Optimal. Leaf size=122 \[ -\frac{a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x)}{d}-a^2 x-\frac{2 a b \csc ^5(c+d x)}{5 d}+\frac{4 a b \csc ^3(c+d x)}{3 d}-\frac{2 a b \csc (c+d x)}{d}-\frac{b^2 \cot ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.134744, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3886, 3473, 8, 2606, 194, 2607, 30} \[ -\frac{a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x)}{d}-a^2 x-\frac{2 a b \csc ^5(c+d x)}{5 d}+\frac{4 a b \csc ^3(c+d x)}{3 d}-\frac{2 a b \csc (c+d x)}{d}-\frac{b^2 \cot ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 3886
Rule 3473
Rule 8
Rule 2606
Rule 194
Rule 2607
Rule 30
Rubi steps
\begin{align*} \int \cot ^6(c+d x) (a+b \sec (c+d x))^2 \, dx &=\int \left (a^2 \cot ^6(c+d x)+2 a b \cot ^5(c+d x) \csc (c+d x)+b^2 \cot ^4(c+d x) \csc ^2(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^6(c+d x) \, dx+(2 a b) \int \cot ^5(c+d x) \csc (c+d x) \, dx+b^2 \int \cot ^4(c+d x) \csc ^2(c+d x) \, dx\\ &=-\frac{a^2 \cot ^5(c+d x)}{5 d}-a^2 \int \cot ^4(c+d x) \, dx-\frac{(2 a b) \operatorname{Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{d}+\frac{b^2 \operatorname{Subst}\left (\int x^4 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{b^2 \cot ^5(c+d x)}{5 d}+a^2 \int \cot ^2(c+d x) \, dx-\frac{(2 a b) \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac{a^2 \cot (c+d x)}{d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{b^2 \cot ^5(c+d x)}{5 d}-\frac{2 a b \csc (c+d x)}{d}+\frac{4 a b \csc ^3(c+d x)}{3 d}-\frac{2 a b \csc ^5(c+d x)}{5 d}-a^2 \int 1 \, dx\\ &=-a^2 x-\frac{a^2 \cot (c+d x)}{d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{b^2 \cot ^5(c+d x)}{5 d}-\frac{2 a b \csc (c+d x)}{d}+\frac{4 a b \csc ^3(c+d x)}{3 d}-\frac{2 a b \csc ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.591659, size = 198, normalized size = 1.62 \[ -\frac{\csc ^5(c+d x) \left (10 \left (5 a^2+3 b^2\right ) \cos (c+d x)+150 a^2 c \sin (c+d x)+150 a^2 d x \sin (c+d x)-75 a^2 c \sin (3 (c+d x))-75 a^2 d x \sin (3 (c+d x))+15 a^2 c \sin (5 (c+d x))+15 a^2 d x \sin (5 (c+d x))-25 a^2 \cos (3 (c+d x))+23 a^2 \cos (5 (c+d x))-80 a b \cos (2 (c+d x))+60 a b \cos (4 (c+d x))+116 a b+15 b^2 \cos (3 (c+d x))+3 b^2 \cos (5 (c+d x))\right )}{240 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 154, normalized size = 1.3 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( -{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5}}+{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3}}-\cot \left ( dx+c \right ) -dx-c \right ) +2\,ab \left ( -1/5\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+1/15\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-1/5\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{\sin \left ( dx+c \right ) }}-1/5\, \left ( 8/3+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+4/3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) \right ) -{\frac{{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.499, size = 130, normalized size = 1.07 \begin{align*} -\frac{{\left (15 \, d x + 15 \, c + \frac{15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{2} + \frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{4} - 10 \, \sin \left (d x + c\right )^{2} + 3\right )} a b}{\sin \left (d x + c\right )^{5}} + \frac{3 \, b^{2}}{\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 [A] time = 0.781459, size = 382, normalized size = 3.13 \begin{align*} -\frac{30 \, a b \cos \left (d x + c\right )^{4} +{\left (23 \, a^{2} + 3 \, b^{2}\right )} \cos \left (d x + c\right )^{5} - 35 \, a^{2} \cos \left (d x + c\right )^{3} - 40 \, a b \cos \left (d x + c\right )^{2} + 15 \, a^{2} \cos \left (d x + c\right ) + 16 \, a b + 15 \,{\left (a^{2} d x \cos \left (d x + c\right )^{4} - 2 \, a^{2} d x \cos \left (d x + c\right )^{2} + a^{2} d x\right )} \sin \left (d x + c\right )}{15 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + 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(-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 [B] time = 1.4415, size = 369, normalized size = 3.02 \begin{align*} \frac{3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 6 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 35 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 50 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 15 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 480 \,{\left (d x + c\right )} a^{2} + 330 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 300 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 30 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{330 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 300 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 30 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 35 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 50 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 3 \, a^{2} + 6 \, a b + 3 \, b^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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