Optimal. Leaf size=49 \[ -\frac{4 a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc (c+d x)}{d}+\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{d}+a^3 (-x) \]
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Rubi [A] time = 0.0970836, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {3886, 3473, 8, 2606, 3767, 2621, 321, 207} \[ -\frac{4 a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc (c+d x)}{d}+\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{d}+a^3 (-x) \]
Antiderivative was successfully verified.
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Rule 3886
Rule 3473
Rule 8
Rule 2606
Rule 3767
Rule 2621
Rule 321
Rule 207
Rubi steps
\begin{align*} \int \cot ^2(c+d x) (a+a \sec (c+d x))^3 \, dx &=\int \left (a^3 \cot ^2(c+d x)+3 a^3 \cot (c+d x) \csc (c+d x)+3 a^3 \csc ^2(c+d x)+a^3 \csc ^2(c+d x) \sec (c+d x)\right ) \, dx\\ &=a^3 \int \cot ^2(c+d x) \, dx+a^3 \int \csc ^2(c+d x) \sec (c+d x) \, dx+\left (3 a^3\right ) \int \cot (c+d x) \csc (c+d x) \, dx+\left (3 a^3\right ) \int \csc ^2(c+d x) \, dx\\ &=-\frac{a^3 \cot (c+d x)}{d}-a^3 \int 1 \, dx-\frac{a^3 \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}(\int 1 \, dx,x,\csc (c+d x))}{d}\\ &=-a^3 x-\frac{4 a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc (c+d x)}{d}-\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}\\ &=-a^3 x+\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{4 a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc (c+d x)}{d}\\ \end{align*}
Mathematica [B] time = 0.235522, size = 109, normalized size = 2.22 \[ -\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left (\frac{1}{2} (c+d x)\right ) \left (-4 \csc \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \csc \left (\frac{1}{2} (c+d x)\right )+\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+d x\right )}{8 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 68, normalized size = 1.4 \begin{align*} -{a}^{3}x-4\,{\frac{{a}^{3}\cot \left ( dx+c \right ) }{d}}-{\frac{{a}^{3}c}{d}}-4\,{\frac{{a}^{3}}{d\sin \left ( dx+c \right ) }}+{\frac{{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.77699, size = 115, normalized size = 2.35 \begin{align*} -\frac{2 \,{\left (d x + c + \frac{1}{\tan \left (d x + c\right )}\right )} a^{3} + a^{3}{\left (\frac{2}{\sin \left (d x + c\right )} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + \frac{6 \, a^{3}}{\sin \left (d x + c\right )} + \frac{6 \, a^{3}}{\tan \left (d x + c\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.19995, size = 216, normalized size = 4.41 \begin{align*} -\frac{2 \, a^{3} d x \sin \left (d x + c\right ) - a^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + a^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 8 \, a^{3} \cos \left (d x + c\right ) + 8 \, a^{3}}{2 \, d \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^{3} \left (\int 3 \cot ^{2}{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int 3 \cot ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \cot ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \cot ^{2}{\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 = 1.48206, size = 89, normalized size = 1.82 \begin{align*} -\frac{{\left (d x + c\right )} a^{3} - a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) + a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{4 \, a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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