Optimal. Leaf size=164 \[ \frac{2 \tan ^7(c+d x)}{7 a^2 d}+\frac{7 \tan ^5(c+d x)}{5 a^2 d}+\frac{3 \tan ^3(c+d x)}{a^2 d}+\frac{5 \tan (c+d x)}{a^2 d}-\frac{\cot (c+d x)}{a^2 d}-\frac{2 \sec ^7(c+d x)}{7 a^2 d}-\frac{2 \sec ^5(c+d x)}{5 a^2 d}-\frac{2 \sec ^3(c+d x)}{3 a^2 d}-\frac{2 \sec (c+d x)}{a^2 d}+\frac{2 \tanh ^{-1}(\cos (c+d x))}{a^2 d} \]
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Rubi [A] time = 0.329989, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2875, 2873, 3767, 2622, 302, 207, 2620, 270} \[ \frac{2 \tan ^7(c+d x)}{7 a^2 d}+\frac{7 \tan ^5(c+d x)}{5 a^2 d}+\frac{3 \tan ^3(c+d x)}{a^2 d}+\frac{5 \tan (c+d x)}{a^2 d}-\frac{\cot (c+d x)}{a^2 d}-\frac{2 \sec ^7(c+d x)}{7 a^2 d}-\frac{2 \sec ^5(c+d x)}{5 a^2 d}-\frac{2 \sec ^3(c+d x)}{3 a^2 d}-\frac{2 \sec (c+d x)}{a^2 d}+\frac{2 \tanh ^{-1}(\cos (c+d x))}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 2875
Rule 2873
Rule 3767
Rule 2622
Rule 302
Rule 207
Rule 2620
Rule 270
Rubi steps
\begin{align*} \int \frac{\csc ^2(c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\int \csc ^2(c+d x) \sec ^8(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac{\int \left (a^2 \sec ^8(c+d x)-2 a^2 \csc (c+d x) \sec ^8(c+d x)+a^2 \csc ^2(c+d x) \sec ^8(c+d x)\right ) \, dx}{a^4}\\ &=\frac{\int \sec ^8(c+d x) \, dx}{a^2}+\frac{\int \csc ^2(c+d x) \sec ^8(c+d x) \, dx}{a^2}-\frac{2 \int \csc (c+d x) \sec ^8(c+d x) \, dx}{a^2}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^4}{x^2} \, dx,x,\tan (c+d x)\right )}{a^2 d}-\frac{\operatorname{Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,-\tan (c+d x)\right )}{a^2 d}-\frac{2 \operatorname{Subst}\left (\int \frac{x^8}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=\frac{\tan (c+d x)}{a^2 d}+\frac{\tan ^3(c+d x)}{a^2 d}+\frac{3 \tan ^5(c+d x)}{5 a^2 d}+\frac{\tan ^7(c+d x)}{7 a^2 d}+\frac{\operatorname{Subst}\left (\int \left (4+\frac{1}{x^2}+6 x^2+4 x^4+x^6\right ) \, dx,x,\tan (c+d x)\right )}{a^2 d}-\frac{2 \operatorname{Subst}\left (\int \left (1+x^2+x^4+x^6+\frac{1}{-1+x^2}\right ) \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=-\frac{\cot (c+d x)}{a^2 d}-\frac{2 \sec (c+d x)}{a^2 d}-\frac{2 \sec ^3(c+d x)}{3 a^2 d}-\frac{2 \sec ^5(c+d x)}{5 a^2 d}-\frac{2 \sec ^7(c+d x)}{7 a^2 d}+\frac{5 \tan (c+d x)}{a^2 d}+\frac{3 \tan ^3(c+d x)}{a^2 d}+\frac{7 \tan ^5(c+d x)}{5 a^2 d}+\frac{2 \tan ^7(c+d x)}{7 a^2 d}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=\frac{2 \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{\cot (c+d x)}{a^2 d}-\frac{2 \sec (c+d x)}{a^2 d}-\frac{2 \sec ^3(c+d x)}{3 a^2 d}-\frac{2 \sec ^5(c+d x)}{5 a^2 d}-\frac{2 \sec ^7(c+d x)}{7 a^2 d}+\frac{5 \tan (c+d x)}{a^2 d}+\frac{3 \tan ^3(c+d x)}{a^2 d}+\frac{7 \tan ^5(c+d x)}{5 a^2 d}+\frac{2 \tan ^7(c+d x)}{7 a^2 d}\\ \end{align*}
Mathematica [B] time = 6.09012, size = 442, normalized size = 2.7 \[ \frac{16 \left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{32 d}-\frac{\cot \left (\frac{1}{2} (c+d x)\right )}{32 d}-\frac{\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{8 d}+\frac{\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{8 d}+\frac{13 \sin \left (\frac{1}{2} (c+d x)\right )}{384 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{\sin \left (\frac{1}{2} (c+d x)\right )}{384 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3}+\frac{4777 \sin \left (\frac{1}{2} (c+d x)\right )}{13440 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{1}{768 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}-\frac{997}{26880 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{997 \sin \left (\frac{1}{2} (c+d x)\right )}{13440 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3}-\frac{3}{280 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4}+\frac{3 \sin \left (\frac{1}{2} (c+d x)\right )}{140 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^5}-\frac{1}{448 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6}+\frac{\sin \left (\frac{1}{2} (c+d x)\right )}{224 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^7}\right )}{a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.135, size = 266, normalized size = 1.6 \begin{align*}{\frac{1}{2\,d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{1}{12\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-3}}-{\frac{1}{8\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}-{\frac{5}{8\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{4}{7\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-7}}+2\,{\frac{1}{d{a}^{2} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{6}}}-{\frac{24}{5\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-5}}+7\,{\frac{1}{d{a}^{2} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{4}}}-{\frac{107}{12\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}+{\frac{59}{8\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}-{\frac{75}{8\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-{\frac{1}{2\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-2\,{\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 = 1.06658, size = 649, normalized size = 3.96 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8143, size = 671, normalized size = 4.09 \begin{align*} -\frac{432 \, \cos \left (d x + c\right )^{6} - 660 \, \cos \left (d x + c\right )^{4} + 98 \, \cos \left (d x + c\right )^{2} - 105 \,{\left (2 \, \cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3} +{\left (\cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 105 \,{\left (2 \, \cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3} +{\left (\cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 2 \,{\left (327 \, \cos \left (d x + c\right )^{4} - 41 \, \cos \left (d x + c\right )^{2} - 5\right )} \sin \left (d x + c\right ) + 25}{105 \,{\left (2 \, a^{2} d \cos \left (d x + c\right )^{5} - 2 \, a^{2} d \cos \left (d x + c\right )^{3} +{\left (a^{2} d \cos \left (d x + c\right )^{5} - 2 \, a^{2} d \cos \left (d x + c\right )^{3}\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 = 1.30984, size = 275, normalized size = 1.68 \begin{align*} -\frac{\frac{1680 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac{420 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{2}} - \frac{420 \,{\left (4 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}}{a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )} + \frac{35 \,{\left (15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 27 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 14\right )}}{a^{2}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{3}} + \frac{7875 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 41055 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 94640 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 119630 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 87507 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 34979 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6122}{a^{2}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{7}}}{840 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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