Optimal. Leaf size=62 \[ \frac{\tan (c+d x)}{a d}-\frac{\cot ^5(c+d x)}{5 a d}-\frac{\cot ^3(c+d x)}{a d}-\frac{3 \cot (c+d x)}{a d} \]
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Rubi [A] time = 0.0844312, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3175, 2620, 270} \[ \frac{\tan (c+d x)}{a d}-\frac{\cot ^5(c+d x)}{5 a d}-\frac{\cot ^3(c+d x)}{a d}-\frac{3 \cot (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 2620
Rule 270
Rubi steps
\begin{align*} \int \frac{\csc ^6(c+d x)}{a-a \sin ^2(c+d x)} \, dx &=\frac{\int \csc ^6(c+d x) \sec ^2(c+d x) \, dx}{a}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^3}{x^6} \, dx,x,\tan (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \left (1+\frac{1}{x^6}+\frac{3}{x^4}+\frac{3}{x^2}\right ) \, dx,x,\tan (c+d x)\right )}{a d}\\ &=-\frac{3 \cot (c+d x)}{a d}-\frac{\cot ^3(c+d x)}{a d}-\frac{\cot ^5(c+d x)}{5 a d}+\frac{\tan (c+d x)}{a d}\\ \end{align*}
Mathematica [A] time = 0.0367788, size = 70, normalized size = 1.13 \[ \frac{\frac{\tan (c+d x)}{d}-\frac{11 \cot (c+d x)}{5 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 d}-\frac{3 \cot (c+d x) \csc ^2(c+d x)}{5 d}}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.078, size = 45, normalized size = 0.7 \begin{align*}{\frac{1}{da} \left ( \tan \left ( dx+c \right ) -3\, \left ( \tan \left ( dx+c \right ) \right ) ^{-1}-{\frac{1}{5\, \left ( \tan \left ( dx+c \right ) \right ) ^{5}}}- \left ( \tan \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 = 0.961835, size = 70, normalized size = 1.13 \begin{align*} \frac{\frac{5 \, \tan \left (d x + c\right )}{a} - \frac{15 \, \tan \left (d x + c\right )^{4} + 5 \, \tan \left (d x + c\right )^{2} + 1}{a \tan \left (d x + c\right )^{5}}}{5 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57284, size = 200, normalized size = 3.23 \begin{align*} -\frac{16 \, \cos \left (d x + c\right )^{6} - 40 \, \cos \left (d x + c\right )^{4} + 30 \, \cos \left (d x + c\right )^{2} - 5}{5 \,{\left (a d \cos \left (d x + c\right )^{5} - 2 \, a d \cos \left (d x + c\right )^{3} + a d \cos \left (d x + c\right )\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(-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 = 1.22769, size = 70, normalized size = 1.13 \begin{align*} \frac{\frac{5 \, \tan \left (d x + c\right )}{a} - \frac{15 \, \tan \left (d x + c\right )^{4} + 5 \, \tan \left (d x + c\right )^{2} + 1}{a \tan \left (d x + c\right )^{5}}}{5 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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