Optimal. Leaf size=97 \[ -\frac {2 (a \sin (c+d x)+a)^{15/2}}{15 a^7 d}+\frac {12 (a \sin (c+d x)+a)^{13/2}}{13 a^6 d}-\frac {24 (a \sin (c+d x)+a)^{11/2}}{11 a^5 d}+\frac {16 (a \sin (c+d x)+a)^{9/2}}{9 a^4 d} \]
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Rubi [A] time = 0.08, 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 = {2667, 43} \[ -\frac {2 (a \sin (c+d x)+a)^{15/2}}{15 a^7 d}+\frac {12 (a \sin (c+d x)+a)^{13/2}}{13 a^6 d}-\frac {24 (a \sin (c+d x)+a)^{11/2}}{11 a^5 d}+\frac {16 (a \sin (c+d x)+a)^{9/2}}{9 a^4 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \cos ^7(c+d x) \sqrt {a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^3 (a+x)^{7/2} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {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 = 4.32, size = 74, normalized size = 0.76 \[ \frac {\sqrt {a (\sin (c+d x)+1)} \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^8 (-10755 \sin (c+d x)+429 \sin (3 (c+d x))-3366 \cos (2 (c+d x))+8330)}{12870 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 88, normalized size = 0.91 \[ \frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{6} + 56 \, \cos \left (d x + c\right )^{4} + 128 \, \cos \left (d x + c\right )^{2} + {\left (429 \, \cos \left (d x + c\right )^{6} + 504 \, \cos \left (d x + c\right )^{4} + 640 \, \cos \left (d x + c\right )^{2} + 1024\right )} \sin \left (d x + c\right ) + 1024\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{6435 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.63, size = 249, normalized size = 2.57 \[ \frac {1}{411840} \, \sqrt {2} \sqrt {a} {\left (\frac {495 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{4} \, \pi + \frac {13}{2} \, d x + \frac {13}{2} \, c\right )}{d} + \frac {5005 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{4} \, \pi + \frac {9}{2} \, d x + \frac {9}{2} \, c\right )}{d} + \frac {27027 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{4} \, \pi + \frac {5}{2} \, d x + \frac {5}{2} \, c\right )}{d} + \frac {225225 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d} + \frac {429 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {15}{2} \, d x + \frac {15}{2} \, c\right )}{d} + \frac {4095 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {11}{2} \, d x + \frac {11}{2} \, c\right )}{d} + \frac {19305 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {7}{2} \, d x + \frac {7}{2} \, c\right )}{d} + \frac {75075 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right )}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 57, normalized size = 0.59 \[ \frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {9}{2}} \left (429 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-1683 \left (\cos ^{2}\left (d x +c \right )\right )-2796 \sin \left (d x +c \right )+2924\right )}{6435 a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 72, normalized size = 0.74 \[ -\frac {2 \, {\left (429 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {15}{2}} - 2970 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {13}{2}} a + 7020 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {11}{2}} a^{2} - 5720 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a^{3}\right )}}{6435 \, a^{7} d} \]
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 \[ \int {\cos \left (c+d\,x\right )}^7\,\sqrt {a+a\,\sin \left (c+d\,x\right )} \,d x \]
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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