Optimal. Leaf size=55 \[ -\frac {\sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^2}+\frac {b \tanh ^{-1}(\cos (x))}{a^2}-\frac {\csc (x)}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3103, 3770, 3074, 206} \[ -\frac {\sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^2}+\frac {b \tanh ^{-1}(\cos (x))}{a^2}-\frac {\csc (x)}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 3074
Rule 3103
Rule 3770
Rubi steps
\begin {align*} \int \frac {\csc ^2(x)}{a \cos (x)+b \sin (x)} \, dx &=-\frac {\csc (x)}{a}-\frac {b \int \csc (x) \, dx}{a^2}+\frac {\left (a^2+b^2\right ) \int \frac {1}{a \cos (x)+b \sin (x)} \, dx}{a^2}\\ &=\frac {b \tanh ^{-1}(\cos (x))}{a^2}-\frac {\csc (x)}{a}-\frac {\left (a^2+b^2\right ) \operatorname {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,b \cos (x)-a \sin (x)\right )}{a^2}\\ &=\frac {b \tanh ^{-1}(\cos (x))}{a^2}-\frac {\sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^2}-\frac {\csc (x)}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.13, size = 67, normalized size = 1.22 \[ \frac {2 \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {a \tan \left (\frac {x}{2}\right )-b}{\sqrt {a^2+b^2}}\right )-a \csc (x)+b \left (\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right )}{a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.49, size = 133, normalized size = 2.42 \[ \frac {b \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) \sin \relax (x) - b \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) \sin \relax (x) + \sqrt {a^{2} + b^{2}} \log \left (-\frac {2 \, a b \cos \relax (x) \sin \relax (x) + {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \relax (x) - a \sin \relax (x)\right )}}{2 \, a b \cos \relax (x) \sin \relax (x) + {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + b^{2}}\right ) \sin \relax (x) - 2 \, a}{2 \, a^{2} \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 2.97, size = 108, normalized size = 1.96 \[ -\frac {b \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{a^{2}} - \frac {\tan \left (\frac {1}{2} \, x\right )}{2 \, a} - \frac {\sqrt {a^{2} + b^{2}} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{a^{2}} + \frac {2 \, b \tan \left (\frac {1}{2} \, x\right ) - a}{2 \, a^{2} \tan \left (\frac {1}{2} \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.79, size = 107, normalized size = 1.95 \[ -\frac {\tan \left (\frac {x}{2}\right )}{2 a}-\frac {1}{2 a \tan \left (\frac {x}{2}\right )}-\frac {b \ln \left (\tan \left (\frac {x}{2}\right )\right )}{a^{2}}+\frac {2 \arctanh \left (\frac {2 a \tan \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}+\frac {2 \arctanh \left (\frac {2 a \tan \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right ) b^{2}}{a^{2} \sqrt {a^{2}+b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.43, size = 107, normalized size = 1.95 \[ -\frac {b \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )}{a^{2}} - \frac {\sqrt {a^{2} + b^{2}} \log \left (\frac {b - \frac {a \sin \relax (x)}{\cos \relax (x) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \relax (x)}{\cos \relax (x) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{a^{2}} - \frac {\cos \relax (x) + 1}{2 \, a \sin \relax (x)} - \frac {\sin \relax (x)}{2 \, a {\left (\cos \relax (x) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.68, size = 170, normalized size = 3.09 \[ \frac {2\,\mathrm {atanh}\left (\frac {a^3\,\cos \left (\frac {x}{2}\right )\,\sqrt {a^2+b^2}+4\,b^3\,\sin \left (\frac {x}{2}\right )\,\sqrt {a^2+b^2}+3\,a^2\,b\,\sin \left (\frac {x}{2}\right )\,\sqrt {a^2+b^2}+2\,a\,b^2\,\cos \left (\frac {x}{2}\right )\,\sqrt {a^2+b^2}}{\sin \left (\frac {x}{2}\right )\,a^4+2\,\cos \left (\frac {x}{2}\right )\,a^3\,b+5\,\sin \left (\frac {x}{2}\right )\,a^2\,b^2+2\,\cos \left (\frac {x}{2}\right )\,a\,b^3+4\,\sin \left (\frac {x}{2}\right )\,b^4}\right )\,\sqrt {a^2+b^2}}{a^2}-\frac {1}{a\,\sin \relax (x)}-\frac {b\,\ln \left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )}{a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{2}{\relax (x )}}{a \cos {\relax (x )} + b \sin {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________