Optimal. Leaf size=124 \[ -\frac {152 a^2 \cos ^5(c+d x)}{3465 d (a \sin (c+d x)+a)^{5/2}}-\frac {2 \cos ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{11 a d}+\frac {20 \cos ^5(c+d x)}{99 d \sqrt {a \sin (c+d x)+a}}-\frac {38 a \cos ^5(c+d x)}{693 d (a \sin (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.41, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2877, 2856, 2674, 2673} \[ -\frac {152 a^2 \cos ^5(c+d x)}{3465 d (a \sin (c+d x)+a)^{5/2}}-\frac {2 \cos ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{11 a d}+\frac {20 \cos ^5(c+d x)}{99 d \sqrt {a \sin (c+d x)+a}}-\frac {38 a \cos ^5(c+d x)}{693 d (a \sin (c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2673
Rule 2674
Rule 2856
Rule 2877
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx &=\frac {\cos ^5(c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}-\frac {\int \cos ^4(c+d x) \left (-\frac {a}{2}-4 a \sin (c+d x)\right ) \sqrt {a+a \sin (c+d x)} \, dx}{4 a^2}\\ &=\frac {\cos ^5(c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{11 a d}+\frac {19 \int \cos ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{88 a}\\ &=\frac {20 \cos ^5(c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{11 a d}+\frac {19}{99} \int \frac {\cos ^4(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=-\frac {38 a \cos ^5(c+d x)}{693 d (a+a \sin (c+d x))^{3/2}}+\frac {20 \cos ^5(c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{11 a d}+\frac {1}{693} (76 a) \int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac {152 a^2 \cos ^5(c+d x)}{3465 d (a+a \sin (c+d x))^{5/2}}-\frac {38 a \cos ^5(c+d x)}{693 d (a+a \sin (c+d x))^{3/2}}+\frac {20 \cos ^5(c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{11 a d}\\ \end {align*}
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Mathematica [A] time = 1.70, size = 143, normalized size = 1.15 \[ -\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^5 \left (5773 \sin \left (\frac {1}{2} (c+d x)\right )+3495 \sin \left (\frac {3}{2} (c+d x)\right )-1505 \sin \left (\frac {5}{2} (c+d x)\right )-315 \sin \left (\frac {7}{2} (c+d x)\right )+5773 \cos \left (\frac {1}{2} (c+d x)\right )-3495 \cos \left (\frac {3}{2} (c+d x)\right )-1505 \cos \left (\frac {5}{2} (c+d x)\right )+315 \cos \left (\frac {7}{2} (c+d x)\right )\right )}{13860 d \sqrt {a (\sin (c+d x)+1)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 155, normalized size = 1.25 \[ -\frac {2 \, {\left (315 \, \cos \left (d x + c\right )^{6} - 35 \, \cos \left (d x + c\right )^{5} - 445 \, \cos \left (d x + c\right )^{4} + 19 \, \cos \left (d x + c\right )^{3} - 38 \, \cos \left (d x + c\right )^{2} + {\left (315 \, \cos \left (d x + c\right )^{5} + 350 \, \cos \left (d x + c\right )^{4} - 95 \, \cos \left (d x + c\right )^{3} - 114 \, \cos \left (d x + c\right )^{2} - 152 \, \cos \left (d x + c\right ) - 304\right )} \sin \left (d x + c\right ) + 152 \, \cos \left (d x + c\right ) + 304\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{3465 \, {\left (a d \cos \left (d x + c\right ) + a d \sin \left (d x + c\right ) + a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.81, size = 342, normalized size = 2.76 \[ \frac {8 \, {\left (\frac {76 \, \sqrt {2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{\sqrt {a}} - \frac {\frac {34 \, a^{5}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {187 \, a^{5}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {1155 \, a^{5}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {1287 \, a^{5}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {231 \, a^{5}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {231 \, a^{5}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {1287 \, a^{5}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {1155 \, a^{5}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - 17 \, {\left (\frac {2 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + \frac {11 \, a^{5}}{\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 )^{2}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {11}{2}}}\right )}}{3465 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.35, size = 74, normalized size = 0.60 \[ \frac {2 \left (1+\sin \left (d x +c \right )\right ) \left (\sin \left (d x +c \right )-1\right )^{3} \left (315 \left (\sin ^{3}\left (d x +c \right )\right )+595 \left (\sin ^{2}\left (d x +c \right )\right )+340 \sin \left (d x +c \right )+136\right )}{3465 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
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 \[ \int \frac {\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{2}}{\sqrt {a \sin \left (d x + c\right ) + a}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^2}{\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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