Optimal. Leaf size=97 \[ \frac {16 (a+a \sin (c+d x))^{9/2}}{9 a^4 d}-\frac {24 (a+a \sin (c+d x))^{11/2}}{11 a^5 d}+\frac {12 (a+a \sin (c+d x))^{13/2}}{13 a^6 d}-\frac {2 (a+a \sin (c+d x))^{15/2}}{15 a^7 d} \]
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Rubi [A]
time = 0.06, antiderivative size = 97, 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 = {2746, 45}
\begin {gather*} -\frac {2 (a \sin (c+d x)+a)^{15/2}}{15 a^7 d}+\frac {12 (a \sin (c+d x)+a)^{13/2}}{13 a^6 d}-\frac {24 (a \sin (c+d x)+a)^{11/2}}{11 a^5 d}+\frac {16 (a \sin (c+d x)+a)^{9/2}}{9 a^4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2746
Rubi steps
\begin {align*} \int \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)} \, dx &=\frac {\text {Subst}\left (\int (a-x)^3 (a+x)^{7/2} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\text {Subst}\left (\int \left (8 a^3 (a+x)^{7/2}-12 a^2 (a+x)^{9/2}+6 a (a+x)^{11/2}-(a+x)^{13/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {16 (a+a \sin (c+d x))^{9/2}}{9 a^4 d}-\frac {24 (a+a \sin (c+d x))^{11/2}}{11 a^5 d}+\frac {12 (a+a \sin (c+d x))^{13/2}}{13 a^6 d}-\frac {2 (a+a \sin (c+d x))^{15/2}}{15 a^7 d}\\ \end {align*}
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Mathematica [A]
time = 2.59, size = 74, normalized size = 0.76 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8 \sqrt {a (1+\sin (c+d x))} (8330-3366 \cos (2 (c+d x))-10755 \sin (c+d x)+429 \sin (3 (c+d x)))}{12870 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 57, normalized size = 0.59
method | result | size |
default | \(\frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {9}{2}} \left (429 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-1683 \left (\cos ^{2}\left (d x +c \right )\right )-2796 \sin \left (d x +c \right )+2924\right )}{6435 a^{4} d}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 72, normalized size = 0.74 \begin {gather*} -\frac {2 \, {\left (429 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {15}{2}} - 2970 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {13}{2}} a + 7020 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {11}{2}} a^{2} - 5720 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a^{3}\right )}}{6435 \, a^{7} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 88, normalized size = 0.91 \begin {gather*} \frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{6} + 56 \, \cos \left (d x + c\right )^{4} + 128 \, \cos \left (d x + c\right )^{2} + {\left (429 \, \cos \left (d x + c\right )^{6} + 504 \, \cos \left (d x + c\right )^{4} + 640 \, \cos \left (d x + c\right )^{2} + 1024\right )} \sin \left (d x + c\right ) + 1024\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{6435 \, d} \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 = 7.79, size = 128, normalized size = 1.32 \begin {gather*} -\frac {256 \, \sqrt {2} {\left (429 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 1485 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 1755 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 715 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sqrt {a}}{6435 \, 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 {\cos \left (c+d\,x\right )}^7\,\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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