Optimal. Leaf size=97 \[ -\frac {2 (a \sin (c+d x)+a)^{13/2}}{13 a^7 d}+\frac {12 (a \sin (c+d x)+a)^{11/2}}{11 a^6 d}-\frac {8 (a \sin (c+d x)+a)^{9/2}}{3 a^5 d}+\frac {16 (a \sin (c+d x)+a)^{7/2}}{7 a^4 d} \]
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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} \[ -\frac {2 (a \sin (c+d x)+a)^{13/2}}{13 a^7 d}+\frac {12 (a \sin (c+d x)+a)^{11/2}}{11 a^6 d}-\frac {8 (a \sin (c+d x)+a)^{9/2}}{3 a^5 d}+\frac {16 (a \sin (c+d x)+a)^{7/2}}{7 a^4 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \frac {\cos ^7(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^3 (a+x)^{5/2} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (8 a^3 (a+x)^{5/2}-12 a^2 (a+x)^{7/2}+6 a (a+x)^{9/2}-(a+x)^{11/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {16 (a+a \sin (c+d x))^{7/2}}{7 a^4 d}-\frac {8 (a+a \sin (c+d x))^{9/2}}{3 a^5 d}+\frac {12 (a+a \sin (c+d x))^{11/2}}{11 a^6 d}-\frac {2 (a+a \sin (c+d x))^{13/2}}{13 a^7 d}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 61, normalized size = 0.63 \[ -\frac {2 (\sin (c+d x)+1)^4 \left (231 \sin ^3(c+d x)-945 \sin ^2(c+d x)+1421 \sin (c+d x)-835\right )}{3003 d \sqrt {a (\sin (c+d x)+1)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 82, normalized size = 0.85 \[ \frac {2 \, {\left (231 \, \cos \left (d x + c\right )^{6} + 28 \, \cos \left (d x + c\right )^{4} + 64 \, \cos \left (d x + c\right )^{2} + 4 \, {\left (63 \, \cos \left (d x + c\right )^{4} + 80 \, \cos \left (d x + c\right )^{2} + 128\right )} \sin \left (d x + c\right ) + 512\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{3003 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.70, size = 430, normalized size = 4.43 \[ \frac {2 \, {\left (\frac {835 \, a^{6}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {3003 \, a^{6}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {3926 \, a^{6}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {6006 \, a^{6}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {15301 \, a^{6}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {21021 \, a^{6}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {15444 \, a^{6}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {15444 \, a^{6}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {21021 \, a^{6}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {15301 \, a^{6}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {6006 \, a^{6}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {3926 \, a^{6}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {835 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + \frac {3003 \, a^{6}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{3003 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {13}{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 57, normalized size = 0.59 \[ \frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {7}{2}} \left (231 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-945 \left (\cos ^{2}\left (d x +c \right )\right )-1652 \sin \left (d x +c \right )+1780\right )}{3003 a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 281, normalized size = 2.90 \[ \frac {2 \, {\left (15015 \, \sqrt {a \sin \left (d x + c\right ) + a} - \frac {3003 \, {\left (3 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} - 10 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{2}\right )}}{a^{2}} + \frac {143 \, {\left (35 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} - 180 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a + 378 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{4}\right )}}{a^{4}} - \frac {5 \, {\left (231 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {13}{2}} - 1638 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {11}{2}} a + 5005 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a^{2} - 8580 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{3} + 9009 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{4} - 6006 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{5} + 3003 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{6}\right )}}{a^{6}}\right )}}{15015 \, a d} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^7}{\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \]
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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