Optimal. Leaf size=122 \[ -\frac {4 a \cos (c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos (c+d x) \sin ^3(c+d x)}{7 d \sqrt {a+a \sin (c+d x)}}+\frac {8 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{35 d}-\frac {12 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 a d} \]
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Rubi [A]
time = 0.12, antiderivative size = 122, 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 = {2849, 2838,
2830, 2725} \begin {gather*} -\frac {2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}-\frac {12 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 a d}+\frac {8 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{35 d}-\frac {4 a \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2725
Rule 2830
Rule 2838
Rule 2849
Rubi steps
\begin {align*} \int \sin ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx &=-\frac {2 a \cos (c+d x) \sin ^3(c+d x)}{7 d \sqrt {a+a \sin (c+d x)}}+\frac {6}{7} \int \sin ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {2 a \cos (c+d x) \sin ^3(c+d x)}{7 d \sqrt {a+a \sin (c+d x)}}-\frac {12 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 a d}+\frac {12 \int \left (\frac {3 a}{2}-a \sin (c+d x)\right ) \sqrt {a+a \sin (c+d x)} \, dx}{35 a}\\ &=-\frac {2 a \cos (c+d x) \sin ^3(c+d x)}{7 d \sqrt {a+a \sin (c+d x)}}+\frac {8 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{35 d}-\frac {12 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 a d}+\frac {2}{5} \int \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {4 a \cos (c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos (c+d x) \sin ^3(c+d x)}{7 d \sqrt {a+a \sin (c+d x)}}+\frac {8 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{35 d}-\frac {12 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 a d}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 141, normalized size = 1.16 \begin {gather*} \frac {\sqrt {a (1+\sin (c+d x))} \left (-105 \cos \left (\frac {1}{2} (c+d x)\right )-35 \cos \left (\frac {3}{2} (c+d x)\right )+7 \cos \left (\frac {5}{2} (c+d x)\right )+5 \cos \left (\frac {7}{2} (c+d x)\right )+105 \sin \left (\frac {1}{2} (c+d x)\right )-35 \sin \left (\frac {3}{2} (c+d x)\right )-7 \sin \left (\frac {5}{2} (c+d x)\right )+5 \sin \left (\frac {7}{2} (c+d x)\right )\right )}{140 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.73, size = 73, normalized size = 0.60
method | result | size |
default | \(\frac {2 \left (1+\sin \left (d x +c \right )\right ) a \left (\sin \left (d x +c \right )-1\right ) \left (5 \left (\sin ^{3}\left (d x +c \right )\right )+6 \left (\sin ^{2}\left (d x +c \right )\right )+8 \sin \left (d x +c \right )+16\right )}{35 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(73\) |
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 [A]
time = 0.34, size = 111, normalized size = 0.91 \begin {gather*} \frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{4} + 6 \, \cos \left (d x + c\right )^{3} - 12 \, \cos \left (d x + c\right )^{2} + {\left (5 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} - 13 \, \cos \left (d x + c\right ) + 9\right )} \sin \left (d x + c\right ) - 22 \, \cos \left (d x + c\right ) - 9\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{35 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + 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 = 0.50, size = 120, normalized size = 0.98 \begin {gather*} \frac {\sqrt {2} {\left (105 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 35 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 7 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 5 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, d x + \frac {7}{2} \, c\right )\right )} \sqrt {a}}{140 \, d} \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 = -0.01 \begin {gather*} \int {\sin \left (c+d\,x\right )}^3\,\sqrt {a+a\,\sin \left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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