Optimal. Leaf size=169 \[ \frac {b \cot ^4(c+d x)}{4 a^2 d}-\frac {b \left (a^2+b^2\right )^2 \log (\tan (c+d x))}{a^6 d}+\frac {b \left (a^2+b^2\right )^2 \log (a+b \tan (c+d x))}{a^6 d}-\frac {\left (a^2+b^2\right )^2 \cot (c+d x)}{a^5 d}+\frac {b \left (2 a^2+b^2\right ) \cot ^2(c+d x)}{2 a^4 d}-\frac {\left (2 a^2+b^2\right ) \cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{5 a d} \]
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Rubi [A] time = 0.15, antiderivative size = 169, 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} \[ -\frac {\left (2 a^2+b^2\right ) \cot ^3(c+d x)}{3 a^3 d}+\frac {b \left (2 a^2+b^2\right ) \cot ^2(c+d x)}{2 a^4 d}-\frac {\left (a^2+b^2\right )^2 \cot (c+d x)}{a^5 d}-\frac {b \left (a^2+b^2\right )^2 \log (\tan (c+d x))}{a^6 d}+\frac {b \left (a^2+b^2\right )^2 \log (a+b \tan (c+d x))}{a^6 d}+\frac {b \cot ^4(c+d x)}{4 a^2 d}-\frac {\cot ^5(c+d x)}{5 a d} \]
Antiderivative was successfully verified.
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Rule 894
Rule 3516
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*}
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Mathematica [A] time = 2.10, size = 150, normalized size = 0.89 \[ \frac {15 b \left (a^4 \csc ^4(c+d x)+2 a^2 \left (a^2+b^2\right ) \csc ^2(c+d x)-4 \left (a^2+b^2\right )^2 (\log (\sin (c+d x))-\log (a \cos (c+d x)+b \sin (c+d x)))\right )-4 \cot (c+d x) \left (3 a^5 \csc ^4(c+d x)+8 a^5+25 a^3 b^2+a^3 \left (4 a^2+5 b^2\right ) \csc ^2(c+d x)+15 a b^4\right )}{60 a^6 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 385, normalized size = 2.28 \[ -\frac {4 \, {\left (8 \, a^{5} + 25 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} - 20 \, {\left (4 \, a^{5} + 11 \, a^{3} b^{2} + 6 \, a b^{4}\right )} \cos \left (d x + c\right )^{3} - 30 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) \sin \left (d x + c\right ) + 30 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{4} \, \cos \left (d x + c\right )^{2} + \frac {1}{4}\right ) \sin \left (d x + c\right ) + 60 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right ) - 15 \, {\left (3 \, a^{4} b + 2 \, a^{2} b^{3} - 2 \, {\left (a^{4} b + a^{2} b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{60 \, {\left (a^{6} d \cos \left (d x + c\right )^{4} - 2 \, a^{6} d \cos \left (d x + c\right )^{2} + a^{6} d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.93, size = 251, normalized size = 1.49 \[ -\frac {\frac {60 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{6}} - \frac {60 \, {\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b} - \frac {137 \, a^{4} b \tan \left (d x + c\right )^{5} + 274 \, a^{2} b^{3} \tan \left (d x + c\right )^{5} + 137 \, b^{5} \tan \left (d x + c\right )^{5} - 60 \, a^{5} \tan \left (d x + c\right )^{4} - 120 \, a^{3} b^{2} \tan \left (d x + c\right )^{4} - 60 \, a b^{4} \tan \left (d x + c\right )^{4} + 60 \, a^{4} b \tan \left (d x + c\right )^{3} + 30 \, a^{2} b^{3} \tan \left (d x + c\right )^{3} - 40 \, a^{5} \tan \left (d x + c\right )^{2} - 20 \, a^{3} b^{2} \tan \left (d x + c\right )^{2} + 15 \, a^{4} b \tan \left (d x + c\right ) - 12 \, a^{5}}{a^{6} \tan \left (d x + c\right )^{5}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 273, normalized size = 1.62 \[ \frac {b \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{2} d}+\frac {2 b^{3} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \,a^{4}}+\frac {b^{5} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \,a^{6}}-\frac {1}{5 d a \tan \left (d x +c \right )^{5}}-\frac {2}{3 d a \tan \left (d x +c \right )^{3}}-\frac {b^{2}}{3 d \,a^{3} \tan \left (d x +c \right )^{3}}-\frac {1}{d a \tan \left (d x +c \right )}-\frac {2 b^{2}}{d \,a^{3} \tan \left (d x +c \right )}-\frac {b^{4}}{d \,a^{5} \tan \left (d x +c \right )}+\frac {b}{4 d \,a^{2} \tan \left (d x +c \right )^{4}}+\frac {b}{d \,a^{2} \tan \left (d x +c \right )^{2}}+\frac {b^{3}}{2 d \,a^{4} \tan \left (d x +c \right )^{2}}-\frac {b \ln \left (\tan \left (d x +c \right )\right )}{a^{2} d}-\frac {2 b^{3} \ln \left (\tan \left (d x +c \right )\right )}{d \,a^{4}}-\frac {b^{5} \ln \left (\tan \left (d x +c \right )\right )}{d \,a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 168, normalized size = 0.99 \[ \frac {\frac {60 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6}} - \frac {60 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{6}} + \frac {15 \, a^{3} b \tan \left (d x + c\right ) - 60 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )^{4} - 12 \, a^{4} + 30 \, {\left (2 \, a^{3} b + a b^{3}\right )} \tan \left (d x + c\right )^{3} - 20 \, {\left (2 \, a^{4} + a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2}}{a^{5} \tan \left (d x + c\right )^{5}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.33, size = 167, normalized size = 0.99 \[ \frac {2\,b\,\mathrm {atanh}\left (\frac {b\,{\left (a^2+b^2\right )}^2\,\left (a+2\,b\,\mathrm {tan}\left (c+d\,x\right )\right )}{a\,\left (a^4\,b+2\,a^2\,b^3+b^5\right )}\right )\,{\left (a^2+b^2\right )}^2}{a^6\,d}-\frac {\frac {1}{5\,a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (2\,a^2+b^2\right )}{3\,a^3}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (a^4+2\,a^2\,b^2+b^4\right )}{a^5}-\frac {b\,\mathrm {tan}\left (c+d\,x\right )}{4\,a^2}-\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (2\,a^2+b^2\right )}{2\,a^4}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^5} \]
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 \[ \int \frac {\csc ^{6}{\left (c + d x \right )}}{a + b \tan {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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