Optimal. Leaf size=300 \[ -\frac{b \left (20 a^2 b^2+3 a^4+21 b^4\right )}{a^8 d (a+b \tan (c+d x))}-\frac{b \left (a^2+b^2\right ) \left (a^2+3 b^2\right )}{a^7 d (a+b \tan (c+d x))^2}-\frac{b \left (a^2+b^2\right )^2}{3 a^6 d (a+b \tan (c+d x))^3}-\frac{2 \left (a^2+5 b^2\right ) \cot ^3(c+d x)}{3 a^6 d}+\frac{2 b \left (2 a^2+5 b^2\right ) \cot ^2(c+d x)}{a^7 d}-\frac{\left (20 a^2 b^2+a^4+35 b^4\right ) \cot (c+d x)}{a^8 d}-\frac{4 b \left (10 a^2 b^2+a^4+14 b^4\right ) \log (\tan (c+d x))}{a^9 d}+\frac{4 b \left (10 a^2 b^2+a^4+14 b^4\right ) \log (a+b \tan (c+d x))}{a^9 d}+\frac{b \cot ^4(c+d x)}{a^5 d}-\frac{\cot ^5(c+d x)}{5 a^4 d} \]
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Rubi [A] time = 0.271243, antiderivative size = 300, 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} \[ -\frac{b \left (20 a^2 b^2+3 a^4+21 b^4\right )}{a^8 d (a+b \tan (c+d x))}-\frac{b \left (a^2+b^2\right ) \left (a^2+3 b^2\right )}{a^7 d (a+b \tan (c+d x))^2}-\frac{b \left (a^2+b^2\right )^2}{3 a^6 d (a+b \tan (c+d x))^3}-\frac{2 \left (a^2+5 b^2\right ) \cot ^3(c+d x)}{3 a^6 d}+\frac{2 b \left (2 a^2+5 b^2\right ) \cot ^2(c+d x)}{a^7 d}-\frac{\left (20 a^2 b^2+a^4+35 b^4\right ) \cot (c+d x)}{a^8 d}-\frac{4 b \left (10 a^2 b^2+a^4+14 b^4\right ) \log (\tan (c+d x))}{a^9 d}+\frac{4 b \left (10 a^2 b^2+a^4+14 b^4\right ) \log (a+b \tan (c+d x))}{a^9 d}+\frac{b \cot ^4(c+d x)}{a^5 d}-\frac{\cot ^5(c+d x)}{5 a^4 d} \]
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))^4} \, dx &=\frac{b \operatorname{Subst}\left (\int \frac{\left (b^2+x^2\right )^2}{x^6 (a+x)^4} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac{b \operatorname{Subst}\left (\int \left (\frac{b^4}{a^4 x^6}-\frac{4 b^4}{a^5 x^5}+\frac{2 b^2 \left (a^2+5 b^2\right )}{a^6 x^4}-\frac{4 \left (2 a^2 b^2+5 b^4\right )}{a^7 x^3}+\frac{a^4+20 a^2 b^2+35 b^4}{a^8 x^2}-\frac{4 \left (a^4+10 a^2 b^2+14 b^4\right )}{a^9 x}+\frac{\left (a^2+b^2\right )^2}{a^6 (a+x)^4}+\frac{2 \left (a^4+4 a^2 b^2+3 b^4\right )}{a^7 (a+x)^3}+\frac{3 a^4+20 a^2 b^2+21 b^4}{a^8 (a+x)^2}+\frac{4 \left (a^4+10 a^2 b^2+14 b^4\right )}{a^9 (a+x)}\right ) \, dx,x,b \tan (c+d x)\right )}{d}\\ &=-\frac{\left (a^4+20 a^2 b^2+35 b^4\right ) \cot (c+d x)}{a^8 d}+\frac{2 b \left (2 a^2+5 b^2\right ) \cot ^2(c+d x)}{a^7 d}-\frac{2 \left (a^2+5 b^2\right ) \cot ^3(c+d x)}{3 a^6 d}+\frac{b \cot ^4(c+d x)}{a^5 d}-\frac{\cot ^5(c+d x)}{5 a^4 d}-\frac{4 b \left (a^4+10 a^2 b^2+14 b^4\right ) \log (\tan (c+d x))}{a^9 d}+\frac{4 b \left (a^4+10 a^2 b^2+14 b^4\right ) \log (a+b \tan (c+d x))}{a^9 d}-\frac{b \left (a^2+b^2\right )^2}{3 a^6 d (a+b \tan (c+d x))^3}-\frac{b \left (a^2+b^2\right ) \left (a^2+3 b^2\right )}{a^7 d (a+b \tan (c+d x))^2}-\frac{b \left (3 a^4+20 a^2 b^2+21 b^4\right )}{a^8 d (a+b \tan (c+d x))}\\ \end{align*}
Mathematica [B] time = 1.63369, size = 673, normalized size = 2.24 \[ \frac{\sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left (-7680 b \left (10 a^2 b^2+a^4+14 b^4\right ) \log (\sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3+7680 b \left (10 a^2 b^2+a^4+14 b^4\right ) (a \cos (c+d x)+b \sin (c+d x))^3 \log (a \cos (c+d x)+b \sin (c+d x))+\csc ^5(c+d x) \left (372 a^5 b^3 \sin (2 (c+d x))-2476 a^5 b^3 \sin (4 (c+d x))+2756 a^5 b^3 \sin (6 (c+d x))-922 a^5 b^3 \sin (8 (c+d x))+4830 a^3 b^5 \sin (2 (c+d x))-9730 a^3 b^5 \sin (4 (c+d x))+7670 a^3 b^5 \sin (6 (c+d x))-2095 a^3 b^5 \sin (8 (c+d x))+776 a^6 b^2 \cos (6 (c+d x))-316 a^6 b^2 \cos (8 (c+d x))-1000 a^4 b^4 \cos (6 (c+d x))-70 a^4 b^4 \cos (8 (c+d x))-8540 a^2 b^6 \cos (6 (c+d x))+1645 a^2 b^6 \cos (8 (c+d x))-4 \left (194 a^6 b^2+1510 a^4 b^4+5705 a^2 b^6+52 a^8+4410 b^8\right ) \cos (2 (c+d x))+4 \left (-16 a^6 b^2+1010 a^4 b^4+4585 a^2 b^6+4 a^8+2205 b^8\right ) \cos (4 (c+d x))+380 a^6 b^2+3070 a^4 b^4+11375 a^2 b^6+264 a^7 b \sin (2 (c+d x))+144 a^7 b \sin (4 (c+d x))-24 a^7 b \sin (6 (c+d x))-24 a^7 b \sin (8 (c+d x))+16 a^8 \cos (6 (c+d x))-8 a^8 \cos (8 (c+d x))-200 a^8+1470 a b^7 \sin (2 (c+d x))-1470 a b^7 \sin (4 (c+d x))+630 a b^7 \sin (6 (c+d x))-105 a b^7 \sin (8 (c+d x))-2520 b^8 \cos (6 (c+d x))+315 b^8 \cos (8 (c+d x))+11025 b^8\right )\right )}{1920 a^9 d (a+b \tan (c+d x))^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.165, size = 476, normalized size = 1.6 \begin{align*} -{\frac{1}{5\,d{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{5}}}-{\frac{2}{3\,d{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{3}}}-{\frac{10\,{b}^{2}}{3\,d{a}^{6} \left ( \tan \left ( dx+c \right ) \right ) ^{3}}}-{\frac{1}{d{a}^{4}\tan \left ( dx+c \right ) }}-20\,{\frac{{b}^{2}}{d{a}^{6}\tan \left ( dx+c \right ) }}-35\,{\frac{{b}^{4}}{d{a}^{8}\tan \left ( dx+c \right ) }}+{\frac{b}{d{a}^{5} \left ( \tan \left ( dx+c \right ) \right ) ^{4}}}+4\,{\frac{b}{d{a}^{5} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}}+10\,{\frac{{b}^{3}}{d{a}^{7} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}}-4\,{\frac{b\ln \left ( \tan \left ( dx+c \right ) \right ) }{d{a}^{5}}}-40\,{\frac{{b}^{3}\ln \left ( \tan \left ( dx+c \right ) \right ) }{d{a}^{7}}}-56\,{\frac{{b}^{5}\ln \left ( \tan \left ( dx+c \right ) \right ) }{d{a}^{9}}}-3\,{\frac{b}{d{a}^{4} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-20\,{\frac{{b}^{3}}{d{a}^{6} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-21\,{\frac{{b}^{5}}{d{a}^{8} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-{\frac{b}{3\,{a}^{2}d \left ( a+b\tan \left ( dx+c \right ) \right ) ^{3}}}-{\frac{2\,{b}^{3}}{3\,d{a}^{4} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{3}}}-{\frac{{b}^{5}}{3\,d{a}^{6} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{3}}}-{\frac{b}{{a}^{3}d \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}-4\,{\frac{{b}^{3}}{d{a}^{5} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}-3\,{\frac{{b}^{5}}{d{a}^{7} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}+4\,{\frac{b\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d{a}^{5}}}+40\,{\frac{{b}^{3}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d{a}^{7}}}+56\,{\frac{{b}^{5}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d{a}^{9}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.23988, size = 439, normalized size = 1.46 \begin{align*} \frac{\frac{6 \, a^{6} b \tan \left (d x + c\right ) - 60 \,{\left (a^{4} b^{3} + 10 \, a^{2} b^{5} + 14 \, b^{7}\right )} \tan \left (d x + c\right )^{7} - 3 \, a^{7} - 150 \,{\left (a^{5} b^{2} + 10 \, a^{3} b^{4} + 14 \, a b^{6}\right )} \tan \left (d x + c\right )^{6} - 110 \,{\left (a^{6} b + 10 \, a^{4} b^{3} + 14 \, a^{2} b^{5}\right )} \tan \left (d x + c\right )^{5} - 15 \,{\left (a^{7} + 10 \, a^{5} b^{2} + 14 \, a^{3} b^{4}\right )} \tan \left (d x + c\right )^{4} + 6 \,{\left (5 \, a^{6} b + 7 \, a^{4} b^{3}\right )} \tan \left (d x + c\right )^{3} - 2 \,{\left (5 \, a^{7} + 7 \, a^{5} b^{2}\right )} \tan \left (d x + c\right )^{2}}{a^{8} b^{3} \tan \left (d x + c\right )^{8} + 3 \, a^{9} b^{2} \tan \left (d x + c\right )^{7} + 3 \, a^{10} b \tan \left (d x + c\right )^{6} + a^{11} \tan \left (d x + c\right )^{5}} + \frac{60 \,{\left (a^{4} b + 10 \, a^{2} b^{3} + 14 \, b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{9}} - \frac{60 \,{\left (a^{4} b + 10 \, a^{2} b^{3} + 14 \, b^{5}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{9}}}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 4.43096, size = 3507, normalized size = 11.69 \begin{align*} \text{result too large to display} \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 = 1.36148, size = 578, normalized size = 1.93 \begin{align*} -\frac{\frac{60 \,{\left (a^{4} b + 10 \, a^{2} b^{3} + 14 \, b^{5}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{9}} - \frac{60 \,{\left (a^{4} b^{2} + 10 \, a^{2} b^{4} + 14 \, b^{6}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{9} b} + \frac{5 \,{\left (22 \, a^{4} b^{4} \tan \left (d x + c\right )^{3} + 220 \, a^{2} b^{6} \tan \left (d x + c\right )^{3} + 308 \, b^{8} \tan \left (d x + c\right )^{3} + 75 \, a^{5} b^{3} \tan \left (d x + c\right )^{2} + 720 \, a^{3} b^{5} \tan \left (d x + c\right )^{2} + 987 \, a b^{7} \tan \left (d x + c\right )^{2} + 87 \, a^{6} b^{2} \tan \left (d x + c\right ) + 792 \, a^{4} b^{4} \tan \left (d x + c\right ) + 1059 \, a^{2} b^{6} \tan \left (d x + c\right ) + 35 \, a^{7} b + 294 \, a^{5} b^{3} + 381 \, a^{3} b^{5}\right )}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{3} a^{9}} - \frac{137 \, a^{4} b \tan \left (d x + c\right )^{5} + 1370 \, a^{2} b^{3} \tan \left (d x + c\right )^{5} + 1918 \, b^{5} \tan \left (d x + c\right )^{5} - 15 \, a^{5} \tan \left (d x + c\right )^{4} - 300 \, a^{3} b^{2} \tan \left (d x + c\right )^{4} - 525 \, a b^{4} \tan \left (d x + c\right )^{4} + 60 \, a^{4} b \tan \left (d x + c\right )^{3} + 150 \, a^{2} b^{3} \tan \left (d x + c\right )^{3} - 10 \, a^{5} \tan \left (d x + c\right )^{2} - 50 \, a^{3} b^{2} \tan \left (d x + c\right )^{2} + 15 \, a^{4} b \tan \left (d x + c\right ) - 3 \, a^{5}}{a^{9} \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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