Optimal. Leaf size=68 \[ 8 \tan ^{-1}\left (\frac {2 \tan (x)}{\sqrt {-1-5 \tan ^2(x)}}\right )-\frac {7}{2} \sqrt {5} \tan ^{-1}\left (\frac {\sqrt {5} \tan (x)}{\sqrt {-1-5 \tan ^2(x)}}\right )-\frac {5}{2} \tan (x) \sqrt {-1-5 \tan ^2(x)} \]
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Rubi [A]
time = 0.08, 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 = {4213, 427, 537,
223, 209, 385} \begin {gather*} 8 \text {ArcTan}\left (\frac {2 \tan (x)}{\sqrt {-5 \tan ^2(x)-1}}\right )-\frac {7}{2} \sqrt {5} \text {ArcTan}\left (\frac {\sqrt {5} \tan (x)}{\sqrt {-5 \tan ^2(x)-1}}\right )-\frac {5}{2} \tan (x) \sqrt {-5 \tan ^2(x)-1} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 223
Rule 385
Rule 427
Rule 537
Rule 4213
Rubi steps
\begin {align*} \int \left (4-5 \sec ^2(x)\right )^{3/2} \, dx &=\text {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} \text {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 \text {Subst}\left (\int \frac {1}{\sqrt {-1-5 x^2} \left (1+x^2\right )} \, dx,x,\tan (x)\right )-\frac {35}{2} \text {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 \text {Subst}\left (\int \frac {1}{1+4 x^2} \, dx,x,\frac {\tan (x)}{\sqrt {-1-5 \tan ^2(x)}}\right )-\frac {35}{2} \text {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] Result contains complex when optimal does not.
time = 0.15, size = 115, normalized size = 1.69 \begin {gather*} -\frac {\left (-5+4 \cos ^2(x)\right ) \sec (x) \sqrt {4-5 \sec ^2(x)} \left (7 \sqrt {5} \tan ^{-1}\left (\frac {\sqrt {5} \sin (x)}{\sqrt {-3+2 \cos (2 x)}}\right ) \cos ^2(x)+16 i \cos ^2(x) \log \left (\sqrt {-3+2 \cos (2 x)}+2 i \sin (x)\right )+5 \sqrt {-3+2 \cos (2 x)} \sin (x)\right )}{2 (-3+2 \cos (2 x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.46, size = 754, normalized size = 11.09
method | result | size |
default | \(\text {Expression too large to display}\) | \(754\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 130 vs.
\(2 (54) = 108\).
time = 1.19, size = 130, normalized size = 1.91 \begin {gather*} \frac {7 \, \sqrt {5} \arctan \left (\frac {\sqrt {5} \sqrt {\frac {4 \, \cos \left (x\right )^{2} - 5}{\cos \left (x\right )^{2}}} \cos \left (x\right )}{5 \, \sin \left (x\right )}\right ) \cos \left (x\right ) + 8 \, \arctan \left (\frac {4 \, {\left (8 \, \cos \left (x\right )^{3} - 9 \, \cos \left (x\right )\right )} \sqrt {\frac {4 \, \cos \left (x\right )^{2} - 5}{\cos \left (x\right )^{2}}} \sin \left (x\right ) + \cos \left (x\right ) \sin \left (x\right )}{64 \, \cos \left (x\right )^{4} - 143 \, \cos \left (x\right )^{2} + 80}\right ) \cos \left (x\right ) - 8 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right )}\right ) \cos \left (x\right ) - 5 \, \sqrt {\frac {4 \, \cos \left (x\right )^{2} - 5}{\cos \left (x\right )^{2}}} \sin \left (x\right )}{2 \, \cos \left (x\right )} \end {gather*}
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 \begin {gather*} \int \left (4 - 5 \sec ^{2}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (4-\frac {5}{{\cos \left (x\right )}^2}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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