Optimal. Leaf size=97 \[ -\frac {2 (a \sin (c+d x)+a)^{11/2}}{11 a^7 d}+\frac {4 (a \sin (c+d x)+a)^{9/2}}{3 a^6 d}-\frac {24 (a \sin (c+d x)+a)^{7/2}}{7 a^5 d}+\frac {16 (a \sin (c+d x)+a)^{5/2}}{5 a^4 d} \]
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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} \[ -\frac {2 (a \sin (c+d x)+a)^{11/2}}{11 a^7 d}+\frac {4 (a \sin (c+d x)+a)^{9/2}}{3 a^6 d}-\frac {24 (a \sin (c+d x)+a)^{7/2}}{7 a^5 d}+\frac {16 (a \sin (c+d x)+a)^{5/2}}{5 a^4 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \frac {\cos ^7(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^3 (a+x)^{3/2} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (8 a^3 (a+x)^{3/2}-12 a^2 (a+x)^{5/2}+6 a (a+x)^{7/2}-(a+x)^{9/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {16 (a+a \sin (c+d x))^{5/2}}{5 a^4 d}-\frac {24 (a+a \sin (c+d x))^{7/2}}{7 a^5 d}+\frac {4 (a+a \sin (c+d x))^{9/2}}{3 a^6 d}-\frac {2 (a+a \sin (c+d x))^{11/2}}{11 a^7 d}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 54, normalized size = 0.56 \[ -\frac {2 \left (105 \sin ^3(c+d x)-455 \sin ^2(c+d x)+755 \sin (c+d x)-533\right ) (a (\sin (c+d x)+1))^{5/2}}{1155 a^4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 72, normalized size = 0.74 \[ \frac {2 \, {\left (245 \, \cos \left (d x + c\right )^{4} + 32 \, \cos \left (d x + c\right )^{2} - {\left (105 \, \cos \left (d x + c\right )^{4} - 160 \, \cos \left (d x + c\right )^{2} - 256\right )} \sin \left (d x + c\right ) + 256\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{1155 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.56, size = 370, normalized size = 3.81 \[ \frac {2 \, {\left (\frac {533 \, a^{4}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {1155 \, a^{4}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {1199 \, a^{4}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {3465 \, a^{4}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {5874 \, a^{4}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {4158 \, a^{4}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {4158 \, a^{4}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {5874 \, a^{4}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {3465 \, a^{4}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {1199 \, a^{4}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {533 \, a^{4} \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 {1155 \, a^{4}}{\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 )}}{1155 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {11}{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 \[ \frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {5}{2}} \left (105 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-455 \left (\cos ^{2}\left (d x +c \right )\right )-860 \sin \left (d x +c \right )+988\right )}{1155 a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 72, normalized size = 0.74 \[ -\frac {2 \, {\left (105 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {11}{2}} - 770 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a + 1980 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{2} - 1848 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{3}\right )}}{1155 \, a^{7} d} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^7}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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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