Optimal. Leaf size=169 \[ -\frac{\left (2 a^2+b^2\right ) \cot ^3(c+d x)}{3 a^3 d}+\frac{b \left (2 a^2+b^2\right ) \cot ^2(c+d x)}{2 a^4 d}-\frac{\left (a^2+b^2\right )^2 \cot (c+d x)}{a^5 d}-\frac{b \left (a^2+b^2\right )^2 \log (\tan (c+d x))}{a^6 d}+\frac{b \left (a^2+b^2\right )^2 \log (a+b \tan (c+d x))}{a^6 d}+\frac{b \cot ^4(c+d x)}{4 a^2 d}-\frac{\cot ^5(c+d x)}{5 a d} \]
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Rubi [A] time = 0.150783, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3516, 894} \[ -\frac{\left (2 a^2+b^2\right ) \cot ^3(c+d x)}{3 a^3 d}+\frac{b \left (2 a^2+b^2\right ) \cot ^2(c+d x)}{2 a^4 d}-\frac{\left (a^2+b^2\right )^2 \cot (c+d x)}{a^5 d}-\frac{b \left (a^2+b^2\right )^2 \log (\tan (c+d x))}{a^6 d}+\frac{b \left (a^2+b^2\right )^2 \log (a+b \tan (c+d x))}{a^6 d}+\frac{b \cot ^4(c+d x)}{4 a^2 d}-\frac{\cot ^5(c+d x)}{5 a d} \]
Antiderivative was successfully verified.
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Rule 3516
Rule 894
Rubi steps
\begin{align*} \int \frac{\csc ^6(c+d x)}{a+b \tan (c+d x)} \, dx &=\frac{b \operatorname{Subst}\left (\int \frac{\left (b^2+x^2\right )^2}{x^6 (a+x)} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac{b \operatorname{Subst}\left (\int \left (\frac{b^4}{a x^6}-\frac{b^4}{a^2 x^5}+\frac{2 a^2 b^2+b^4}{a^3 x^4}+\frac{b^2 \left (-2 a^2-b^2\right )}{a^4 x^3}+\frac{\left (a^2+b^2\right )^2}{a^5 x^2}-\frac{\left (a^2+b^2\right )^2}{a^6 x}+\frac{\left (a^2+b^2\right )^2}{a^6 (a+x)}\right ) \, dx,x,b \tan (c+d x)\right )}{d}\\ &=-\frac{\left (a^2+b^2\right )^2 \cot (c+d x)}{a^5 d}+\frac{b \left (2 a^2+b^2\right ) \cot ^2(c+d x)}{2 a^4 d}-\frac{\left (2 a^2+b^2\right ) \cot ^3(c+d x)}{3 a^3 d}+\frac{b \cot ^4(c+d x)}{4 a^2 d}-\frac{\cot ^5(c+d x)}{5 a d}-\frac{b \left (a^2+b^2\right )^2 \log (\tan (c+d x))}{a^6 d}+\frac{b \left (a^2+b^2\right )^2 \log (a+b \tan (c+d x))}{a^6 d}\\ \end{align*}
Mathematica [A] time = 2.16387, size = 150, normalized size = 0.89 \[ \frac{15 b \left (2 a^2 \left (a^2+b^2\right ) \csc ^2(c+d x)-4 \left (a^2+b^2\right )^2 (\log (\sin (c+d x))-\log (a \cos (c+d x)+b \sin (c+d x)))+a^4 \csc ^4(c+d x)\right )-4 \cot (c+d x) \left (a^3 \left (4 a^2+5 b^2\right ) \csc ^2(c+d x)+25 a^3 b^2+3 a^5 \csc ^4(c+d x)+8 a^5+15 a b^4\right )}{60 a^6 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.083, size = 273, normalized size = 1.6 \begin{align*} -{\frac{1}{5\,ad \left ( \tan \left ( dx+c \right ) \right ) ^{5}}}-{\frac{2}{3\,ad \left ( \tan \left ( dx+c \right ) \right ) ^{3}}}-{\frac{{b}^{2}}{3\,d{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{3}}}-{\frac{1}{ad\tan \left ( dx+c \right ) }}-2\,{\frac{{b}^{2}}{d{a}^{3}\tan \left ( dx+c \right ) }}-{\frac{{b}^{4}}{d{a}^{5}\tan \left ( dx+c \right ) }}+{\frac{b}{4\,{a}^{2}d \left ( \tan \left ( dx+c \right ) \right ) ^{4}}}+{\frac{b}{{a}^{2}d \left ( \tan \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{b}^{3}}{2\,d{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}}-{\frac{b\ln \left ( \tan \left ( dx+c \right ) \right ) }{{a}^{2}d}}-2\,{\frac{{b}^{3}\ln \left ( \tan \left ( dx+c \right ) \right ) }{d{a}^{4}}}-{\frac{{b}^{5}\ln \left ( \tan \left ( dx+c \right ) \right ) }{d{a}^{6}}}+{\frac{b\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{{a}^{2}d}}+2\,{\frac{{b}^{3}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d{a}^{4}}}+{\frac{{b}^{5}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d{a}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09497, size = 227, normalized size = 1.34 \begin{align*} \frac{\frac{60 \,{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6}} - \frac{60 \,{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{6}} + \frac{15 \, a^{3} b \tan \left (d x + c\right ) - 60 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )^{4} - 12 \, a^{4} + 30 \,{\left (2 \, a^{3} b + a b^{3}\right )} \tan \left (d x + c\right )^{3} - 20 \,{\left (2 \, a^{4} + a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2}}{a^{5} \tan \left (d x + c\right )^{5}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.41224, size = 902, normalized size = 5.34 \begin{align*} -\frac{4 \,{\left (8 \, a^{5} + 25 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} - 20 \,{\left (4 \, a^{5} + 11 \, a^{3} b^{2} + 6 \, a b^{4}\right )} \cos \left (d x + c\right )^{3} - 30 \,{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5} +{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \,{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) \sin \left (d x + c\right ) + 30 \,{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5} +{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \,{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac{1}{4} \, \cos \left (d x + c\right )^{2} + \frac{1}{4}\right ) \sin \left (d x + c\right ) + 60 \,{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right ) - 15 \,{\left (3 \, a^{4} b + 2 \, a^{2} b^{3} - 2 \,{\left (a^{4} b + a^{2} b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{60 \,{\left (a^{6} d \cos \left (d x + c\right )^{4} - 2 \, a^{6} d \cos \left (d x + c\right )^{2} + a^{6} 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 [A] time = 1.24413, size = 339, normalized size = 2.01 \begin{align*} -\frac{\frac{60 \,{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{6}} - \frac{60 \,{\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b} - \frac{137 \, a^{4} b \tan \left (d x + c\right )^{5} + 274 \, a^{2} b^{3} \tan \left (d x + c\right )^{5} + 137 \, b^{5} \tan \left (d x + c\right )^{5} - 60 \, a^{5} \tan \left (d x + c\right )^{4} - 120 \, a^{3} b^{2} \tan \left (d x + c\right )^{4} - 60 \, a b^{4} \tan \left (d x + c\right )^{4} + 60 \, a^{4} b \tan \left (d x + c\right )^{3} + 30 \, a^{2} b^{3} \tan \left (d x + c\right )^{3} - 40 \, a^{5} \tan \left (d x + c\right )^{2} - 20 \, a^{3} b^{2} \tan \left (d x + c\right )^{2} + 15 \, a^{4} b \tan \left (d x + c\right ) - 12 \, a^{5}}{a^{6} \tan \left (d x + c\right )^{5}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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