Optimal. Leaf size=143 \[ \frac{4 \cot ^7(c+d x)}{7 a^3 d}-\frac{\cot ^5(c+d x)}{5 a^3 d}+\frac{\cot ^3(c+d x)}{3 a^3 d}-\frac{\cot (c+d x)}{a^3 d}-\frac{4 \csc ^7(c+d x)}{7 a^3 d}+\frac{11 \csc ^5(c+d x)}{5 a^3 d}-\frac{10 \csc ^3(c+d x)}{3 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}-\frac{x}{a^3} \]
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Rubi [A] time = 0.236258, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {3888, 3886, 3473, 8, 2606, 194, 2607, 30, 270} \[ \frac{4 \cot ^7(c+d x)}{7 a^3 d}-\frac{\cot ^5(c+d x)}{5 a^3 d}+\frac{\cot ^3(c+d x)}{3 a^3 d}-\frac{\cot (c+d x)}{a^3 d}-\frac{4 \csc ^7(c+d x)}{7 a^3 d}+\frac{11 \csc ^5(c+d x)}{5 a^3 d}-\frac{10 \csc ^3(c+d x)}{3 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}-\frac{x}{a^3} \]
Antiderivative was successfully verified.
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Rule 3888
Rule 3886
Rule 3473
Rule 8
Rule 2606
Rule 194
Rule 2607
Rule 30
Rule 270
Rubi steps
\begin{align*} \int \frac{\cot ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=\frac{\int \cot ^8(c+d x) (-a+a \sec (c+d x))^3 \, dx}{a^6}\\ &=\frac{\int \left (-a^3 \cot ^8(c+d x)+3 a^3 \cot ^7(c+d x) \csc (c+d x)-3 a^3 \cot ^6(c+d x) \csc ^2(c+d x)+a^3 \cot ^5(c+d x) \csc ^3(c+d x)\right ) \, dx}{a^6}\\ &=-\frac{\int \cot ^8(c+d x) \, dx}{a^3}+\frac{\int \cot ^5(c+d x) \csc ^3(c+d x) \, dx}{a^3}+\frac{3 \int \cot ^7(c+d x) \csc (c+d x) \, dx}{a^3}-\frac{3 \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx}{a^3}\\ &=\frac{\cot ^7(c+d x)}{7 a^3 d}+\frac{\int \cot ^6(c+d x) \, dx}{a^3}-\frac{\operatorname{Subst}\left (\int x^2 \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac{3 \operatorname{Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac{3 \operatorname{Subst}\left (\int \left (-1+x^2\right )^3 \, dx,x,\csc (c+d x)\right )}{a^3 d}\\ &=-\frac{\cot ^5(c+d x)}{5 a^3 d}+\frac{4 \cot ^7(c+d x)}{7 a^3 d}-\frac{\int \cot ^4(c+d x) \, dx}{a^3}-\frac{\operatorname{Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac{3 \operatorname{Subst}\left (\int \left (-1+3 x^2-3 x^4+x^6\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}\\ &=\frac{\cot ^3(c+d x)}{3 a^3 d}-\frac{\cot ^5(c+d x)}{5 a^3 d}+\frac{4 \cot ^7(c+d x)}{7 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}-\frac{10 \csc ^3(c+d x)}{3 a^3 d}+\frac{11 \csc ^5(c+d x)}{5 a^3 d}-\frac{4 \csc ^7(c+d x)}{7 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{4 \cot ^7(c+d x)}{7 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}-\frac{10 \csc ^3(c+d x)}{3 a^3 d}+\frac{11 \csc ^5(c+d x)}{5 a^3 d}-\frac{4 \csc ^7(c+d x)}{7 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}+\frac{4 \cot ^7(c+d x)}{7 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}-\frac{10 \csc ^3(c+d x)}{3 a^3 d}+\frac{11 \csc ^5(c+d x)}{5 a^3 d}-\frac{4 \csc ^7(c+d x)}{7 a^3 d}\\ \end{align*}
Mathematica [A] time = 1.27293, size = 252, normalized size = 1.76 \[ \frac{\csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) \csc \left (\frac{1}{2} (c+d x)\right ) \sec ^7\left (\frac{1}{2} (c+d x)\right ) (-23282 \sin (c+d x)-23282 \sin (2 (c+d x))-9978 \sin (3 (c+d x))-1663 \sin (4 (c+d x))+13720 \sin (2 c+d x)+15512 \sin (c+2 d x)+9240 \sin (3 c+2 d x)+8088 \sin (2 c+3 d x)+2520 \sin (4 c+3 d x)+1768 \sin (3 c+4 d x)+5880 d x \cos (2 c+d x)-5880 d x \cos (c+2 d x)+5880 d x \cos (3 c+2 d x)-2520 d x \cos (2 c+3 d x)+2520 d x \cos (4 c+3 d x)-420 d x \cos (3 c+4 d x)+420 d x \cos (5 c+4 d x)+4200 \sin (c)+11032 \sin (d x)-5880 d x \cos (d x))}{215040 a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 113, normalized size = 0.8 \begin{align*} -{\frac{1}{112\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{3}{40\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{1}{3\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{13}{8\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{3}}}-{\frac{1}{16\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.64185, size = 180, normalized size = 1.26 \begin{align*} \frac{\frac{\frac{2730 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{560 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{126 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{3}} - \frac{3360 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac{105 \,{\left (\cos \left (d x + c\right ) + 1\right )}}{a^{3} \sin \left (d x + c\right )}}{1680 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.2438, size = 377, normalized size = 2.64 \begin{align*} -\frac{221 \, \cos \left (d x + c\right )^{4} + 348 \, \cos \left (d x + c\right )^{3} - 25 \, \cos \left (d x + c\right )^{2} + 105 \,{\left (d x \cos \left (d x + c\right )^{3} + 3 \, d x \cos \left (d x + c\right )^{2} + 3 \, d x \cos \left (d x + c\right ) + d x\right )} \sin \left (d x + c\right ) - 303 \, \cos \left (d x + c\right ) - 136}{105 \,{\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + 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{\cot ^{2}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec{\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 [A] time = 1.48011, size = 134, normalized size = 0.94 \begin{align*} -\frac{\frac{1680 \,{\left (d x + c\right )}}{a^{3}} + \frac{105}{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )} + \frac{15 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 126 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 560 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2730 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{21}}}{1680 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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