Optimal. Leaf size=58 \[ \frac{2 a^2 \tan (c+d x)}{d}-\frac{a^2 \cot (c+d x)}{d}+\frac{2 a^2 \sec (c+d x)}{d}-\frac{2 a^2 \tanh ^{-1}(\cos (c+d x))}{d} \]
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Rubi [A] time = 0.222525, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2873, 3767, 8, 2622, 321, 207, 2620, 14} \[ \frac{2 a^2 \tan (c+d x)}{d}-\frac{a^2 \cot (c+d x)}{d}+\frac{2 a^2 \sec (c+d x)}{d}-\frac{2 a^2 \tanh ^{-1}(\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 2873
Rule 3767
Rule 8
Rule 2622
Rule 321
Rule 207
Rule 2620
Rule 14
Rubi steps
\begin{align*} \int \csc ^2(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^2 \, dx &=\int \left (a^2 \sec ^2(c+d x)+2 a^2 \csc (c+d x) \sec ^2(c+d x)+a^2 \csc ^2(c+d x) \sec ^2(c+d x)\right ) \, dx\\ &=a^2 \int \sec ^2(c+d x) \, dx+a^2 \int \csc ^2(c+d x) \sec ^2(c+d x) \, dx+\left (2 a^2\right ) \int \csc (c+d x) \sec ^2(c+d x) \, dx\\ &=-\frac{a^2 \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1+x^2}{x^2} \, dx,x,\tan (c+d x)\right )}{d}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac{2 a^2 \sec (c+d x)}{d}+\frac{a^2 \tan (c+d x)}{d}+\frac{a^2 \operatorname{Subst}\left (\int \left (1+\frac{1}{x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac{2 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^2 \cot (c+d x)}{d}+\frac{2 a^2 \sec (c+d x)}{d}+\frac{2 a^2 \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.37997, size = 96, normalized size = 1.66 \[ \frac{a^2 \left (\tan \left (\frac{1}{2} (c+d x)\right )-\cot \left (\frac{1}{2} (c+d x)\right )+4 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-4 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{8 \sin \left (\frac{1}{2} (c+d x)\right )}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.094, size = 92, normalized size = 1.6 \begin{align*}{\frac{{a}^{2}\tan \left ( dx+c \right ) }{d}}+2\,{\frac{{a}^{2}}{d\cos \left ( dx+c \right ) }}+2\,{\frac{{a}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}}+{\frac{{a}^{2}}{d\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }}-2\,{\frac{{a}^{2}\cot \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16119, size = 97, normalized size = 1.67 \begin{align*} \frac{a^{2}{\left (\frac{2}{\cos \left (d x + c\right )} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - a^{2}{\left (\frac{1}{\tan \left (d x + c\right )} - \tan \left (d x + c\right )\right )} + a^{2} \tan \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.12086, size = 473, normalized size = 8.16 \begin{align*} -\frac{3 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right ) - 2 \, a^{2} +{\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2} +{\left (a^{2} \cos \left (d x + c\right ) + a^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) -{\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2} +{\left (a^{2} \cos \left (d x + c\right ) + a^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) -{\left (3 \, a^{2} \cos \left (d x + c\right ) + 2 \, a^{2}\right )} \sin \left (d x + c\right )}{d \cos \left (d x + c\right )^{2} +{\left (d \cos \left (d x + c\right ) + d\right )} \sin \left (d x + c\right ) - d} \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 = 1.27156, size = 132, normalized size = 2.28 \begin{align*} \frac{4 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{2 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 7 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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