Optimal. Leaf size=59 \[ \frac{\frac{1}{a^2}-\frac{1}{b^2}}{a+b \tan (x)}-\frac{\log (a+b \tan (x))}{a^3}+\frac{\log (\tan (x))}{a^3}+\frac{\frac{a}{b^2}+\frac{1}{a}}{2 (a+b \tan (x))^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0819996, antiderivative size = 59, normalized size of antiderivative = 1., 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.
[In]
[Out]
Rule 3087
Rule 894
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*}
Mathematica [A] time = 0.196871, 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.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.12, size = 73, normalized size = 1.2 \begin{align*}{\frac{a}{2\,{b}^{2} \left ( a+b\tan \left ( x \right ) \right ) ^{2}}}+{\frac{1}{2\,a \left ( a+b\tan \left ( x \right ) \right ) ^{2}}}-{\frac{1}{{b}^{2} \left ( a+b\tan \left ( x \right ) \right ) }}+{\frac{1}{{a}^{2} \left ( a+b\tan \left ( x \right ) \right ) }}-{\frac{\ln \left ( a+b\tan \left ( x \right ) \right ) }{{a}^{3}}}+{\frac{\ln \left ( \tan \left ( x \right ) \right ) }{{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.24345, size = 232, normalized size = 3.93 \begin{align*} -\frac{2 \,{\left (\frac{2 \, a b \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac{2 \, a b \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac{{\left (a^{2} - 3 \, b^{2}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}}{a^{5} + \frac{4 \, a^{4} b \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac{4 \, a^{4} b \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{a^{5} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - \frac{2 \,{\left (a^{5} - 2 \, a^{3} b^{2}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}} - \frac{\log \left (-a - \frac{2 \, b \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}{a^{3}} + \frac{\log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 0.553722, size = 524, normalized size = 8.88 \begin{align*} \frac{4 \, a^{2} b^{2} \cos \left (x\right )^{2} + a^{4} - a^{2} b^{2} - 2 \,{\left (a^{3} b - a b^{3}\right )} \cos \left (x\right ) \sin \left (x\right ) -{\left (a^{2} b^{2} + b^{4} +{\left (a^{4} - b^{4}\right )} \cos \left (x\right )^{2} + 2 \,{\left (a^{3} b + a b^{3}\right )} \cos \left (x\right ) \sin \left (x\right )\right )} \log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}\right ) +{\left (a^{2} b^{2} + b^{4} +{\left (a^{4} - b^{4}\right )} \cos \left (x\right )^{2} + 2 \,{\left (a^{3} b + a b^{3}\right )} \cos \left (x\right ) \sin \left (x\right )\right )} \log \left (-\frac{1}{4} \, \cos \left (x\right )^{2} + \frac{1}{4}\right )}{2 \,{\left (a^{5} b^{2} + a^{3} b^{4} +{\left (a^{7} - a^{3} b^{4}\right )} \cos \left (x\right )^{2} + 2 \,{\left (a^{6} b + a^{4} b^{3}\right )} \cos \left (x\right ) \sin \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc{\left (x \right )}}{\left (a \cos{\left (x \right )} + b \sin{\left (x \right )}\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.18307, size = 104, normalized size = 1.76 \begin{align*} -\frac{\log \left ({\left | b \tan \left (x\right ) + a \right |}\right )}{a^{3}} + \frac{\log \left ({\left | \tan \left (x\right ) \right |}\right )}{a^{3}} + \frac{3 \, b^{4} \tan \left (x\right )^{2} - 2 \, a^{3} b \tan \left (x\right ) + 8 \, a b^{3} \tan \left (x\right ) - a^{4} + 6 \, a^{2} b^{2}}{2 \,{\left (b \tan \left (x\right ) + a\right )}^{2} a^{3} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]