/2} \, dx}{7 a}\\ &=-\frac {4 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac {2 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 a d}+\frac {19}{35} \int (a+a \cos (c+d x))^{3/2} \, dx\\ &=\frac {38 a \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{105 d}-\frac {4 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac {2 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 a d}+\frac {1}{105} (76 a) \int \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {152 a^2 \sin (c+d x)}{105 d \sqrt {a+a \cos (c+d x)}}+\frac {38 a \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{105 d}-\frac {4 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac {2 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 a d}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 81, normalized size = 0.70 \[ \frac {a \left (735 \sin \left (\frac {1}{2} (c+d x)\right )+175 \sin \left (\frac {3}{2} (c+d x)\right )+63 \sin \left (\frac {5}{2} (c+d x)\right )+15 \sin \left (\frac {7}{2} (c+d x)\right )\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cos (c+d x)+1)}}{420 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 67, normalized size = 0.58 \[ \frac {2 \, {\left (15 \, a \cos \left (d x + c\right )^{3} + 39 \, a \cos \left (d x + c\right )^{2} + 52 \, a \cos \left (d x + c\right ) + 104 \, a\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{105 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 109, normalized size = 0.94 \[ \frac {1}{420} \, \sqrt {2} {\left (\frac {15 \, a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )}{d} + \frac {63 \, a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )}{d} + \frac {175 \, a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )}{d} + \frac {735 \, a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 86, normalized size = 0.74 \[ \frac {4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (60 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+19 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+38\right ) \sqrt {2}}{105 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 69, normalized size = 0.59 \[ \frac {{\left (15 \, \sqrt {2} a \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 63 \, \sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 175 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 735 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{420 \, d} \]
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 \[ \int {\cos \left (c+d\,x\right )}^2\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2} \,d x \]
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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