Optimal. Leaf size=119 \[ -\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 a d}+\frac{4 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x)}{d \sqrt{a \sin (c+d x)+a}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{\sqrt{a} d} \]
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Rubi [A] time = 0.503369, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258, Rules used = {2881, 2759, 2751, 2649, 206, 3044, 2985, 2773} \[ -\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 a d}+\frac{4 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x)}{d \sqrt{a \sin (c+d x)+a}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{\sqrt{a} d} \]
Antiderivative was successfully verified.
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Rule 2881
Rule 2759
Rule 2751
Rule 2649
Rule 206
Rule 3044
Rule 2985
Rule 2773
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx &=\int \frac{\sin ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx+\int \frac{\csc ^2(c+d x) \left (1-2 \sin ^2(c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=-\frac{\cot (c+d x)}{d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{3 a d}+\frac{2 \int \frac{\frac{a}{2}-a \sin (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{3 a}+\frac{\int \frac{\csc (c+d x) \left (-\frac{a}{2}-\frac{3}{2} a \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{a}\\ &=\frac{4 \cos (c+d x)}{3 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x)}{d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{3 a d}-\frac{\int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{2 a}\\ &=\frac{4 \cos (c+d x)}{3 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x)}{d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{3 a d}+\frac{\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{\tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{\sqrt{a} d}+\frac{4 \cos (c+d x)}{3 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x)}{d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{3 a d}\\ \end{align*}
Mathematica [A] time = 0.361274, size = 190, normalized size = 1.6 \[ \frac{\left (\tan \left (\frac{1}{2} (c+d x)\right )+1\right ) \csc \left (\frac{1}{4} (c+d x)\right ) \sec \left (\frac{1}{4} (c+d x)\right ) \left (10 \sin \left (\frac{1}{2} (c+d x)\right )+3 \sin \left (\frac{3}{2} (c+d x)\right )-\sin \left (\frac{5}{2} (c+d x)\right )-10 \cos \left (\frac{1}{2} (c+d x)\right )+3 \cos \left (\frac{3}{2} (c+d x)\right )+\cos \left (\frac{5}{2} (c+d x)\right )+3 \sin (c+d x) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-3 \sin (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{24 d \sqrt{a (\sin (c+d x)+1)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.967, size = 126, normalized size = 1.1 \begin{align*} -{\frac{1+\sin \left ( dx+c \right ) }{3\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) d}\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( \sin \left ( dx+c \right ) \left ( -2\,\sqrt{a} \left ( a-a\sin \left ( dx+c \right ) \right ) ^{3/2}-3\,{\it Artanh} \left ({\frac{\sqrt{a-a\sin \left ( dx+c \right ) }}{\sqrt{a}}} \right ){a}^{2} \right ) +3\,\sqrt{a-a\sin \left ( dx+c \right ) }{a}^{3/2} \right ){a}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{2}}{\sqrt{a \sin \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.21375, size = 818, normalized size = 6.87 \begin{align*} \frac{3 \,{\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 (2 \, \cos \left (d x + c\right )^{3} + 4 \, \cos \left (d x + c\right )^{2} -{\left (2 \, \cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) - 7\right )} \sin \left (d x + c\right ) - 5 \, \cos \left (d x + c\right ) - 7\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{12 \,{\left (a d \cos \left (d x + c\right )^{2} - a d -{\left (a d \cos \left (d x + c\right ) + a d\right )} \sin \left (d x + c\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.24795, size = 649, normalized size = 5.45 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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