Optimal. Leaf size=203 \[ -\frac {284 a^3 \cos (c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}-\frac {710 a^3 \cos (c+d x) \sin ^3(c+d x)}{693 d \sqrt {a+a \sin (c+d x)}}-\frac {46 a^3 \cos (c+d x) \sin ^4(c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}+\frac {568 a^2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{693 d}-\frac {2 a^2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}-\frac {284 a \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{231 d} \]
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Rubi [A]
time = 0.24, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2842, 3060,
2849, 2838, 2830, 2725} \begin {gather*} -\frac {46 a^3 \sin ^4(c+d x) \cos (c+d x)}{99 d \sqrt {a \sin (c+d x)+a}}-\frac {710 a^3 \sin ^3(c+d x) \cos (c+d x)}{693 d \sqrt {a \sin (c+d x)+a}}-\frac {284 a^3 \cos (c+d x)}{99 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a^2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{11 d}+\frac {568 a^2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{693 d}-\frac {284 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{231 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2725
Rule 2830
Rule 2838
Rule 2842
Rule 2849
Rule 3060
Rubi steps
\begin {align*} \int \sin ^3(c+d x) (a+a \sin (c+d x))^{5/2} \, dx &=-\frac {2 a^2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}+\frac {2}{11} \int \sin ^3(c+d x) \sqrt {a+a \sin (c+d x)} \left (\frac {19 a^2}{2}+\frac {23}{2} a^2 \sin (c+d x)\right ) \, dx\\ &=-\frac {46 a^3 \cos (c+d x) \sin ^4(c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a^2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}+\frac {1}{99} \left (355 a^2\right ) \int \sin ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {710 a^3 \cos (c+d x) \sin ^3(c+d x)}{693 d \sqrt {a+a \sin (c+d x)}}-\frac {46 a^3 \cos (c+d x) \sin ^4(c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a^2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}+\frac {1}{231} \left (710 a^2\right ) \int \sin ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {710 a^3 \cos (c+d x) \sin ^3(c+d x)}{693 d \sqrt {a+a \sin (c+d x)}}-\frac {46 a^3 \cos (c+d x) \sin ^4(c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a^2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}-\frac {284 a \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{231 d}+\frac {1}{231} (284 a) \int \left (\frac {3 a}{2}-a \sin (c+d x)\right ) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {710 a^3 \cos (c+d x) \sin ^3(c+d x)}{693 d \sqrt {a+a \sin (c+d x)}}-\frac {46 a^3 \cos (c+d x) \sin ^4(c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}+\frac {568 a^2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{693 d}-\frac {2 a^2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}-\frac {284 a \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{231 d}+\frac {1}{99} \left (142 a^2\right ) \int \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {284 a^3 \cos (c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}-\frac {710 a^3 \cos (c+d x) \sin ^3(c+d x)}{693 d \sqrt {a+a \sin (c+d x)}}-\frac {46 a^3 \cos (c+d x) \sin ^4(c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}+\frac {568 a^2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{693 d}-\frac {2 a^2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}-\frac {284 a \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{231 d}\\ \end {align*}
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Mathematica [A]
time = 0.80, size = 189, normalized size = 0.93 \begin {gather*} -\frac {(a (1+\sin (c+d x)))^{5/2} \left (31878 \cos \left (\frac {1}{2} (c+d x)\right )+8778 \cos \left (\frac {3}{2} (c+d x)\right )-3465 \cos \left (\frac {5}{2} (c+d x)\right )-1287 \cos \left (\frac {7}{2} (c+d x)\right )+385 \cos \left (\frac {9}{2} (c+d x)\right )+63 \cos \left (\frac {11}{2} (c+d x)\right )-31878 \sin \left (\frac {1}{2} (c+d x)\right )+8778 \sin \left (\frac {3}{2} (c+d x)\right )+3465 \sin \left (\frac {5}{2} (c+d x)\right )-1287 \sin \left (\frac {7}{2} (c+d x)\right )-385 \sin \left (\frac {9}{2} (c+d x)\right )+63 \sin \left (\frac {11}{2} (c+d x)\right )\right )}{11088 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.86, size = 95, normalized size = 0.47
method | result | size |
default | \(\frac {2 \left (1+\sin \left (d x +c \right )\right ) a^{3} \left (\sin \left (d x +c \right )-1\right ) \left (63 \left (\sin ^{5}\left (d x +c \right )\right )+224 \left (\sin ^{4}\left (d x +c \right )\right )+355 \left (\sin ^{3}\left (d x +c \right )\right )+426 \left (\sin ^{2}\left (d x +c \right )\right )+568 \sin \left (d x +c \right )+1136\right )}{693 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(95\) |
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 = 0.33, size = 192, normalized size = 0.95 \begin {gather*} -\frac {2 \, {\left (63 \, a^{2} \cos \left (d x + c\right )^{6} + 224 \, a^{2} \cos \left (d x + c\right )^{5} - 320 \, a^{2} \cos \left (d x + c\right )^{4} - 874 \, a^{2} \cos \left (d x + c\right )^{3} + 593 \, a^{2} \cos \left (d x + c\right )^{2} + 1786 \, a^{2} \cos \left (d x + c\right ) + 800 \, a^{2} + {\left (63 \, a^{2} \cos \left (d x + c\right )^{5} - 161 \, a^{2} \cos \left (d x + c\right )^{4} - 481 \, a^{2} \cos \left (d x + c\right )^{3} + 393 \, a^{2} \cos \left (d x + c\right )^{2} + 986 \, a^{2} \cos \left (d x + c\right ) - 800 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{693 \, {\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 = 192, normalized size = 0.95 \begin {gather*} \frac {\sqrt {2} {\left (31878 \, a^{2} \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 ) + 8778 \, a^{2} \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 ) + 3465 \, a^{2} \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 ) + 1287 \, a^{2} \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 ) + 385 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 63 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {11}{4} \, \pi + \frac {11}{2} \, d x + \frac {11}{2} \, c\right )\right )} \sqrt {a}}{11088 \, 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 = -0.00 \begin {gather*} \int {\sin \left (c+d\,x\right )}^3\,{\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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