Optimal. Leaf size=140 \[ -\frac {2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{a^{3/2} d}-\frac {4 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{5 a^2 d}-\frac {2 \cos ^3(c+d x)}{5 a d \sqrt {a \sin (c+d x)+a}}+\frac {18 \cos (c+d x)}{5 a d \sqrt {a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.35, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2878, 2858, 2751, 2649, 206} \[ -\frac {4 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{5 a^2 d}-\frac {2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{a^{3/2} d}-\frac {2 \cos ^3(c+d x)}{5 a d \sqrt {a \sin (c+d x)+a}}+\frac {18 \cos (c+d x)}{5 a d \sqrt {a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rule 2751
Rule 2858
Rule 2878
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=-\frac {2 \cos ^3(c+d x)}{5 a d \sqrt {a+a \sin (c+d x)}}+\frac {2 \int \frac {\cos ^2(c+d x) \left (-\frac {a}{2}-3 a \sin (c+d x)\right )}{(a+a \sin (c+d x))^{3/2}} \, dx}{5 a}\\ &=-\frac {2 \cos ^3(c+d x)}{5 a d \sqrt {a+a \sin (c+d x)}}-\frac {4 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{5 a^2 d}-\frac {4 \int \frac {-\frac {3 a^2}{4}+\frac {27}{4} a^2 \sin (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{15 a^3}\\ &=\frac {18 \cos (c+d x)}{5 a d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos ^3(c+d x)}{5 a d \sqrt {a+a \sin (c+d x)}}-\frac {4 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{5 a^2 d}+\frac {2 \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx}{a}\\ &=\frac {18 \cos (c+d x)}{5 a d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos ^3(c+d x)}{5 a d \sqrt {a+a \sin (c+d x)}}-\frac {4 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{5 a^2 d}-\frac {4 \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{a d}\\ &=-\frac {2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{a^{3/2} d}+\frac {18 \cos (c+d x)}{5 a d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos ^3(c+d x)}{5 a d \sqrt {a+a \sin (c+d x)}}-\frac {4 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{5 a^2 d}\\ \end {align*}
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Mathematica [C] time = 0.29, size = 150, normalized size = 1.07 \[ \frac {\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (-30 \sin \left (\frac {1}{2} (c+d x)\right )-5 \sin \left (\frac {3}{2} (c+d x)\right )+\sin \left (\frac {5}{2} (c+d x)\right )+30 \cos \left (\frac {1}{2} (c+d x)\right )-5 \cos \left (\frac {3}{2} (c+d x)\right )-\cos \left (\frac {5}{2} (c+d x)\right )+(40+40 i) (-1)^{3/4} \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac {1}{4} (c+d x)\right )-1\right )\right )\right )}{10 d (a (\sin (c+d x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 236, normalized size = 1.69 \[ \frac {\frac {5 \, \sqrt {2} {\left (a \cos \left (d x + c\right ) + a \sin \left (d x + c\right ) + a\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) - \frac {2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )}}{\sqrt {a}} + 3 \, \cos \left (d x + c\right ) + 2}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right )}{\sqrt {a}} - 2 \, {\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) - 9\right )} \sin \left (d x + c\right ) - 7 \, \cos \left (d x + c\right ) - 9\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{5 \, {\left (a^{2} d \cos \left (d x + c\right ) + a^{2} d \sin \left (d x + c\right ) + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.67, size = 303, normalized size = 2.16 \[ -\frac {2 \, {\left (\frac {\sqrt {2} {\left (10 \, a \arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right ) + 9 \, \sqrt {-a} \sqrt {a}\right )} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{\sqrt {-a} a^{2}} - \frac {10 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} + \sqrt {a}\right )}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + \frac {2 \, {\left ({\left ({\left ({\left ({\left (\frac {3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - \frac {5 \, a}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {10 \, a}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {10 \, a}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {5 \, a}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {3 \, a}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {5}{2}}}\right )}}{5 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.68, size = 112, normalized size = 0.80 \[ \frac {2 \left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (-5 a^{\frac {5}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )+\left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}}+5 a^{2} \sqrt {a -a \sin \left (d x +c \right )}\right )}{5 d \,a^{4} \cos \left (d x +c \right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}} \]
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 \[ \int \frac {\cos \left (d x + c\right )^{2} \sin \left (d x + c\right )^{2}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^2\,{\sin \left (c+d\,x\right )}^2}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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