Optimal. Leaf size=73 \[ -\frac {b \cos (x)-a \sin (x)}{2 \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))^2}-\frac {\tanh ^{-1}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3076, 3074, 206} \[ -\frac {b \cos (x)-a \sin (x)}{2 \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))^2}-\frac {\tanh ^{-1}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 3074
Rule 3076
Rubi steps
\begin {align*} \int \frac {1}{(a \cos (x)+b \sin (x))^3} \, dx &=-\frac {b \cos (x)-a \sin (x)}{2 \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))^2}+\frac {\int \frac {1}{a \cos (x)+b \sin (x)} \, dx}{2 \left (a^2+b^2\right )}\\ &=-\frac {b \cos (x)-a \sin (x)}{2 \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))^2}-\frac {\operatorname {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,b \cos (x)-a \sin (x)\right )}{2 \left (a^2+b^2\right )}\\ &=-\frac {\tanh ^{-1}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (x)-a \sin (x)}{2 \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))^2}\\ \end {align*}
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Mathematica [C] time = 0.17, size = 101, normalized size = 1.38 \[ \frac {\left (a^2+b^2\right ) (a \sin (x)-b \cos (x))+2 \sqrt {a^2+b^2} (a \cos (x)+b \sin (x))^2 \tanh ^{-1}\left (\frac {a \tan \left (\frac {x}{2}\right )-b}{\sqrt {a^2+b^2}}\right )}{2 (a-i b)^2 (a+i b)^2 (a \cos (x)+b \sin (x))^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 225, normalized size = 3.08 \[ \frac {{\left (2 \, a b \cos \relax (x) \sin \relax (x) + {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + b^{2}\right )} \sqrt {a^{2} + b^{2}} \log \left (-\frac {2 \, a b \cos \relax (x) \sin \relax (x) + {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \relax (x) - a \sin \relax (x)\right )}}{2 \, a b \cos \relax (x) \sin \relax (x) + {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + b^{2}}\right ) - 2 \, {\left (a^{2} b + b^{3}\right )} \cos \relax (x) + 2 \, {\left (a^{3} + a b^{2}\right )} \sin \relax (x)}{4 \, {\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6} + {\left (a^{6} + a^{4} b^{2} - a^{2} b^{4} - b^{6}\right )} \cos \relax (x)^{2} + 2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \cos \relax (x) \sin \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.38, size = 166, normalized size = 2.27 \[ -\frac {\log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{2 \, {\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} + \frac {a^{3} \tan \left (\frac {1}{2} \, x\right )^{3} + 2 \, a b^{2} \tan \left (\frac {1}{2} \, x\right )^{3} + a^{2} b \tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, b^{3} \tan \left (\frac {1}{2} \, x\right )^{2} + a^{3} \tan \left (\frac {1}{2} \, x\right ) - 2 \, a b^{2} \tan \left (\frac {1}{2} \, x\right ) - a^{2} b}{{\left (a^{4} + a^{2} b^{2}\right )} {\left (a \tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, b \tan \left (\frac {1}{2} \, x\right ) - a\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.54, size = 157, normalized size = 2.15 \[ -\frac {2 \left (-\frac {\left (a^{2}+2 b^{2}\right ) \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{2 \left (a^{2}+b^{2}\right ) a}-\frac {b \left (a^{2}-2 b^{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2 a^{2} \left (a^{2}+b^{2}\right )}-\frac {\left (a^{2}-2 b^{2}\right ) \tan \left (\frac {x}{2}\right )}{2 \left (a^{2}+b^{2}\right ) a}+\frac {b}{2 a^{2}+2 b^{2}}\right )}{\left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-2 \tan \left (\frac {x}{2}\right ) b -a \right )^{2}}+\frac {\arctanh \left (\frac {2 a \tan \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 250, normalized size = 3.42 \[ -\frac {a^{2} b - \frac {{\left (a^{3} - 2 \, a b^{2}\right )} \sin \relax (x)}{\cos \relax (x) + 1} - \frac {{\left (a^{2} b - 2 \, b^{3}\right )} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} - \frac {{\left (a^{3} + 2 \, a b^{2}\right )} \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}}}{a^{6} + a^{4} b^{2} + \frac {4 \, {\left (a^{5} b + a^{3} b^{3}\right )} \sin \relax (x)}{\cos \relax (x) + 1} - \frac {2 \, {\left (a^{6} - a^{4} b^{2} - 2 \, a^{2} b^{4}\right )} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} - \frac {4 \, {\left (a^{5} b + a^{3} b^{3}\right )} \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {{\left (a^{6} + a^{4} b^{2}\right )} \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}}} - \frac {\log \left (\frac {b - \frac {a \sin \relax (x)}{\cos \relax (x) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \relax (x)}{\cos \relax (x) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{2 \, {\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.76, size = 216, normalized size = 2.96 \[ \frac {\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^3\,\left (a^2+2\,b^2\right )}{a\,\left (a^2+b^2\right )}-\frac {b}{a^2+b^2}+\frac {\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a^2-2\,b^2\right )}{a\,\left (a^2+b^2\right )}+\frac {b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (a^2-2\,b^2\right )}{a^2\,\left (a^2+b^2\right )}}{a^2-{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (2\,a^2-4\,b^2\right )+a^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+4\,a\,b\,\mathrm {tan}\left (\frac {x}{2}\right )-4\,a\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}-\frac {\mathrm {atanh}\left (-\frac {\left (2\,a\,\mathrm {tan}\left (\frac {x}{2}\right )-\frac {2\,a^2\,b+2\,b^3}{a^2+b^2}\right )\,\left (\frac {a^2}{2}+\frac {b^2}{2}\right )}{{\left (a^2+b^2\right )}^{3/2}}\right )}{{\left (a^2+b^2\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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