Optimal. Leaf size=55 \[ \frac{\cot ^5(c+d x)}{5 a^3 d}-\frac{\cot ^3(c+d x)}{3 a^3 d}+\frac{\cot (c+d x)}{a^3 d}+\frac{x}{a^3} \]
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Rubi [A] time = 0.0387751, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4120, 3473, 8} \[ \frac{\cot ^5(c+d x)}{5 a^3 d}-\frac{\cot ^3(c+d x)}{3 a^3 d}+\frac{\cot (c+d x)}{a^3 d}+\frac{x}{a^3} \]
Antiderivative was successfully verified.
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Rule 4120
Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{\left (a-a \sec ^2(c+d x)\right )^3} \, dx &=-\frac{\int \cot ^6(c+d x) \, dx}{a^3}\\ &=\frac{\cot ^5(c+d x)}{5 a^3 d}+\frac{\int \cot ^4(c+d x) \, dx}{a^3}\\ &=-\frac{\cot ^3(c+d x)}{3 a^3 d}+\frac{\cot ^5(c+d x)}{5 a^3 d}-\frac{\int \cot ^2(c+d x) \, dx}{a^3}\\ &=\frac{\cot (c+d x)}{a^3 d}-\frac{\cot ^3(c+d x)}{3 a^3 d}+\frac{\cot ^5(c+d x)}{5 a^3 d}+\frac{\int 1 \, dx}{a^3}\\ &=\frac{x}{a^3}+\frac{\cot (c+d x)}{a^3 d}-\frac{\cot ^3(c+d x)}{3 a^3 d}+\frac{\cot ^5(c+d x)}{5 a^3 d}\\ \end{align*}
Mathematica [C] time = 0.0473571, size = 36, normalized size = 0.65 \[ \frac{\cot ^5(c+d x) \text{Hypergeometric2F1}\left (-\frac{5}{2},1,-\frac{3}{2},-\tan ^2(c+d x)\right )}{5 a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 63, normalized size = 1.2 \begin{align*}{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d{a}^{3}}}-{\frac{1}{3\,d{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{3}}}+{\frac{1}{5\,d{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{5}}}+{\frac{1}{d{a}^{3}\tan \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51894, size = 68, normalized size = 1.24 \begin{align*} \frac{\frac{15 \,{\left (d x + c\right )}}{a^{3}} + \frac{15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{a^{3} \tan \left (d x + c\right )^{5}}}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.484161, size = 274, normalized size = 4.98 \begin{align*} \frac{23 \, \cos \left (d x + c\right )^{5} - 35 \, \cos \left (d x + c\right )^{3} + 15 \,{\left (d x \cos \left (d x + c\right )^{4} - 2 \, d x \cos \left (d x + c\right )^{2} + d x\right )} \sin \left (d x + c\right ) + 15 \, \cos \left (d x + c\right )}{15 \,{\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} 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*} - \frac{\int \frac{1}{\sec ^{6}{\left (c + d x \right )} - 3 \sec ^{4}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} - 1}\, dx}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.30958, size = 150, normalized size = 2.73 \begin{align*} \frac{\frac{480 \,{\left (d x + c\right )}}{a^{3}} + \frac{330 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 35 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 3}{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}} - \frac{3 \, a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 35 \, a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 330 \, a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{15}}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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