Optimal. Leaf size=112 \[ \frac {5 \sin ^{-1}\left (2 \sqrt {\frac {2}{7}} \sin (x)\right )}{4 \sqrt {2}}+\frac {3}{4} \sin ^{-1}\left (\frac {2 \sin (x)}{\sqrt {3}}\right )-\frac {1}{2} \sin (x) \sqrt {4 \cos ^2(x)-1}-\frac {1}{2} \sin (x) \sqrt {8 \cos ^2(x)-1}-\frac {3}{4} \tan ^{-1}\left (\frac {\sin (x)}{\sqrt {4 \cos ^2(x)-1}}\right )-\frac {3}{4} \tan ^{-1}\left (\frac {\sin (x)}{\sqrt {8 \cos ^2(x)-1}}\right ) \]
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Rubi [A] time = 0.44, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 6, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6742, 402, 216, 377, 204, 195} \[ \frac {5 \sin ^{-1}\left (2 \sqrt {\frac {2}{7}} \sin (x)\right )}{4 \sqrt {2}}+\frac {3}{4} \sin ^{-1}\left (\frac {2 \sin (x)}{\sqrt {3}}\right )-\frac {1}{2} \sin (x) \sqrt {7-8 \sin ^2(x)}-\frac {1}{2} \sin (x) \sqrt {3-4 \sin ^2(x)}-\frac {3}{4} \tan ^{-1}\left (\frac {\sin (x)}{\sqrt {7-8 \sin ^2(x)}}\right )-\frac {3}{4} \tan ^{-1}\left (\frac {\sin (x)}{\sqrt {3-4 \sin ^2(x)}}\right ) \]
Antiderivative was successfully verified.
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Rule 195
Rule 204
Rule 216
Rule 377
Rule 402
Rule 6742
Rubi steps
\begin {align*} \int \frac {\cos (3 x)}{-\sqrt {-1+8 \cos ^2(x)}+\sqrt {3 \cos ^2(x)-\sin ^2(x)}} \, dx &=\operatorname {Subst}\left (\int \frac {-1+4 x^2}{\sqrt {7-8 x^2}-\sqrt {3-4 x^2}} \, dx,x,\sin (x)\right )\\ &=\operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt {7-8 x^2}-\sqrt {3-4 x^2}}+\frac {4 x^2}{\sqrt {7-8 x^2}-\sqrt {3-4 x^2}}\right ) \, dx,x,\sin (x)\right )\\ &=4 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {7-8 x^2}-\sqrt {3-4 x^2}} \, dx,x,\sin (x)\right )-\operatorname {Subst}\left (\int \frac {1}{\sqrt {7-8 x^2}-\sqrt {3-4 x^2}} \, dx,x,\sin (x)\right )\\ &=4 \operatorname {Subst}\left (\int \left (-\frac {1}{4} \sqrt {7-8 x^2}-\frac {1}{4} \sqrt {3-4 x^2}-\frac {\sqrt {7-8 x^2}}{4 \left (-1+x^2\right )}-\frac {\sqrt {3-4 x^2}}{4 \left (-1+x^2\right )}\right ) \, dx,x,\sin (x)\right )-\operatorname {Subst}\left (\int \left (-\frac {\sqrt {7-8 x^2}}{4 \left (-1+x^2\right )}-\frac {\sqrt {3-4 x^2}}{4 \left (-1+x^2\right )}\right ) \, dx,x,\sin (x)\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {\sqrt {7-8 x^2}}{-1+x^2} \, dx,x,\sin (x)\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {\sqrt {3-4 x^2}}{-1+x^2} \, dx,x,\sin (x)\right )-\operatorname {Subst}\left (\int \sqrt {7-8 x^2} \, dx,x,\sin (x)\right )-\operatorname {Subst}\left (\int \sqrt {3-4 x^2} \, dx,x,\sin (x)\right )-\operatorname {Subst}\left (\int \frac {\sqrt {7-8 x^2}}{-1+x^2} \, dx,x,\sin (x)\right )-\operatorname {Subst}\left (\int \frac {\sqrt {3-4 x^2}}{-1+x^2} \, dx,x,\sin (x)\right )\\ &=-\frac {1}{2} \sin (x) \sqrt {7-8 \sin ^2(x)}-\frac {1}{2} \sin (x) \sqrt {3-4 \sin ^2(x)}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {7-8 x^2} \left (-1+x^2\right )} \, dx,x,\sin (x)\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {3-4 x^2} \left (-1+x^2\right )} \, dx,x,\sin (x)\right )-\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {3-4 x^2}} \, dx,x,\sin (x)\right )-2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {7-8 x^2}} \, dx,x,\sin (x)\right )-\frac {7}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {7-8 x^2}} \, dx,x,\sin (x)\right )+4 \operatorname {Subst}\left (\int \frac {1}{\sqrt {3-4 x^2}} \, dx,x,\sin (x)\right )+8 \operatorname {Subst}\left (\int \frac {1}{\sqrt {7-8 x^2}} \, dx,x,\sin (x)\right )-\operatorname {Subst}\left (\int \frac {1}{\sqrt {3-4 x^2}} \, dx,x,\sin (x)\right )+\operatorname {Subst}\left (\int \frac {1}{\sqrt {7-8 x^2} \left (-1+x^2\right )} \, dx,x,\sin (x)\right )+\operatorname {Subst}\left (\int \frac {1}{\sqrt {3-4 x^2} \left (-1+x^2\right )} \, dx,x,\sin (x)\right )\\ &=-\frac {11 \sin ^{-1}\left (2 \sqrt {\frac {2}{7}} \sin (x)\right )}{4 \sqrt {2}}+2 \sqrt {2} \sin ^{-1}\left (2 \sqrt {\frac {2}{7}} \sin (x)\right )+\frac {3}{4} \sin ^{-1}\left (\frac {2 \sin (x)}{\sqrt {3}}\right )-\frac {1}{2} \sin (x) \sqrt {7-8 \sin ^2(x)}-\frac {1}{2} \sin (x) \sqrt {3-4 \sin ^2(x)}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\frac {\sin (x)}{\sqrt {7-8 \sin ^2(x)}}\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\frac {\sin (x)}{\sqrt {3-4 \sin ^2(x)}}\right )+\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\frac {\sin (x)}{\sqrt {7-8 \sin ^2(x)}}\right )+\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\frac {\sin (x)}{\sqrt {3-4 \sin ^2(x)}}\right )\\ &=-\frac {11 \sin ^{-1}\left (2 \sqrt {\frac {2}{7}} \sin (x)\right )}{4 \sqrt {2}}+2 \sqrt {2} \sin ^{-1}\left (2 \sqrt {\frac {2}{7}} \sin (x)\right )+\frac {3}{4} \sin ^{-1}\left (\frac {2 \sin (x)}{\sqrt {3}}\right )-\frac {3}{4} \tan ^{-1}\left (\frac {\sin (x)}{\sqrt {7-8 \sin ^2(x)}}\right )-\frac {3}{4} \tan ^{-1}\left (\frac {\sin (x)}{\sqrt {3-4 \sin ^2(x)}}\right )-\frac {1}{2} \sin (x) \sqrt {7-8 \sin ^2(x)}-\frac {1}{2} \sin (x) \sqrt {3-4 \sin ^2(x)}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 156, normalized size = 1.39 \[ \frac {1}{8} \left (5 \sqrt {2} \sin ^{-1}\left (2 \sqrt {\frac {2}{7}} \sin (x)\right )+6 \sin ^{-1}\left (\frac {2 \sin (x)}{\sqrt {3}}\right )-4 \sin (x) \sqrt {2 \cos (2 x)+1}-4 \sin (x) \sqrt {4 \cos (2 x)+3}+3 \tan ^{-1}\left (\frac {7-8 \sin (x)}{\sqrt {4 \cos (2 x)+3}}\right )+3 \tan ^{-1}\left (\frac {3-4 \sin (x)}{\sqrt {2 \cos (2 x)+1}}\right )-3 \tan ^{-1}\left (\frac {4 \sin (x)+3}{\sqrt {2 \cos (2 x)+1}}\right )-3 \tan ^{-1}\left (\frac {8 \sin (x)+7}{\sqrt {4 \cos (2 x)+3}}\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos (3 x)}{-\sqrt {-1+8 \cos ^2(x)}+\sqrt {3 \cos ^2(x)-\sin ^2(x)}} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.51, size = 195, normalized size = 1.74 \[ -\frac {5}{32} \, \sqrt {2} \arctan \left (\frac {{\left (512 \, \sqrt {2} \cos \relax (x)^{4} - 576 \, \sqrt {2} \cos \relax (x)^{2} + 113 \, \sqrt {2}\right )} \sqrt {8 \, \cos \relax (x)^{2} - 1}}{16 \, {\left (128 \, \cos \relax (x)^{4} - 88 \, \cos \relax (x)^{2} + 9\right )} \sin \relax (x)}\right ) - \frac {1}{2} \, \sqrt {8 \, \cos \relax (x)^{2} - 1} \sin \relax (x) - \frac {1}{2} \, \sqrt {4 \, \cos \relax (x)^{2} - 1} \sin \relax (x) + \frac {3}{8} \, \arctan \left (\frac {4 \, {\left (8 \, \cos \relax (x)^{2} - 5\right )} \sqrt {4 \, \cos \relax (x)^{2} - 1} \sin \relax (x) - 9 \, \cos \relax (x) \sin \relax (x)}{64 \, \cos \relax (x)^{4} - 71 \, \cos \relax (x)^{2} + 16}\right ) + \frac {3}{8} \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x)}\right ) + \frac {3}{8} \, \arctan \left (\frac {9 \, \cos \relax (x)^{2} - 2}{2 \, \sqrt {8 \, \cos \relax (x)^{2} - 1} \sin \relax (x)}\right ) + \frac {3}{4} \, \arctan \left (\frac {\sqrt {4 \, \cos \relax (x)^{2} - 1}}{\sin \relax (x)}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {\cos \left (3 \, x\right )}{\sqrt {8 \, \cos \relax (x)^{2} - 1} - \sqrt {3 \, \cos \relax (x)^{2} - \sin \relax (x)^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\cos \left (3 x \right )}{-\sqrt {-1+8 \left (\cos ^{2}\relax (x )\right )}+\sqrt {3 \left (\cos ^{2}\relax (x )\right )-\left (\sin ^{2}\relax (x )\right )}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int -\frac {\cos \left (3\,x\right )}{\sqrt {3\,{\cos \relax (x)}^2-{\sin \relax (x)}^2}-\sqrt {8\,{\cos \relax (x)}^2-1}} \,d x \]
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 \[ \int \frac {\cos {\left (3 x \right )}}{\sqrt {- \sin ^{2}{\relax (x )} + 3 \cos ^{2}{\relax (x )}} - \sqrt {8 \cos ^{2}{\relax (x )} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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