Optimal. Leaf size=59 \[ -\frac {\log (a+b \tan (x))}{a^3}+\frac {\log (\tan (x))}{a^3}+\frac {\frac {1}{a^2}-\frac {1}{b^2}}{a+b \tan (x)}+\frac {\frac {a}{b^2}+\frac {1}{a}}{2 (a+b \tan (x))^2} \]
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Rubi [A] time = 0.08, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3087, 894} \[ \frac {\frac {1}{a^2}-\frac {1}{b^2}}{a+b \tan (x)}-\frac {\log (a+b \tan (x))}{a^3}+\frac {\log (\tan (x))}{a^3}+\frac {\frac {a}{b^2}+\frac {1}{a}}{2 (a+b \tan (x))^2} \]
Antiderivative was successfully verified.
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Rule 894
Rule 3087
Rubi steps
\begin {align*} \int \frac {\csc (x)}{(a \cos (x)+b \sin (x))^3} \, dx &=\operatorname {Subst}\left (\int \frac {1+x^2}{x (a+b x)^3} \, dx,x,\tan (x)\right )\\ &=\operatorname {Subst}\left (\int \left (\frac {1}{a^3 x}+\frac {-a^2-b^2}{a b (a+b x)^3}+\frac {a^2-b^2}{a^2 b (a+b x)^2}-\frac {b}{a^3 (a+b x)}\right ) \, dx,x,\tan (x)\right )\\ &=\frac {\log (\tan (x))}{a^3}-\frac {\log (a+b \tan (x))}{a^3}+\frac {\frac {1}{a}+\frac {a}{b^2}}{2 (a+b \tan (x))^2}+\frac {\frac {1}{a^2}-\frac {1}{b^2}}{a+b \tan (x)}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 96, normalized size = 1.63 \[ \frac {2 a^2 \cot ^2(x) (\log (\sin (x))-\log (a \cos (x)+b \sin (x)))+a^2 \csc ^2(x)+2 b^2 (-\log (a \cos (x)+b \sin (x))+\log (\sin (x))-1)+2 a b \cot (x) (-2 \log (a \cos (x)+b \sin (x))+2 \log (\sin (x))-1)}{2 a^3 (a \cot (x)+b)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 220, normalized size = 3.73 \[ \frac {4 \, a^{2} b^{2} \cos \relax (x)^{2} + a^{4} - a^{2} b^{2} - 2 \, {\left (a^{3} b - a b^{3}\right )} \cos \relax (x) \sin \relax (x) - {\left (a^{2} b^{2} + b^{4} + {\left (a^{4} - b^{4}\right )} \cos \relax (x)^{2} + 2 \, {\left (a^{3} b + a b^{3}\right )} \cos \relax (x) \sin \relax (x)\right )} \log \left (2 \, a b \cos \relax (x) \sin \relax (x) + {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + b^{2}\right ) + {\left (a^{2} b^{2} + b^{4} + {\left (a^{4} - b^{4}\right )} \cos \relax (x)^{2} + 2 \, {\left (a^{3} b + a b^{3}\right )} \cos \relax (x) \sin \relax (x)\right )} \log \left (-\frac {1}{4} \, \cos \relax (x)^{2} + \frac {1}{4}\right )}{2 \, {\left (a^{5} b^{2} + a^{3} b^{4} + {\left (a^{7} - a^{3} b^{4}\right )} \cos \relax (x)^{2} + 2 \, {\left (a^{6} b + a^{4} b^{3}\right )} \cos \relax (x) \sin \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.02, size = 77, normalized size = 1.31 \[ -\frac {\log \left ({\left | b \tan \relax (x) + a \right |}\right )}{a^{3}} + \frac {\log \left ({\left | \tan \relax (x) \right |}\right )}{a^{3}} + \frac {3 \, b^{4} \tan \relax (x)^{2} - 2 \, a^{3} b \tan \relax (x) + 8 \, a b^{3} \tan \relax (x) - a^{4} + 6 \, a^{2} b^{2}}{2 \, {\left (b \tan \relax (x) + a\right )}^{2} a^{3} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.67, size = 73, normalized size = 1.24 \[ \frac {a}{2 b^{2} \left (a +b \tan \relax (x )\right )^{2}}+\frac {1}{2 a \left (a +b \tan \relax (x )\right )^{2}}-\frac {1}{b^{2} \left (a +b \tan \relax (x )\right )}+\frac {1}{a^{2} \left (a +b \tan \relax (x )\right )}-\frac {\ln \left (a +b \tan \relax (x )\right )}{a^{3}}+\frac {\ln \left (\tan \relax (x )\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 172, normalized size = 2.92 \[ -\frac {2 \, {\left (\frac {2 \, a b \sin \relax (x)}{\cos \relax (x) + 1} - \frac {2 \, a b \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} - \frac {{\left (a^{2} - 3 \, b^{2}\right )} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}}\right )}}{a^{5} + \frac {4 \, a^{4} b \sin \relax (x)}{\cos \relax (x) + 1} - \frac {4 \, a^{4} b \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {a^{5} \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} - \frac {2 \, {\left (a^{5} - 2 \, a^{3} b^{2}\right )} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}}} - \frac {\log \left (-a - \frac {2 \, b \sin \relax (x)}{\cos \relax (x) + 1} + \frac {a \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}}\right )}{a^{3}} + \frac {\log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.71, size = 131, normalized size = 2.22 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{a^3}-\frac {\ln \left (-a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )+a\right )}{a^3}+\frac {\frac {2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (a^2-3\,b^2\right )}{a^3}+\frac {4\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{a^2}-\frac {4\,b\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^2}}{a^2-{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (2\,a^2-4\,b^2\right )+a^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+4\,a\,b\,\mathrm {tan}\left (\frac {x}{2}\right )-4\,a\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3} \]
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 \[ \int \frac {\csc {\relax (x )}}{\left (a \cos {\relax (x )} + b \sin {\relax (x )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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