Optimal. Leaf size=144 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{8 a^{3/2} d}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{3 a^2 d}-\frac {\cot (c+d x)}{8 a d \sqrt {a \sin (c+d x)+a}}+\frac {11 \cot (c+d x) \csc (c+d x)}{12 a d \sqrt {a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.54, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2717, 2772, 2773, 206, 3044, 2980} \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{8 a^{3/2} d}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{3 a^2 d}-\frac {\cot (c+d x)}{8 a d \sqrt {a \sin (c+d x)+a}}+\frac {11 \cot (c+d x) \csc (c+d x)}{12 a d \sqrt {a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2717
Rule 2772
Rule 2773
Rule 2980
Rule 3044
Rubi steps
\begin {align*} \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=\frac {\int \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \left (1+\sin ^2(c+d x)\right ) \, dx}{a^2}-\frac {2 \int \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{a^2}\\ &=\frac {\cot (c+d x) \csc (c+d x)}{a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{3 a^2 d}+\frac {\int \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \left (\frac {a}{2}+\frac {9}{2} a \sin (c+d x)\right ) \, dx}{3 a^3}-\frac {3 \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{2 a^2}\\ &=\frac {3 \cot (c+d x)}{2 a d \sqrt {a+a \sin (c+d x)}}+\frac {11 \cot (c+d x) \csc (c+d x)}{12 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{3 a^2 d}-\frac {3 \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{4 a^2}+\frac {13 \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{8 a^2}\\ &=-\frac {\cot (c+d x)}{8 a d \sqrt {a+a \sin (c+d x)}}+\frac {11 \cot (c+d x) \csc (c+d x)}{12 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{3 a^2 d}+\frac {13 \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{16 a^2}+\frac {3 \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 )}{2 a d}\\ &=\frac {3 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{2 a^{3/2} d}-\frac {\cot (c+d x)}{8 a d \sqrt {a+a \sin (c+d x)}}+\frac {11 \cot (c+d x) \csc (c+d x)}{12 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{3 a^2 d}-\frac {13 \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 )}{8 a d}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 a^{3/2} d}-\frac {\cot (c+d x)}{8 a d \sqrt {a+a \sin (c+d x)}}+\frac {11 \cot (c+d x) \csc (c+d x)}{12 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{3 a^2 d}\\ \end {align*}
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Mathematica [B] time = 0.79, size = 294, normalized size = 2.04 \[ \frac {\csc ^9\left (\frac {1}{2} (c+d x)\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (132 \sin \left (\frac {1}{2} (c+d x)\right )+62 \sin \left (\frac {3}{2} (c+d x)\right )-6 \sin \left (\frac {5}{2} (c+d x)\right )-132 \cos \left (\frac {1}{2} (c+d x)\right )+62 \cos \left (\frac {3}{2} (c+d x)\right )+6 \cos \left (\frac {5}{2} (c+d x)\right )-9 \sin (c+d x) \log \left (-\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )+1\right )+9 \sin (c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )-\cos \left (\frac {1}{2} (c+d x)\right )+1\right )+3 \sin (3 (c+d x)) \log \left (-\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )+1\right )-3 \sin (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )-\cos \left (\frac {1}{2} (c+d x)\right )+1\right )\right )}{24 d (a (\sin (c+d x)+1))^{3/2} \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^3} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.53, size = 383, normalized size = 2.66 \[ \frac {3 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right ) + 1\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \, {\left (3 \, \cos \left (d x + c\right )^{3} + 17 \, \cos \left (d x + c\right )^{2} - {\left (3 \, \cos \left (d x + c\right )^{2} - 14 \, \cos \left (d x + c\right ) - 25\right )} \sin \left (d x + c\right ) - 11 \, \cos \left (d x + c\right ) - 25\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{96 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d - {\left (a^{2} d \cos \left (d x + c\right )^{3} + a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.95, size = 589, normalized size = 4.09 \[ \frac {\sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} {\left ({\left (\frac {2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - \frac {9}{a^{2} \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 {14}{a^{2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}\right )} - \frac {{\left (30 \, \sqrt {2} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a} + \sqrt {a}}{\sqrt {-a}}\right ) - 15 \, \sqrt {2} \sqrt {-a} \log \left (\sqrt {2} \sqrt {a} + \sqrt {a}\right ) + 42 \, \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a} + \sqrt {a}}{\sqrt {-a}}\right ) - 21 \, \sqrt {-a} \log \left (\sqrt {2} \sqrt {a} + \sqrt {a}\right ) + 280 \, \sqrt {2} \sqrt {-a} + 402 \, \sqrt {-a}\right )} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{5 \, \sqrt {2} \sqrt {-a} a^{\frac {3}{2}} + 7 \, \sqrt {-a} a^{\frac {3}{2}}} + \frac {6 \, \arctan \left (-\frac {\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 )}{\sqrt {-a} a \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - \frac {3 \, \log \left ({\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} \right |}\right )}{a^{\frac {3}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - \frac {2 \, {\left (9 \, {\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}\right )}^{5} - 18 \, {\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}\right )}^{4} \sqrt {a} + 24 \, {\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}\right )}^{2} a^{\frac {3}{2}} - 9 \, {\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}\right )} a^{2} - 14 \, a^{\frac {5}{2}}\right )}}{{\left ({\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}\right )}^{2} - a\right )}^{3} a \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.24, size = 144, normalized size = 1.00 \[ -\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (3 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {3}{2}}+3 \arctanh \left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{4} \left (\sin ^{3}\left (d x +c \right )\right )+8 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {5}{2}}-3 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {7}{2}}\right )}{24 a^{\frac {11}{2}} \sin \left (d x +c \right )^{3} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\cos \left (c+d\,x\right )}^4}{{\sin \left (c+d\,x\right )}^4\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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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