Optimal. Leaf size=40 \[ \frac {1}{8} \tan ^{-1}\left (\frac {2 \tan (x)}{\sqrt {-5 \tan ^2(x)-1}}\right )-\frac {5 \tan (x)}{4 \sqrt {-5 \tan ^2(x)-1}} \]
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Rubi [A] time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4128, 382, 377, 203} \[ \frac {1}{8} \tan ^{-1}\left (\frac {2 \tan (x)}{\sqrt {-5 \tan ^2(x)-1}}\right )-\frac {5 \tan (x)}{4 \sqrt {-5 \tan ^2(x)-1}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 377
Rule 382
Rule 4128
Rubi steps
\begin {align*} \int \frac {1}{\left (4-5 \sec ^2(x)\right )^{3/2}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\left (-1-5 x^2\right )^{3/2} \left (1+x^2\right )} \, dx,x,\tan (x)\right )\\ &=-\frac {5 \tan (x)}{4 \sqrt {-1-5 \tan ^2(x)}}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-5 x^2} \left (1+x^2\right )} \, dx,x,\tan (x)\right )\\ &=-\frac {5 \tan (x)}{4 \sqrt {-1-5 \tan ^2(x)}}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1+4 x^2} \, dx,x,\frac {\tan (x)}{\sqrt {-1-5 \tan ^2(x)}}\right )\\ &=\frac {1}{8} \tan ^{-1}\left (\frac {2 \tan (x)}{\sqrt {-1-5 \tan ^2(x)}}\right )-\frac {5 \tan (x)}{4 \sqrt {-1-5 \tan ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 79, normalized size = 1.98 \[ -\frac {(2 \cos (2 x)-3)^{3/2} \sec ^3(x) \left (10 \sin (x) \sqrt {3-2 \cos (2 x)}+(2 \cos (2 x)-3) \sinh ^{-1}(2 \sin (x))\right )}{8 \sqrt {-\left (4 \sin ^2(x)+1\right )^2} \left (4-5 \sec ^2(x)\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (4-5 \sec ^2(x)\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.93, size = 115, normalized size = 2.88 \[ -\frac {20 \, \sqrt {\frac {4 \, \cos \relax (x)^{2} - 5}{\cos \relax (x)^{2}}} \cos \relax (x) \sin \relax (x) - {\left (4 \, \cos \relax (x)^{2} - 5\right )} \arctan \left (\frac {4 \, {\left (8 \, \cos \relax (x)^{3} - 9 \, \cos \relax (x)\right )} \sqrt {\frac {4 \, \cos \relax (x)^{2} - 5}{\cos \relax (x)^{2}}} \sin \relax (x) + \cos \relax (x) \sin \relax (x)}{64 \, \cos \relax (x)^{4} - 143 \, \cos \relax (x)^{2} + 80}\right ) + {\left (4 \, \cos \relax (x)^{2} - 5\right )} \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x)}\right )}{16 \, {\left (4 \, \cos \relax (x)^{2} - 5\right )}} \]
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 \[ \int \frac {1}{{\left (-5 \, \sec \relax (x)^{2} + 4\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.69, size = 473, normalized size = 11.82
method | result | size |
default | \(-\frac {i \left (4 \left (\cos ^{2}\relax (x )\right )-5\right ) \left (2 i \sin \relax (x ) \sqrt {2}\, \sqrt {-\frac {2 \left (2 \cos \relax (x ) \sqrt {5}+4 \cos \relax (x )-2 \sqrt {5}-5\right )}{1+\cos \relax (x )}}\, \sqrt {\frac {2 \cos \relax (x ) \sqrt {5}-4 \cos \relax (x )-2 \sqrt {5}+5}{1+\cos \relax (x )}}\, \sqrt {5}\, \EllipticPi \left (\frac {\sqrt {-9-4 \sqrt {5}}\, \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, \frac {1}{9+4 \sqrt {5}}, \frac {\sqrt {-9+4 \sqrt {5}}}{\sqrt {-9-4 \sqrt {5}}}\right )-i \sin \relax (x ) \sqrt {2}\, \sqrt {-\frac {2 \left (2 \cos \relax (x ) \sqrt {5}+4 \cos \relax (x )-2 \sqrt {5}-5\right )}{1+\cos \relax (x )}}\, \sqrt {\frac {2 \cos \relax (x ) \sqrt {5}-4 \cos \relax (x )-2 \sqrt {5}+5}{1+\cos \relax (x )}}\, \sqrt {5}\, \EllipticF \left (\frac {i \left (-1+\cos \relax (x )\right ) \left (\sqrt {5}+2\right )}{\sin \relax (x )}, 9-4 \sqrt {5}\right )+4 i \sin \relax (x ) \sqrt {2}\, \sqrt {-\frac {2 \left (2 \cos \relax (x ) \sqrt {5}+4 \cos \relax (x )-2 \sqrt {5}-5\right )}{1+\cos \relax (x )}}\, \sqrt {\frac {2 \cos \relax (x ) \sqrt {5}-4 \cos \relax (x )-2 \sqrt {5}+5}{1+\cos \relax (x )}}\, \EllipticPi \left (\frac {\sqrt {-9-4 \sqrt {5}}\, \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, \frac {1}{9+4 \sqrt {5}}, \frac {\sqrt {-9+4 \sqrt {5}}}{\sqrt {-9-4 \sqrt {5}}}\right )-2 i \sin \relax (x ) \sqrt {2}\, \sqrt {-\frac {2 \left (2 \cos \relax (x ) \sqrt {5}+4 \cos \relax (x )-2 \sqrt {5}-5\right )}{1+\cos \relax (x )}}\, \sqrt {\frac {2 \cos \relax (x ) \sqrt {5}-4 \cos \relax (x )-2 \sqrt {5}+5}{1+\cos \relax (x )}}\, \EllipticF \left (\frac {i \left (-1+\cos \relax (x )\right ) \left (\sqrt {5}+2\right )}{\sin \relax (x )}, 9-4 \sqrt {5}\right )+20 \cos \relax (x ) \sqrt {5}+45 \cos \relax (x )-20 \sqrt {5}-45\right ) \sin \relax (x )}{4 \sqrt {-9-4 \sqrt {5}}\, \left (\sqrt {5}+2\right ) \left (-1+\cos \relax (x )\right ) \cos \relax (x )^{3} \left (\frac {4 \left (\cos ^{2}\relax (x )\right )-5}{\cos \relax (x )^{2}}\right )^{\frac {3}{2}}}\) | \(473\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-5 \, \sec \relax (x)^{2} + 4\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (4-\frac {5}{{\cos \relax (x)}^2}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (4 - 5 \sec ^{2}{\relax (x )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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