Optimal. Leaf size=101 \[ 2 \sqrt [4]{5 \sin ^2(x)-1}-\frac {\sqrt [4]{5 \sin ^2(x)-1}}{2 \left (\sqrt {5 \sin ^2(x)-1}+2\right )}-\frac {3 \tan ^{-1}\left (\frac {\sqrt [4]{5 \sin ^2(x)-1}}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{5 \sin ^2(x)-1}}{\sqrt {2}}\right )}{2 \sqrt {2}} \]
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Rubi [A] time = 1.41, antiderivative size = 126, normalized size of antiderivative = 1.25, number of steps used = 14, number of rules used = 10, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {4361, 6742, 6697, 341, 50, 63, 203, 470, 522, 207} \[ 2 \sqrt [4]{4-5 \cos ^2(x)}-\frac {\sqrt [4]{4-5 \cos ^2(x)}}{2 \left (\sqrt {4-5 \cos ^2(x)}+2\right )}-2 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt [4]{4-5 \cos ^2(x)}}{\sqrt {2}}\right )+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{4-5 \cos ^2(x)}}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{4-5 \cos ^2(x)}}{\sqrt {2}}\right )}{2 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 203
Rule 207
Rule 341
Rule 470
Rule 522
Rule 4361
Rule 6697
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (5 \cos ^2(x)-\sqrt {-1+5 \sin ^2(x)}\right ) \tan (x)}{\sqrt [4]{-1+5 \sin ^2(x)} \left (2+\sqrt {-1+5 \sin ^2(x)}\right )} \, dx &=-\operatorname {Subst}\left (\int \frac {5 x^2-\sqrt {4-5 x^2}}{\sqrt [4]{4-5 x^2} \left (2 x+x \sqrt {4-5 x^2}\right )} \, dx,x,\cos (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {5 x}{\sqrt [4]{4-5 x^2} \left (2+\sqrt {4-5 x^2}\right )}-\frac {\sqrt [4]{4-5 x^2}}{x \left (2+\sqrt {4-5 x^2}\right )}\right ) \, dx,x,\cos (x)\right )\\ &=-\left (5 \operatorname {Subst}\left (\int \frac {x}{\sqrt [4]{4-5 x^2} \left (2+\sqrt {4-5 x^2}\right )} \, dx,x,\cos (x)\right )\right )+\operatorname {Subst}\left (\int \frac {\sqrt [4]{4-5 x^2}}{x \left (2+\sqrt {4-5 x^2}\right )} \, dx,x,\cos (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt [4]{4-5 x}}{\left (2+\sqrt {4-5 x}\right ) x} \, dx,x,\cos ^2(x)\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\left (2+\sqrt {x}\right ) \sqrt [4]{x}} \, dx,x,4-5 \cos ^2(x)\right )\\ &=2 \operatorname {Subst}\left (\int \frac {x^4}{\left (-2+x^2\right ) \left (2+x^2\right )^2} \, dx,x,\sqrt [4]{4-5 \cos ^2(x)}\right )+\operatorname {Subst}\left (\int \frac {\sqrt {x}}{2+x} \, dx,x,\sqrt {4-5 \cos ^2(x)}\right )\\ &=2 \sqrt [4]{4-5 \cos ^2(x)}-\frac {\sqrt [4]{4-5 \cos ^2(x)}}{2 \left (2+\sqrt {4-5 \cos ^2(x)}\right )}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {-4+6 x^2}{\left (-2+x^2\right ) \left (2+x^2\right )} \, dx,x,\sqrt [4]{4-5 \cos ^2(x)}\right )-2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} (2+x)} \, dx,x,\sqrt {4-5 \cos ^2(x)}\right )\\ &=2 \sqrt [4]{4-5 \cos ^2(x)}-\frac {\sqrt [4]{4-5 \cos ^2(x)}}{2 \left (2+\sqrt {4-5 \cos ^2(x)}\right )}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-2+x^2} \, dx,x,\sqrt [4]{4-5 \cos ^2(x)}\right )-4 \operatorname {Subst}\left (\int \frac {1}{2+x^2} \, dx,x,\sqrt [4]{4-5 \cos ^2(x)}\right )+\operatorname {Subst}\left (\int \frac {1}{2+x^2} \, dx,x,\sqrt [4]{4-5 \cos ^2(x)}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{4-5 \cos ^2(x)}}{\sqrt {2}}\right )}{\sqrt {2}}-2 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt [4]{4-5 \cos ^2(x)}}{\sqrt {2}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{4-5 \cos ^2(x)}}{\sqrt {2}}\right )}{2 \sqrt {2}}+2 \sqrt [4]{4-5 \cos ^2(x)}-\frac {\sqrt [4]{4-5 \cos ^2(x)}}{2 \left (2+\sqrt {4-5 \cos ^2(x)}\right )}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 89, normalized size = 0.88 \[ \frac {1}{4} \left (-2 \sqrt [4]{4-5 \cos ^2(x)} \left (\frac {1}{\sqrt {4-5 \cos ^2(x)}+2}-4\right )-6 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt [4]{3-5 \cos (2 x)}}{2^{3/4}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt [4]{3-5 \cos (2 x)}}{2^{3/4}}\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (5 \cos ^2(x)-\sqrt {-1+5 \sin ^2(x)}\right ) \tan (x)}{\sqrt [4]{-1+5 \sin ^2(x)} \left (2+\sqrt {-1+5 \sin ^2(x)}\right )} \, dx \]
Verification is Not applicable to the result.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (5 \, \cos \relax (x)^{2} - \sqrt {5 \, \sin \relax (x)^{2} - 1}\right )} \tan \relax (x)}{{\left (5 \, \sin \relax (x)^{2} - 1\right )}^{\frac {1}{4}} {\left (\sqrt {5 \, \sin \relax (x)^{2} - 1} + 2\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.08, size = 0, normalized size = 0.00 \[\int \frac {\left (5 \left (\cos ^{2}\relax (x )\right )-\sqrt {-1+5 \left (\sin ^{2}\relax (x )\right )}\right ) \tan \relax (x )}{\left (-1+5 \left (\sin ^{2}\relax (x )\right )\right )^{\frac {1}{4}} \left (2+\sqrt {-1+5 \left (\sin ^{2}\relax (x )\right )}\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 100, normalized size = 0.99 \[ -\frac {3}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (5 \, \sin \relax (x)^{2} - 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - {\left (5 \, \sin \relax (x)^{2} - 1\right )}^{\frac {1}{4}}}{\sqrt {2} + {\left (5 \, \sin \relax (x)^{2} - 1\right )}^{\frac {1}{4}}}\right ) + 2 \, {\left (5 \, \sin \relax (x)^{2} - 1\right )}^{\frac {1}{4}} - \frac {{\left (5 \, \sin \relax (x)^{2} - 1\right )}^{\frac {1}{4}}}{2 \, {\left (\sqrt {5 \, \sin \relax (x)^{2} - 1} + 2\right )}} \]
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 \[ \int \frac {\mathrm {tan}\relax (x)\,\left (5\,{\cos \relax (x)}^2-\sqrt {5\,{\sin \relax (x)}^2-1}\right )}{{\left (5\,{\sin \relax (x)}^2-1\right )}^{1/4}\,\left (\sqrt {5\,{\sin \relax (x)}^2-1}+2\right )} \,d x \]
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 \[ \int \frac {\left (- \sqrt {5 \sin ^{2}{\relax (x )} - 1} + 5 \cos ^{2}{\relax (x )}\right ) \tan {\relax (x )}}{\left (\sqrt {5 \sin ^{2}{\relax (x )} - 1} + 2\right ) \sqrt [4]{5 \sin ^{2}{\relax (x )} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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