Optimal. Leaf size=69 \[ -\frac{2 a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}+\frac{2 a^2 \csc (c+d x)}{d}+a^2 x \]
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Rubi [A] time = 0.110513, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3886, 3473, 8, 2606, 2607, 30} \[ -\frac{2 a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}+\frac{2 a^2 \csc (c+d x)}{d}+a^2 x \]
Antiderivative was successfully verified.
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Rule 3886
Rule 3473
Rule 8
Rule 2606
Rule 2607
Rule 30
Rubi steps
\begin{align*} \int \cot ^4(c+d x) (a+a \sec (c+d x))^2 \, dx &=\int \left (a^2 \cot ^4(c+d x)+2 a^2 \cot ^3(c+d x) \csc (c+d x)+a^2 \cot ^2(c+d x) \csc ^2(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^4(c+d x) \, dx+a^2 \int \cot ^2(c+d x) \csc ^2(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^3(c+d x) \csc (c+d x) \, dx\\ &=-\frac{a^2 \cot ^3(c+d x)}{3 d}-a^2 \int \cot ^2(c+d x) \, dx+\frac{a^2 \operatorname{Subst}\left (\int x^2 \, dx,x,-\cot (c+d x)\right )}{d}-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=\frac{a^2 \cot (c+d x)}{d}-\frac{2 a^2 \cot ^3(c+d x)}{3 d}+\frac{2 a^2 \csc (c+d x)}{d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}+a^2 \int 1 \, dx\\ &=a^2 x+\frac{a^2 \cot (c+d x)}{d}-\frac{2 a^2 \cot ^3(c+d x)}{3 d}+\frac{2 a^2 \csc (c+d x)}{d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.25178, size = 112, normalized size = 1.62 \[ \frac{a^2 \csc \left (\frac{c}{2}\right ) \csc ^3\left (\frac{1}{2} (c+d x)\right ) \left (-12 \sin \left (c+\frac{d x}{2}\right )+10 \sin \left (c+\frac{3 d x}{2}\right )-9 d x \cos \left (c+\frac{d x}{2}\right )-3 d x \cos \left (c+\frac{3 d x}{2}\right )+3 d x \cos \left (2 c+\frac{3 d x}{2}\right )-18 \sin \left (\frac{d x}{2}\right )+9 d x \cos \left (\frac{d x}{2}\right )\right )}{24 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.068, size = 112, normalized size = 1.6 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( -{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3}}+\cot \left ( dx+c \right ) +dx+c \right ) +2\,{a}^{2} \left ( -1/3\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+1/3\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{\sin \left ( dx+c \right ) }}+1/3\, \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) \right ) -{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.73252, size = 104, normalized size = 1.51 \begin{align*} \frac{{\left (3 \, d x + 3 \, c + \frac{3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a^{2} + \frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{2} - 1\right )} a^{2}}{\sin \left (d x + c\right )^{3}} - \frac{a^{2}}{\tan \left (d x + c\right )^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.940468, size = 190, normalized size = 2.75 \begin{align*} \frac{5 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right ) - 4 \, a^{2} + 3 \,{\left (a^{2} d x \cos \left (d x + c\right ) - a^{2} d x\right )} \sin \left (d x + c\right )}{3 \,{\left (d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{2} \left (\int 2 \cot ^{4}{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int \cot ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \cot ^{4}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.50626, size = 68, normalized size = 0.99 \begin{align*} \frac{6 \,{\left (d x + c\right )} a^{2} + \frac{9 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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