Optimal. Leaf size=96 \[ \frac{2 \cot ^3(c+d x)}{3 a^2 d}+\frac{2 \cot (c+d x)}{a^2 d}-\frac{7 \tanh ^{-1}(\cos (c+d x))}{8 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac{7 \cot (c+d x) \csc (c+d x)}{8 a^2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.187241, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {2869, 2757, 3768, 3770, 3767} \[ \frac{2 \cot ^3(c+d x)}{3 a^2 d}+\frac{2 \cot (c+d x)}{a^2 d}-\frac{7 \tanh ^{-1}(\cos (c+d x))}{8 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac{7 \cot (c+d x) \csc (c+d x)}{8 a^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2869
Rule 2757
Rule 3768
Rule 3770
Rule 3767
Rubi steps
\begin{align*} \int \frac{\cot ^4(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\int \csc ^5(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac{\int \left (a^2 \csc ^3(c+d x)-2 a^2 \csc ^4(c+d x)+a^2 \csc ^5(c+d x)\right ) \, dx}{a^4}\\ &=\frac{\int \csc ^3(c+d x) \, dx}{a^2}+\frac{\int \csc ^5(c+d x) \, dx}{a^2}-\frac{2 \int \csc ^4(c+d x) \, dx}{a^2}\\ &=-\frac{\cot (c+d x) \csc (c+d x)}{2 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac{\int \csc (c+d x) \, dx}{2 a^2}+\frac{3 \int \csc ^3(c+d x) \, dx}{4 a^2}+\frac{2 \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a^2 d}\\ &=-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^2 d}+\frac{2 \cot (c+d x)}{a^2 d}+\frac{2 \cot ^3(c+d x)}{3 a^2 d}-\frac{7 \cot (c+d x) \csc (c+d x)}{8 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac{3 \int \csc (c+d x) \, dx}{8 a^2}\\ &=-\frac{7 \tanh ^{-1}(\cos (c+d x))}{8 a^2 d}+\frac{2 \cot (c+d x)}{a^2 d}+\frac{2 \cot ^3(c+d x)}{3 a^2 d}-\frac{7 \cot (c+d x) \csc (c+d x)}{8 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}\\ \end{align*}
Mathematica [A] time = 1.46709, size = 116, normalized size = 1.21 \[ -\frac{\left (\csc \left (\frac{1}{2} (c+d x)\right )+\sec \left (\frac{1}{2} (c+d x)\right )\right )^4 \left (-48 \sin (2 (c+d x))+45 \cos (c+d x)+(32 \sin (c+d x)-21) \cos (3 (c+d x))+84 \sin ^4(c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{1536 a^2 d (\sin (c+d x)+1)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.171, size = 170, normalized size = 1.8 \begin{align*}{\frac{1}{64\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}}-{\frac{1}{12\,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{3}{4\,d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{3}{4\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{1}{64\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-4}}+{\frac{7}{8\,d{a}^{2}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }+{\frac{1}{12\,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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.11237, size = 263, normalized size = 2.74 \begin{align*} -\frac{\frac{\frac{144 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{48 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{16 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{3 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{a^{2}} - \frac{168 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} - \frac{{\left (\frac{16 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{48 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{144 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 3\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}{a^{2} \sin \left (d x + c\right )^{4}}}{192 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.12775, size = 406, normalized size = 4.23 \begin{align*} \frac{42 \, \cos \left (d x + c\right )^{3} - 21 \,{\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 21 \,{\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 32 \,{\left (2 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 54 \, \cos \left (d x + c\right )}{48 \,{\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\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.35478, size = 212, normalized size = 2.21 \begin{align*} \frac{\frac{168 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac{350 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 144 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 48 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 16 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3}{a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4}} + \frac{3 \, a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 16 \, a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 48 \, a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 144 \, a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{8}}}{192 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]