Optimal. Leaf size=87 \[ -\frac{10 a^2 \cot (c+d x)}{3 d}-\frac{2 a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a^4 \cot (c+d x)}{3 d (a-a \sin (c+d x))^2}+\frac{2 a^2 \cot (c+d x)}{d (1-\sin (c+d x))} \]
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Rubi [A] time = 0.27391, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2869, 2766, 2978, 2748, 3767, 8, 3770} \[ -\frac{10 a^2 \cot (c+d x)}{3 d}-\frac{2 a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a^4 \cot (c+d x)}{3 d (a-a \sin (c+d x))^2}+\frac{2 a^2 \cot (c+d x)}{d (1-\sin (c+d x))} \]
Antiderivative was successfully verified.
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Rule 2869
Rule 2766
Rule 2978
Rule 2748
Rule 3767
Rule 8
Rule 3770
Rubi steps
\begin{align*} \int \csc ^2(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^2 \, dx &=a^4 \int \frac{\csc ^2(c+d x)}{(a-a \sin (c+d x))^2} \, dx\\ &=\frac{a^4 \cot (c+d x)}{3 d (a-a \sin (c+d x))^2}+\frac{1}{3} a^2 \int \frac{\csc ^2(c+d x) (4 a+2 a \sin (c+d x))}{a-a \sin (c+d x)} \, dx\\ &=\frac{2 a^2 \cot (c+d x)}{d (1-\sin (c+d x))}+\frac{a^4 \cot (c+d x)}{3 d (a-a \sin (c+d x))^2}+\frac{1}{3} \int \csc ^2(c+d x) \left (10 a^2+6 a^2 \sin (c+d x)\right ) \, dx\\ &=\frac{2 a^2 \cot (c+d x)}{d (1-\sin (c+d x))}+\frac{a^4 \cot (c+d x)}{3 d (a-a \sin (c+d x))^2}+\left (2 a^2\right ) \int \csc (c+d x) \, dx+\frac{1}{3} \left (10 a^2\right ) \int \csc ^2(c+d x) \, dx\\ &=-\frac{2 a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{2 a^2 \cot (c+d x)}{d (1-\sin (c+d x))}+\frac{a^4 \cot (c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac{\left (10 a^2\right ) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{3 d}\\ &=-\frac{2 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{10 a^2 \cot (c+d x)}{3 d}+\frac{2 a^2 \cot (c+d x)}{d (1-\sin (c+d x))}+\frac{a^4 \cot (c+d x)}{3 d (a-a \sin (c+d x))^2}\\ \end{align*}
Mathematica [A] time = 0.932694, size = 135, normalized size = 1.55 \[ \frac{a^2 \left (3 \tan \left (\frac{1}{2} (c+d x)\right )-3 \cot \left (\frac{1}{2} (c+d x)\right )+12 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-12 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{4 \sin \left (\frac{1}{2} (c+d x)\right ) (7 \sin (c+d x)-8)}{\left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )\right )^3}+\frac{2}{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}\right )}{6 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.122, size = 156, normalized size = 1.8 \begin{align*}{\frac{2\,{a}^{2}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{{a}^{2}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{2\,{a}^{2}}{3\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+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}}{3\,d\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{4\,{a}^{2}}{3\,d\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }}-{\frac{8\,{a}^{2}\cot \left ( dx+c \right ) }{3\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15638, size = 144, normalized size = 1.66 \begin{align*} \frac{{\left (\tan \left (d x + c\right )^{3} - \frac{3}{\tan \left (d x + c\right )} + 6 \, \tan \left (d x + c\right )\right )} a^{2} +{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{2} + a^{2}{\left (\frac{2 \,{\left (3 \, \cos \left (d x + c\right )^{2} + 1\right )}}{\cos \left (d x + c\right )^{3}} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.42352, size = 810, normalized size = 9.31 \begin{align*} \frac{10 \, a^{2} \cos \left (d x + c\right )^{3} - 4 \, a^{2} \cos \left (d x + c\right )^{2} - 13 \, a^{2} \cos \left (d x + c\right ) + a^{2} - 3 \,{\left (a^{2} \cos \left (d x + c\right )^{3} + 2 \, 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} \cos \left (d x + c\right ) - 2 \, 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 ) + 3 \,{\left (a^{2} \cos \left (d x + c\right )^{3} + 2 \, 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} \cos \left (d x + c\right ) - 2 \, 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 (10 \, a^{2} \cos \left (d x + c\right )^{2} + 14 \, a^{2} \cos \left (d x + c\right ) + a^{2}\right )} \sin \left (d x + c\right )}{3 \,{\left (d \cos \left (d x + c\right )^{3} + 2 \, d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) -{\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) - 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\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 = 1.32088, size = 159, normalized size = 1.83 \begin{align*} \frac{12 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{3 \,{\left (4 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a^{2}\right )}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )} - \frac{4 \,{\left (9 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 8 \, a^{2}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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