Optimal. Leaf size=265 \[ \frac {3 b \cot ^4(c+d x)}{4 a^4 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}-\frac {2 b \left (a^2+b^2\right ) \left (a^2+3 b^2\right )}{a^7 d (a+b \tan (c+d x))}-\frac {b \left (a^2+b^2\right )^2}{2 a^6 d (a+b \tan (c+d x))^2}+\frac {b \left (3 a^2+5 b^2\right ) \cot ^2(c+d x)}{a^6 d}-\frac {2 \left (a^2+3 b^2\right ) \cot ^3(c+d x)}{3 a^5 d}-\frac {b \left (3 a^4+20 a^2 b^2+21 b^4\right ) \log (\tan (c+d x))}{a^8 d}+\frac {b \left (3 a^4+20 a^2 b^2+21 b^4\right ) \log (a+b \tan (c+d x))}{a^8 d}-\frac {\left (a^4+12 a^2 b^2+15 b^4\right ) \cot (c+d x)}{a^7 d} \]
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Rubi [A] time = 0.24, antiderivative size = 265, normalized size of antiderivative = 1.00, 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} \[ -\frac {2 b \left (a^2+b^2\right ) \left (a^2+3 b^2\right )}{a^7 d (a+b \tan (c+d x))}-\frac {b \left (a^2+b^2\right )^2}{2 a^6 d (a+b \tan (c+d x))^2}-\frac {2 \left (a^2+3 b^2\right ) \cot ^3(c+d x)}{3 a^5 d}+\frac {b \left (3 a^2+5 b^2\right ) \cot ^2(c+d x)}{a^6 d}-\frac {\left (12 a^2 b^2+a^4+15 b^4\right ) \cot (c+d x)}{a^7 d}-\frac {b \left (20 a^2 b^2+3 a^4+21 b^4\right ) \log (\tan (c+d x))}{a^8 d}+\frac {b \left (20 a^2 b^2+3 a^4+21 b^4\right ) \log (a+b \tan (c+d x))}{a^8 d}+\frac {3 b \cot ^4(c+d x)}{4 a^4 d}-\frac {\cot ^5(c+d x)}{5 a^3 d} \]
Antiderivative was successfully verified.
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Rule 894
Rule 3516
Rubi steps
\begin {align*} \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx &=\frac {b \operatorname {Subst}\left (\int \frac {\left (b^2+x^2\right )^2}{x^6 (a+x)^3} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac {b \operatorname {Subst}\left (\int \left (\frac {b^4}{a^3 x^6}-\frac {3 b^4}{a^4 x^5}+\frac {2 b^2 \left (a^2+3 b^2\right )}{a^5 x^4}-\frac {2 \left (3 a^2 b^2+5 b^4\right )}{a^6 x^3}+\frac {a^4+12 a^2 b^2+15 b^4}{a^7 x^2}+\frac {-3 a^4-20 a^2 b^2-21 b^4}{a^8 x}+\frac {\left (a^2+b^2\right )^2}{a^6 (a+x)^3}+\frac {2 \left (a^4+4 a^2 b^2+3 b^4\right )}{a^7 (a+x)^2}+\frac {3 a^4+20 a^2 b^2+21 b^4}{a^8 (a+x)}\right ) \, dx,x,b \tan (c+d x)\right )}{d}\\ &=-\frac {\left (a^4+12 a^2 b^2+15 b^4\right ) \cot (c+d x)}{a^7 d}+\frac {b \left (3 a^2+5 b^2\right ) \cot ^2(c+d x)}{a^6 d}-\frac {2 \left (a^2+3 b^2\right ) \cot ^3(c+d x)}{3 a^5 d}+\frac {3 b \cot ^4(c+d x)}{4 a^4 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}-\frac {b \left (3 a^4+20 a^2 b^2+21 b^4\right ) \log (\tan (c+d x))}{a^8 d}+\frac {b \left (3 a^4+20 a^2 b^2+21 b^4\right ) \log (a+b \tan (c+d x))}{a^8 d}-\frac {b \left (a^2+b^2\right )^2}{2 a^6 d (a+b \tan (c+d x))^2}-\frac {2 b \left (a^2+b^2\right ) \left (a^2+3 b^2\right )}{a^7 d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [A] time = 4.76, size = 494, normalized size = 1.86 \[ -\frac {\csc ^5(c+d x) \left (960 b \left (3 a^4+20 a^2 b^2+21 b^4\right ) \sin ^5(c+d x) (a+b \tan (c+d x))^2 (\log (\sin (c+d x))-\log (a \cos (c+d x)+b \sin (c+d x)))+5 \sec (c+d x) \left (40 a^7-27 a^5 b^2-42 a^3 b^4-3 b \left (8 a^6+89 a^4 b^2+345 a^2 b^4+210 b^6\right ) \tan (c+d x)+135 a b^6\right )+\sec ^2(c+d x) \left (8 a^7 \cos (7 (c+d x))-126 a^6 b \sin (3 (c+d x))+10 a^6 b \sin (5 (c+d x))+16 a^6 b \sin (7 (c+d x))+187 a^5 b^2 \cos (7 (c+d x))+1665 a^4 b^3 \sin (3 (c+d x))-1215 a^4 b^3 \sin (5 (c+d x))+345 a^4 b^3 \sin (7 (c+d x))+210 a^3 b^4 \cos (7 (c+d x))+4635 a^2 b^5 \sin (3 (c+d x))-2565 a^2 b^5 \sin (5 (c+d x))+585 a^2 b^5 \sin (7 (c+d x))+\left (8 a^7+567 a^5 b^2+630 a^3 b^4-1215 a b^6\right ) \cos (3 (c+d x))-\left (24 a^7+619 a^5 b^2+630 a^3 b^4-675 a b^6\right ) \cos (5 (c+d x))-135 a b^6 \cos (7 (c+d x))+1890 b^7 \sin (3 (c+d x))-630 b^7 \sin (5 (c+d x))+90 b^7 \sin (7 (c+d x))\right )\right )}{960 a^8 d (a+b \tan (c+d x))^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 1018, normalized size = 3.84 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.36, size = 382, normalized size = 1.44 \[ -\frac {\frac {60 \, {\left (3 \, a^{4} b + 20 \, a^{2} b^{3} + 21 \, b^{5}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{8}} - \frac {60 \, {\left (3 \, a^{4} b^{2} + 20 \, a^{2} b^{4} + 21 \, b^{6}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b} + \frac {30 \, {\left (9 \, a^{4} b^{3} \tan \left (d x + c\right )^{2} + 60 \, a^{2} b^{5} \tan \left (d x + c\right )^{2} + 63 \, b^{7} \tan \left (d x + c\right )^{2} + 22 \, a^{5} b^{2} \tan \left (d x + c\right ) + 136 \, a^{3} b^{4} \tan \left (d x + c\right ) + 138 \, a b^{6} \tan \left (d x + c\right ) + 14 \, a^{6} b + 78 \, a^{4} b^{3} + 76 \, a^{2} b^{5}\right )}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{2} a^{8}} - \frac {411 \, a^{4} b \tan \left (d x + c\right )^{5} + 2740 \, a^{2} b^{3} \tan \left (d x + c\right )^{5} + 2877 \, b^{5} \tan \left (d x + c\right )^{5} - 60 \, a^{5} \tan \left (d x + c\right )^{4} - 720 \, a^{3} b^{2} \tan \left (d x + c\right )^{4} - 900 \, a b^{4} \tan \left (d x + c\right )^{4} + 180 \, a^{4} b \tan \left (d x + c\right )^{3} + 300 \, a^{2} b^{3} \tan \left (d x + c\right )^{3} - 40 \, a^{5} \tan \left (d x + c\right )^{2} - 120 \, a^{3} b^{2} \tan \left (d x + c\right )^{2} + 45 \, a^{4} b \tan \left (d x + c\right ) - 12 \, a^{5}}{a^{8} \tan \left (d x + c\right )^{5}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.57, size = 410, normalized size = 1.55 \[ \frac {3 b \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{4} d}+\frac {20 b^{3} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \,a^{6}}+\frac {21 b^{5} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \,a^{8}}-\frac {b}{2 a^{2} d \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {b^{3}}{d \,a^{4} \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {b^{5}}{2 d \,a^{6} \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {2 b}{a^{3} d \left (a +b \tan \left (d x +c \right )\right )}-\frac {8 b^{3}}{d \,a^{5} \left (a +b \tan \left (d x +c \right )\right )}-\frac {6 b^{5}}{d \,a^{7} \left (a +b \tan \left (d x +c \right )\right )}-\frac {1}{5 d \,a^{3} \tan \left (d x +c \right )^{5}}-\frac {2}{3 d \,a^{3} \tan \left (d x +c \right )^{3}}-\frac {2 b^{2}}{d \,a^{5} \tan \left (d x +c \right )^{3}}-\frac {1}{d \,a^{3} \tan \left (d x +c \right )}-\frac {12 b^{2}}{d \,a^{5} \tan \left (d x +c \right )}-\frac {15 b^{4}}{d \,a^{7} \tan \left (d x +c \right )}+\frac {3 b}{4 d \,a^{4} \tan \left (d x +c \right )^{4}}+\frac {3 b}{d \,a^{4} \tan \left (d x +c \right )^{2}}+\frac {5 b^{3}}{d \,a^{6} \tan \left (d x +c \right )^{2}}-\frac {3 b \ln \left (\tan \left (d x +c \right )\right )}{a^{4} d}-\frac {20 b^{3} \ln \left (\tan \left (d x +c \right )\right )}{d \,a^{6}}-\frac {21 b^{5} \ln \left (\tan \left (d x +c \right )\right )}{d \,a^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.78, size = 281, normalized size = 1.06 \[ \frac {\frac {21 \, a^{5} b \tan \left (d x + c\right ) - 60 \, {\left (3 \, a^{4} b^{2} + 20 \, a^{2} b^{4} + 21 \, b^{6}\right )} \tan \left (d x + c\right )^{6} - 12 \, a^{6} - 90 \, {\left (3 \, a^{5} b + 20 \, a^{3} b^{3} + 21 \, a b^{5}\right )} \tan \left (d x + c\right )^{5} - 20 \, {\left (3 \, a^{6} + 20 \, a^{4} b^{2} + 21 \, a^{2} b^{4}\right )} \tan \left (d x + c\right )^{4} + 5 \, {\left (20 \, a^{5} b + 21 \, a^{3} b^{3}\right )} \tan \left (d x + c\right )^{3} - 2 \, {\left (20 \, a^{6} + 21 \, a^{4} b^{2}\right )} \tan \left (d x + c\right )^{2}}{a^{7} b^{2} \tan \left (d x + c\right )^{7} + 2 \, a^{8} b \tan \left (d x + c\right )^{6} + a^{9} \tan \left (d x + c\right )^{5}} + \frac {60 \, {\left (3 \, a^{4} b + 20 \, a^{2} b^{3} + 21 \, b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8}} - \frac {60 \, {\left (3 \, a^{4} b + 20 \, a^{2} b^{3} + 21 \, b^{5}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{8}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.32, size = 297, normalized size = 1.12 \[ \frac {2\,b\,\mathrm {atanh}\left (\frac {b\,\left (a+2\,b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (3\,a^4+20\,a^2\,b^2+21\,b^4\right )}{a\,\left (3\,a^4\,b+20\,a^2\,b^3+21\,b^5\right )}\right )\,\left (3\,a^4+20\,a^2\,b^2+21\,b^4\right )}{a^8\,d}-\frac {\frac {1}{5\,a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (3\,a^4+20\,a^2\,b^2+21\,b^4\right )}{3\,a^5}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (20\,a^2+21\,b^2\right )}{30\,a^3}-\frac {7\,b\,\mathrm {tan}\left (c+d\,x\right )}{20\,a^2}+\frac {b^2\,{\mathrm {tan}\left (c+d\,x\right )}^6\,\left (3\,a^4+20\,a^2\,b^2+21\,b^4\right )}{a^7}+\frac {3\,b\,{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (3\,a^4+20\,a^2\,b^2+21\,b^4\right )}{2\,a^6}-\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (20\,a^2+21\,b^2\right )}{12\,a^4}}{d\,\left (a^2\,{\mathrm {tan}\left (c+d\,x\right )}^5+2\,a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^6+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^7\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{6}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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