Optimal. Leaf size=186 \[ \frac {9 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{4 d}+\frac {73 a^2 \cos (c+d x)}{20 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{5 d}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}-\frac {3 a \cot (c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}-\frac {\cot (c+d x) \csc (c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d} \]
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Rubi [A] time = 0.57, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2881, 2751, 2647, 2646, 3044, 2975, 2981, 2773, 206} \[ \frac {73 a^2 \cos (c+d x)}{20 d \sqrt {a \sin (c+d x)+a}}+\frac {9 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{4 d}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{5 d}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}-\frac {3 a \cot (c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}-\frac {\cot (c+d x) \csc (c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2646
Rule 2647
Rule 2751
Rule 2773
Rule 2881
Rule 2975
Rule 2981
Rule 3044
Rubi steps
\begin {align*} \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\int \sin (c+d x) (a+a \sin (c+d x))^{3/2} \, dx+\int \csc ^3(c+d x) (a+a \sin (c+d x))^{3/2} \left (1-2 \sin ^2(c+d x)\right ) \, dx\\ &=-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac {\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}+\frac {3}{5} \int (a+a \sin (c+d x))^{3/2} \, dx+\frac {\int \csc ^2(c+d x) \left (\frac {3 a}{2}-\frac {9}{2} a \sin (c+d x)\right ) (a+a \sin (c+d x))^{3/2} \, dx}{2 a}\\ &=-\frac {2 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}-\frac {3 a \cot (c+d x) \sqrt {a+a \sin (c+d x)}}{4 d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac {\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}+\frac {\int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \left (-\frac {9 a^2}{4}-\frac {21}{4} a^2 \sin (c+d x)\right ) \, dx}{2 a}+\frac {1}{5} (4 a) \int \sqrt {a+a \sin (c+d x)} \, dx\\ &=\frac {73 a^2 \cos (c+d x)}{20 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}-\frac {3 a \cot (c+d x) \sqrt {a+a \sin (c+d x)}}{4 d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac {\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}-\frac {1}{8} (9 a) \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=\frac {73 a^2 \cos (c+d x)}{20 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}-\frac {3 a \cot (c+d x) \sqrt {a+a \sin (c+d x)}}{4 d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac {\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}+\frac {\left (9 a^2\right ) \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 )}{4 d}\\ &=\frac {9 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{4 d}+\frac {73 a^2 \cos (c+d x)}{20 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}-\frac {3 a \cot (c+d x) \sqrt {a+a \sin (c+d x)}}{4 d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac {\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}\\ \end {align*}
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Mathematica [A] time = 1.07, size = 322, normalized size = 1.73 \[ -\frac {a \csc ^7\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\sin (c+d x)+1)} \left (118 \sin \left (\frac {1}{2} (c+d x)\right )+130 \sin \left (\frac {3}{2} (c+d x)\right )-36 \sin \left (\frac {5}{2} (c+d x)\right )-10 \sin \left (\frac {7}{2} (c+d x)\right )-2 \sin \left (\frac {9}{2} (c+d x)\right )-118 \cos \left (\frac {1}{2} (c+d x)\right )+130 \cos \left (\frac {3}{2} (c+d x)\right )+36 \cos \left (\frac {5}{2} (c+d x)\right )-10 \cos \left (\frac {7}{2} (c+d x)\right )+2 \cos \left (\frac {9}{2} (c+d x)\right )+45 \cos (2 (c+d x)) \log \left (-\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )+1\right )-45 \log \left (-\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )+1\right )-45 \cos (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )-\cos \left (\frac {1}{2} (c+d x)\right )+1\right )+45 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )-\cos \left (\frac {1}{2} (c+d x)\right )+1\right )\right )}{20 d \left (\cot \left (\frac {1}{2} (c+d x)\right )+1\right ) \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 404, normalized size = 2.17 \[ \frac {45 \, {\left (a \cos \left (d x + c\right )^{3} + a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right ) - a\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \, {\left (8 \, a \cos \left (d x + c\right )^{5} - 16 \, a \cos \left (d x + c\right )^{4} + 16 \, a \cos \left (d x + c\right )^{3} + 99 \, a \cos \left (d x + c\right )^{2} - 14 \, a \cos \left (d x + c\right ) - {\left (8 \, a \cos \left (d x + c\right )^{4} + 24 \, a \cos \left (d x + c\right )^{3} + 40 \, a \cos \left (d x + c\right )^{2} - 59 \, a \cos \left (d x + c\right ) - 73 \, a\right )} \sin \left (d x + c\right ) - 73 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{80 \, {\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right ) - d\right )}} \]
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 = 1.17, size = 178, normalized size = 0.96 \[ \frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (40 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {3}{2}} \left (\sin ^{2}\left (d x +c \right )\right )-8 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} \left (\sin ^{2}\left (d x +c \right )\right ) \sqrt {a}+45 \arctanh \left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{3} \left (\sin ^{2}\left (d x +c \right )\right )-45 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {5}{2}}+35 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {3}{2}}\right )}{20 a^{\frac {3}{2}} \sin \left (d x +c \right )^{2} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
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 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{3}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^4\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}}{{\sin \left (c+d\,x\right )}^3} \,d x \]
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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