Optimal. Leaf size=68 \[ -\frac {5}{2} \tan (x) \sqrt {-5 \tan ^2(x)-1}+8 \tan ^{-1}\left (\frac {2 \tan (x)}{\sqrt {-5 \tan ^2(x)-1}}\right )-\frac {7}{2} \sqrt {5} \tan ^{-1}\left (\frac {\sqrt {5} \tan (x)}{\sqrt {-5 \tan ^2(x)-1}}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4128, 416, 523, 217, 203, 377} \[ 8 \tan ^{-1}\left (\frac {2 \tan (x)}{\sqrt {-5 \tan ^2(x)-1}}\right )-\frac {7}{2} \sqrt {5} \tan ^{-1}\left (\frac {\sqrt {5} \tan (x)}{\sqrt {-5 \tan ^2(x)-1}}\right )-\frac {5}{2} \tan (x) \sqrt {-5 \tan ^2(x)-1} \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 377
Rule 416
Rule 523
Rule 4128
Rubi steps
\begin {align*} \int \left (4-5 \sec ^2(x)\right )^{3/2} \, dx &=\operatorname {Subst}\left (\int \frac {\left (-1-5 x^2\right )^{3/2}}{1+x^2} \, dx,x,\tan (x)\right )\\ &=-\frac {5}{2} \tan (x) \sqrt {-1-5 \tan ^2(x)}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {-3-35 x^2}{\sqrt {-1-5 x^2} \left (1+x^2\right )} \, dx,x,\tan (x)\right )\\ &=-\frac {5}{2} \tan (x) \sqrt {-1-5 \tan ^2(x)}+16 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-5 x^2} \left (1+x^2\right )} \, dx,x,\tan (x)\right )-\frac {35}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-5 x^2}} \, dx,x,\tan (x)\right )\\ &=-\frac {5}{2} \tan (x) \sqrt {-1-5 \tan ^2(x)}+16 \operatorname {Subst}\left (\int \frac {1}{1+4 x^2} \, dx,x,\frac {\tan (x)}{\sqrt {-1-5 \tan ^2(x)}}\right )-\frac {35}{2} \operatorname {Subst}\left (\int \frac {1}{1+5 x^2} \, dx,x,\frac {\tan (x)}{\sqrt {-1-5 \tan ^2(x)}}\right )\\ &=8 \tan ^{-1}\left (\frac {2 \tan (x)}{\sqrt {-1-5 \tan ^2(x)}}\right )-\frac {7}{2} \sqrt {5} \tan ^{-1}\left (\frac {\sqrt {5} \tan (x)}{\sqrt {-1-5 \tan ^2(x)}}\right )-\frac {5}{2} \tan (x) \sqrt {-1-5 \tan ^2(x)}\\ \end {align*}
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Mathematica [C] time = 0.20, size = 115, normalized size = 1.69 \[ -\frac {\left (4 \cos ^2(x)-5\right ) \sec (x) \sqrt {4-5 \sec ^2(x)} \left (5 \sin (x) \sqrt {2 \cos (2 x)-3}+16 i \cos ^2(x) \log \left (\sqrt {2 \cos (2 x)-3}+2 i \sin (x)\right )+7 \sqrt {5} \cos ^2(x) \tan ^{-1}\left (\frac {\sqrt {5} \sin (x)}{\sqrt {2 \cos (2 x)-3}}\right )\right )}{2 (2 \cos (2 x)-3)^{3/2}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (4-5 \sec ^2(x)\right )^{3/2} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.80, size = 130, normalized size = 1.91 \[ \frac {7 \, \sqrt {5} \arctan \left (\frac {\sqrt {5} \sqrt {\frac {4 \, \cos \relax (x)^{2} - 5}{\cos \relax (x)^{2}}} \cos \relax (x)}{5 \, \sin \relax (x)}\right ) \cos \relax (x) + 8 \, \arctan \left (\frac {4 \, {\left (8 \, \cos \relax (x)^{3} - 9 \, \cos \relax (x)\right )} \sqrt {\frac {4 \, \cos \relax (x)^{2} - 5}{\cos \relax (x)^{2}}} \sin \relax (x) + \cos \relax (x) \sin \relax (x)}{64 \, \cos \relax (x)^{4} - 143 \, \cos \relax (x)^{2} + 80}\right ) \cos \relax (x) - 8 \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x)}\right ) \cos \relax (x) - 5 \, \sqrt {\frac {4 \, \cos \relax (x)^{2} - 5}{\cos \relax (x)^{2}}} \sin \relax (x)}{2 \, \cos \relax (x)} \]
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 \[ \int {\left (-5 \, \sec \relax (x)^{2} + 4\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.70, size = 754, normalized size = 11.09
method | result | size |
default | \(-\frac {i \left (-70 i \left (\cos ^{2}\relax (x )\right ) \sin \relax (x ) \sqrt {2}\, \sqrt {5}\, \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 )+64 i \left (\cos ^{2}\relax (x )\right ) \sin \relax (x ) \sqrt {2}\, \sqrt {5}\, \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 )+3 i \left (\cos ^{2}\relax (x )\right ) \sin \relax (x ) \sqrt {2}\, \sqrt {5}\, \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 )-140 i \left (\cos ^{2}\relax (x )\right ) \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 )+128 i \left (\cos ^{2}\relax (x )\right ) \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 )+6 i \left (\cos ^{2}\relax (x )\right ) \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 )+80 \left (\cos ^{3}\relax (x )\right ) \sqrt {5}+180 \left (\cos ^{3}\relax (x )\right )-80 \left (\cos ^{2}\relax (x )\right ) \sqrt {5}-180 \left (\cos ^{2}\relax (x )\right )-100 \cos \relax (x ) \sqrt {5}-225 \cos \relax (x )+100 \sqrt {5}+225\right ) \cos \relax (x ) \sin \relax (x ) \left (\frac {4 \left (\cos ^{2}\relax (x )\right )-5}{\cos \relax (x )^{2}}\right )^{\frac {3}{2}}}{2 \left (\sqrt {5}+2\right ) \sqrt {-9-4 \sqrt {5}}\, \left (-1+\cos \relax (x )\right ) \left (4 \left (\cos ^{2}\relax (x )\right )-5\right )^{2}}\) | \(754\) |
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 \[ \int {\left (-5 \, \sec \relax (x)^{2} + 4\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (4-\frac {5}{{\cos \relax (x)}^2}\right )}^{3/2} \,d x \]
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 \[ \int \left (4 - 5 \sec ^{2}{\relax (x )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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