Optimal. Leaf size=91 \[ -\frac{\cot ^3(c+d x)}{3 a^2 d}-\frac{3 \cot (c+d x)}{a^2 d}-\frac{2 \cos (c+d x)}{a^2 d (\sin (c+d x)+1)}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{\cot (c+d x) \csc (c+d x)}{a^2 d} \]
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Rubi [A] time = 0.251215, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2874, 2966, 3770, 3767, 8, 3768, 2648} \[ -\frac{\cot ^3(c+d x)}{3 a^2 d}-\frac{3 \cot (c+d x)}{a^2 d}-\frac{2 \cos (c+d x)}{a^2 d (\sin (c+d x)+1)}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{\cot (c+d x) \csc (c+d x)}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 2874
Rule 2966
Rule 3770
Rule 3767
Rule 8
Rule 3768
Rule 2648
Rubi steps
\begin{align*} \int \frac{\cot ^2(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\int \frac{\csc ^4(c+d x) (a-a \sin (c+d x))}{a+a \sin (c+d x)} \, dx}{a^2}\\ &=\frac{\int \left (-2 \csc (c+d x)+2 \csc ^2(c+d x)-2 \csc ^3(c+d x)+\csc ^4(c+d x)+\frac{2}{1+\sin (c+d x)}\right ) \, dx}{a^2}\\ &=\frac{\int \csc ^4(c+d x) \, dx}{a^2}-\frac{2 \int \csc (c+d x) \, dx}{a^2}+\frac{2 \int \csc ^2(c+d x) \, dx}{a^2}-\frac{2 \int \csc ^3(c+d x) \, dx}{a^2}+\frac{2 \int \frac{1}{1+\sin (c+d x)} \, dx}{a^2}\\ &=\frac{2 \tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{\cot (c+d x) \csc (c+d x)}{a^2 d}-\frac{2 \cos (c+d x)}{a^2 d (1+\sin (c+d x))}-\frac{\int \csc (c+d x) \, dx}{a^2}-\frac{\operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a^2 d}-\frac{2 \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^2 d}\\ &=\frac{3 \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{3 \cot (c+d x)}{a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{\cot (c+d x) \csc (c+d x)}{a^2 d}-\frac{2 \cos (c+d x)}{a^2 d (1+\sin (c+d x))}\\ \end{align*}
Mathematica [B] time = 1.24768, size = 472, normalized size = 5.19 \[ \frac{\left (\csc \left (\frac{1}{2} (c+d x)\right )+\sec \left (\frac{1}{2} (c+d x)\right )\right )^3 \left (12 \sin \left (\frac{1}{2} (c+d x)\right )-6 \sin \left (\frac{3}{2} (c+d x)\right )-2 \sin \left (\frac{5}{2} (c+d x)\right )+8 \sin \left (\frac{7}{2} (c+d x)\right )-10 \cos \left (\frac{5}{2} (c+d x)\right )+20 \cos \left (\frac{7}{2} (c+d x)\right )-27 \sin \left (\frac{1}{2} (c+d x)\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-27 \sin \left (\frac{3}{2} (c+d x)\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+9 \sin \left (\frac{5}{2} (c+d x)\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+9 \sin \left (\frac{7}{2} (c+d x)\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-9 \cos \left (\frac{5}{2} (c+d x)\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+9 \cos \left (\frac{7}{2} (c+d x)\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+9 \cos \left (\frac{5}{2} (c+d x)\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+3 \cos \left (\frac{1}{2} (c+d x)\right ) \left (-9 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+9 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+8\right )-3 \cos \left (\frac{3}{2} (c+d x)\right ) \left (-9 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+9 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+14\right )-9 \cos \left (\frac{7}{2} (c+d x)\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+27 \sin \left (\frac{1}{2} (c+d x)\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+27 \sin \left (\frac{3}{2} (c+d x)\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-9 \sin \left (\frac{5}{2} (c+d x)\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-9 \sin \left (\frac{7}{2} (c+d x)\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{192 a^2 d (\sin (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.171, size = 153, normalized size = 1.7 \begin{align*}{\frac{1}{24\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{1}{4\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}+{\frac{11}{8\,d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-4\,{\frac{1}{d{a}^{2} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) }}-{\frac{1}{24\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}+{\frac{1}{4\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}-{\frac{11}{8\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-3\,{\frac{\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.19624, size = 269, normalized size = 2.96 \begin{align*} \frac{\frac{\frac{5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{27 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{129 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 1}{\frac{a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac{\frac{33 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{6 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac{72 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.71775, size = 801, normalized size = 8.8 \begin{align*} \frac{28 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3} - 42 \, \cos \left (d x + c\right )^{2} + 9 \,{\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 )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 9 \,{\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 )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 2 \,{\left (14 \, \cos \left (d x + c\right )^{3} + 9 \, \cos \left (d x + c\right )^{2} - 12 \, \cos \left (d x + c\right ) - 6\right )} \sin \left (d x + c\right ) - 12 \, \cos \left (d x + c\right ) + 12}{6 \,{\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 )}} \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 [A] time = 1.34138, size = 197, normalized size = 2.16 \begin{align*} -\frac{\frac{72 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{2}} + \frac{96}{a^{2}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}} - \frac{132 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 33 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 6 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1}{a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3}} - \frac{a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 6 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 33 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{6}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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