Optimal. Leaf size=118 \[ -\frac {2 b^2 \tanh ^{-1}(\cos (x))}{a^4}+\frac {2 b \csc (x)}{a^3}-\frac {\tanh ^{-1}(\cos (x))}{2 a^2}-\frac {\cot (x) \csc (x)}{2 a^2}-\frac {\left (a^2+b^2\right ) \tanh ^{-1}(\cos (x))}{a^4}+\frac {3 b \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^4}+\frac {a^2+b^2}{a^3 (a \cos (x)+b \sin (x))} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.18, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3105, 3093, 3770, 3074, 206, 3768, 3103} \[ \frac {a^2+b^2}{a^3 (a \cos (x)+b \sin (x))}-\frac {2 b^2 \tanh ^{-1}(\cos (x))}{a^4}-\frac {\left (a^2+b^2\right ) \tanh ^{-1}(\cos (x))}{a^4}+\frac {3 b \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^4}+\frac {2 b \csc (x)}{a^3}-\frac {\tanh ^{-1}(\cos (x))}{2 a^2}-\frac {\cot (x) \csc (x)}{2 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 3074
Rule 3093
Rule 3103
Rule 3105
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\csc ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx &=\frac {\int \csc ^3(x) \, dx}{a^2}-\frac {(2 b) \int \frac {\csc ^2(x)}{a \cos (x)+b \sin (x)} \, dx}{a^2}+\frac {\left (a^2+b^2\right ) \int \frac {\csc (x)}{(a \cos (x)+b \sin (x))^2} \, dx}{a^2}\\ &=\frac {2 b \csc (x)}{a^3}-\frac {\cot (x) \csc (x)}{2 a^2}+\frac {a^2+b^2}{a^3 (a \cos (x)+b \sin (x))}+\frac {\int \csc (x) \, dx}{2 a^2}+\frac {\left (2 b^2\right ) \int \csc (x) \, dx}{a^4}+\frac {\left (a^2+b^2\right ) \int \csc (x) \, dx}{a^4}-\frac {\left (b \left (a^2+b^2\right )\right ) \int \frac {1}{a \cos (x)+b \sin (x)} \, dx}{a^4}-\frac {\left (2 b \left (a^2+b^2\right )\right ) \int \frac {1}{a \cos (x)+b \sin (x)} \, dx}{a^4}\\ &=-\frac {\tanh ^{-1}(\cos (x))}{2 a^2}-\frac {2 b^2 \tanh ^{-1}(\cos (x))}{a^4}-\frac {\left (a^2+b^2\right ) \tanh ^{-1}(\cos (x))}{a^4}+\frac {2 b \csc (x)}{a^3}-\frac {\cot (x) \csc (x)}{2 a^2}+\frac {a^2+b^2}{a^3 (a \cos (x)+b \sin (x))}+\frac {\left (b \left (a^2+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,b \cos (x)-a \sin (x)\right )}{a^4}+\frac {\left (2 b \left (a^2+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,b \cos (x)-a \sin (x)\right )}{a^4}\\ &=-\frac {\tanh ^{-1}(\cos (x))}{2 a^2}-\frac {2 b^2 \tanh ^{-1}(\cos (x))}{a^4}-\frac {\left (a^2+b^2\right ) \tanh ^{-1}(\cos (x))}{a^4}+\frac {3 b \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^4}+\frac {2 b \csc (x)}{a^3}-\frac {\cot (x) \csc (x)}{2 a^2}+\frac {a^2+b^2}{a^3 (a \cos (x)+b \sin (x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 1.91, size = 270, normalized size = 2.29 \[ \frac {8 a^3 \csc (x)+a^3 \cot (x) \sec ^2\left (\frac {x}{2}\right )-12 a^3 \cot (x) \log \left (\cos \left (\frac {x}{2}\right )\right )+12 a^3 \cot (x) \log \left (\sin \left (\frac {x}{2}\right )\right )-48 b \sqrt {a^2+b^2} (a \cot (x)+b) \tanh ^{-1}\left (\frac {a \tan \left (\frac {x}{2}\right )-b}{\sqrt {a^2+b^2}}\right )+a^2 b \sec ^2\left (\frac {x}{2}\right )+12 a^2 b \log \left (\sin \left (\frac {x}{2}\right )\right )-12 a^2 b \log \left (\cos \left (\frac {x}{2}\right )\right )+8 a^2 b \tan \left (\frac {x}{2}\right ) \cot (x)-a \csc ^2\left (\frac {x}{2}\right ) \left (a^2 \cot (x)+b (a-4 b \sin (x))-4 a b \cos (x)\right )+8 a b^2 \tan \left (\frac {x}{2}\right )+8 a b^2 \csc (x)-24 a b^2 \cot (x) \log \left (\cos \left (\frac {x}{2}\right )\right )+24 a b^2 \cot (x) \log \left (\sin \left (\frac {x}{2}\right )\right )+24 b^3 \log \left (\sin \left (\frac {x}{2}\right )\right )-24 b^3 \log \left (\cos \left (\frac {x}{2}\right )\right )}{8 a^4 (a \cot (x)+b)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.60, size = 345, normalized size = 2.92 \[ -\frac {6 \, a^{2} b \cos \relax (x) \sin \relax (x) + 4 \, a^{3} + 12 \, a b^{2} - 6 \, {\left (a^{3} + 2 \, a b^{2}\right )} \cos \relax (x)^{2} - 6 \, {\left (a b \cos \relax (x)^{3} - a b \cos \relax (x) + {\left (b^{2} \cos \relax (x)^{2} - b^{2}\right )} \sin \relax (x)\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \relax (x) \sin \relax (x) + {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} - 2 \, a^{2} - b^{2} - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \relax (x) - a \sin \relax (x)\right )}}{2 \, a b \cos \relax (x) \sin \relax (x) + {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + b^{2}}\right ) + 3 \, {\left ({\left (a^{3} + 2 \, a b^{2}\right )} \cos \relax (x)^{3} - {\left (a^{3} + 2 \, a b^{2}\right )} \cos \relax (x) - {\left (a^{2} b + 2 \, b^{3} - {\left (a^{2} b + 2 \, b^{3}\right )} \cos \relax (x)^{2}\right )} \sin \relax (x)\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - 3 \, {\left ({\left (a^{3} + 2 \, a b^{2}\right )} \cos \relax (x)^{3} - {\left (a^{3} + 2 \, a b^{2}\right )} \cos \relax (x) - {\left (a^{2} b + 2 \, b^{3} - {\left (a^{2} b + 2 \, b^{3}\right )} \cos \relax (x)^{2}\right )} \sin \relax (x)\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right )}{4 \, {\left (a^{5} \cos \relax (x)^{3} - a^{5} \cos \relax (x) + {\left (a^{4} b \cos \relax (x)^{2} - a^{4} b\right )} \sin \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 8.78, size = 215, normalized size = 1.82 \[ \frac {3 \, {\left (a^{2} + 2 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{2 \, a^{4}} + \frac {3 \, {\left (a^{2} b + b^{3}\right )} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a^{4}} + \frac {a^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 8 \, a b \tan \left (\frac {1}{2} \, x\right )}{8 \, a^{4}} - \frac {2 \, {\left (a^{2} b \tan \left (\frac {1}{2} \, x\right ) + b^{3} \tan \left (\frac {1}{2} \, x\right ) + a^{3} + a b^{2}\right )}}{{\left (a \tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, b \tan \left (\frac {1}{2} \, x\right ) - a\right )} a^{4}} - \frac {18 \, a^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 36 \, b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} - 8 \, a b \tan \left (\frac {1}{2} \, x\right ) + a^{2}}{8 \, a^{4} \tan \left (\frac {1}{2} \, x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.72, size = 224, normalized size = 1.90 \[ \frac {\tan ^{2}\left (\frac {x}{2}\right )}{8 a^{2}}+\frac {\tan \left (\frac {x}{2}\right ) b}{a^{3}}-\frac {1}{8 a^{2} \tan \left (\frac {x}{2}\right )^{2}}+\frac {3 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{2 a^{2}}+\frac {3 \ln \left (\tan \left (\frac {x}{2}\right )\right ) b^{2}}{a^{4}}+\frac {b}{a^{3} \tan \left (\frac {x}{2}\right )}-\frac {2 \tan \left (\frac {x}{2}\right ) b}{a^{2} \left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-2 \tan \left (\frac {x}{2}\right ) b -a \right )}-\frac {2 \tan \left (\frac {x}{2}\right ) b^{3}}{a^{4} \left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-2 \tan \left (\frac {x}{2}\right ) b -a \right )}-\frac {2}{a \left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-2 \tan \left (\frac {x}{2}\right ) b -a \right )}-\frac {2 b^{2}}{a^{3} \left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-2 \tan \left (\frac {x}{2}\right ) b -a \right )}-\frac {6 b \sqrt {a^{2}+b^{2}}\, \arctanh \left (\frac {2 a \tan \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.43, size = 242, normalized size = 2.05 \[ -\frac {a^{3} - \frac {6 \, a^{2} b \sin \relax (x)}{\cos \relax (x) + 1} - \frac {{\left (17 \, a^{3} + 32 \, a b^{2}\right )} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} - \frac {8 \, {\left (a^{2} b + 2 \, b^{3}\right )} \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}}}{8 \, {\left (\frac {a^{5} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {2 \, a^{4} b \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} - \frac {a^{5} \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}}\right )}} + \frac {\frac {8 \, b \sin \relax (x)}{\cos \relax (x) + 1} + \frac {a \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}}}{8 \, a^{3}} + \frac {3 \, {\left (a^{2} + 2 \, b^{2}\right )} \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )}{2 \, a^{4}} + \frac {3 \, {\left (a^{2} b + b^{3}\right )} \log \left (\frac {b - \frac {a \sin \relax (x)}{\cos \relax (x) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \relax (x)}{\cos \relax (x) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.74, size = 511, normalized size = 4.33 \[ \frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (\frac {17\,a^2}{2}+16\,b^2\right )-\frac {a^2}{2}+3\,a\,b\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {4\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3\,\left (a^2\,b+2\,b^3\right )}{a}}{-4\,a^4\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+4\,a^4\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+8\,b\,a^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{8\,a^2}+\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )\,\left (3\,a^2+6\,b^2\right )}{2\,a^4}+\frac {b\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^3}-\frac {6\,b\,\mathrm {atanh}\left (\frac {54\,b^2\,\sqrt {a^2+b^2}}{18\,a^2\,b+90\,b^3+\frac {72\,b^5}{a^2}+\frac {216\,b^4\,\mathrm {tan}\left (\frac {x}{2}\right )}{a}+\frac {144\,b^6\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^3}+72\,a\,b^2\,\mathrm {tan}\left (\frac {x}{2}\right )}+\frac {72\,b^4\,\sqrt {a^2+b^2}}{18\,a^4\,b+72\,b^5+90\,a^2\,b^3+72\,a^3\,b^2\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {144\,b^6\,\mathrm {tan}\left (\frac {x}{2}\right )}{a}+216\,a\,b^4\,\mathrm {tan}\left (\frac {x}{2}\right )}+\frac {144\,b^3\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {a^2+b^2}}{216\,b^4\,\mathrm {tan}\left (\frac {x}{2}\right )+90\,a\,b^3+18\,a^3\,b+\frac {72\,b^5}{a}+72\,a^2\,b^2\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {144\,b^6\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^2}}+\frac {144\,b^5\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {a^2+b^2}}{18\,a^5\,b+72\,\mathrm {tan}\left (\frac {x}{2}\right )\,a^4\,b^2+90\,a^3\,b^3+216\,\mathrm {tan}\left (\frac {x}{2}\right )\,a^2\,b^4+72\,a\,b^5+144\,\mathrm {tan}\left (\frac {x}{2}\right )\,b^6}+\frac {18\,b\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {a^2+b^2}}{18\,a\,b+72\,b^2\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {90\,b^3}{a}+\frac {72\,b^5}{a^3}+\frac {216\,b^4\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^2}+\frac {144\,b^6\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^4}}\right )\,\sqrt {a^2+b^2}}{a^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{3}{\relax (x )}}{\left (a \cos {\relax (x )} + b \sin {\relax (x )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________