Optimal. Leaf size=148 \[ -\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 a d}+\frac {4 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{15 d}+\frac {61 a \cos (c+d x)}{15 d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \sqrt {a \sin (c+d x)+a}}{d}-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d} \]
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Rubi [A] time = 0.48, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {2881, 2759, 2751, 2646, 3044, 2981, 2773, 206} \[ -\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 a d}+\frac {4 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{15 d}+\frac {61 a \cos (c+d x)}{15 d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \sqrt {a \sin (c+d x)+a}}{d}-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2646
Rule 2751
Rule 2759
Rule 2773
Rule 2881
Rule 2981
Rule 3044
Rubi steps
\begin {align*} \int \cos ^2(c+d x) \cot ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx &=\int \sin ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx+\int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \left (1-2 \sin ^2(c+d x)\right ) \, dx\\ &=-\frac {\cot (c+d x) \sqrt {a+a \sin (c+d x)}}{d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 a d}+\frac {2 \int \left (\frac {3 a}{2}-a \sin (c+d x)\right ) \sqrt {a+a \sin (c+d x)} \, dx}{5 a}+\frac {\int \csc (c+d x) \left (\frac {a}{2}-\frac {5}{2} a \sin (c+d x)\right ) \sqrt {a+a \sin (c+d x)} \, dx}{a}\\ &=\frac {5 a \cos (c+d x)}{d \sqrt {a+a \sin (c+d x)}}+\frac {4 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{15 d}-\frac {\cot (c+d x) \sqrt {a+a \sin (c+d x)}}{d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 a d}+\frac {7}{15} \int \sqrt {a+a \sin (c+d x)} \, dx+\frac {1}{2} \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=\frac {61 a \cos (c+d x)}{15 d \sqrt {a+a \sin (c+d x)}}+\frac {4 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{15 d}-\frac {\cot (c+d x) \sqrt {a+a \sin (c+d x)}}{d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 a d}-\frac {a \operatorname {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\frac {a \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{d}\\ &=-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{d}+\frac {61 a \cos (c+d x)}{15 d \sqrt {a+a \sin (c+d x)}}+\frac {4 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{15 d}-\frac {\cot (c+d x) \sqrt {a+a \sin (c+d x)}}{d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 a d}\\ \end {align*}
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Mathematica [A] time = 0.74, size = 258, normalized size = 1.74 \[ \frac {\csc ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\sin (c+d x)+1)} \left (155 \sin \left (\frac {1}{2} (c+d x)\right )+87 \sin \left (\frac {3}{2} (c+d x)\right )-5 \sin \left (\frac {5}{2} (c+d x)\right )+3 \sin \left (\frac {7}{2} (c+d x)\right )-155 \cos \left (\frac {1}{2} (c+d x)\right )+87 \cos \left (\frac {3}{2} (c+d x)\right )+5 \cos \left (\frac {5}{2} (c+d x)\right )+3 \cos \left (\frac {7}{2} (c+d x)\right )-30 \sin (c+d x) \log \left (-\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )+1\right )+30 \sin (c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )-\cos \left (\frac {1}{2} (c+d x)\right )+1\right )\right )}{30 d \left (\cot \left (\frac {1}{2} (c+d x)\right )+1\right ) \left (\csc \left (\frac {1}{4} (c+d x)\right )-\sec \left (\frac {1}{4} (c+d x)\right )\right ) \left (\csc \left (\frac {1}{4} (c+d x)\right )+\sec \left (\frac {1}{4} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 320, normalized size = 2.16 \[ \frac {15 \, {\left (\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right ) - 1\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) - 4 \, {\left (6 \, \cos \left (d x + c\right )^{4} + 8 \, \cos \left (d x + c\right )^{3} + 40 \, \cos \left (d x + c\right )^{2} + {\left (6 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + 38 \, \cos \left (d x + c\right ) + 61\right )} \sin \left (d x + c\right ) - 23 \, \cos \left (d x + c\right ) - 61\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{60 \, {\left (d \cos \left (d x + c\right )^{2} - {\left (d \cos \left (d x + c\right ) + d\right )} \sin \left (d x + c\right ) - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 1.28, size = 162, normalized size = 1.09 \[ -\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (\sin \left (d x +c \right ) \left (6 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} a^{\frac {3}{2}}-20 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} a^{\frac {5}{2}}-30 \sqrt {a -a \sin \left (d x +c \right )}\, a^{\frac {7}{2}}+15 \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}}{\sqrt {a}}\right ) a^{4}\right )+15 \sqrt {a -a \sin \left (d x +c \right )}\, a^{\frac {7}{2}}\right )}{15 a^{\frac {7}{2}} \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
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 \[ \int \sqrt {a \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{2}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^4\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{{\sin \left (c+d\,x\right )}^2} \,d x \]
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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