Optimal. Leaf size=56 \[ \frac {\sqrt {-1-\cos ^2(x)} E\left (\left .\frac {\pi }{2}+x\right |-1\right )}{2 \sqrt {1+\cos ^2(x)}}+\frac {\cos (x) \sin (x)}{2 \sqrt {-1-\cos ^2(x)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 56, 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 = {3263, 21, 3257,
3256} \begin {gather*} \frac {\sin (x) \cos (x)}{2 \sqrt {-\cos ^2(x)-1}}+\frac {\sqrt {-\cos ^2(x)-1} E\left (\left .x+\frac {\pi }{2}\right |-1\right )}{2 \sqrt {\cos ^2(x)+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 3256
Rule 3257
Rule 3263
Rubi steps
\begin {align*} \int \frac {1}{\left (-1-\cos ^2(x)\right )^{3/2}} \, dx &=\frac {\cos (x) \sin (x)}{2 \sqrt {-1-\cos ^2(x)}}-\frac {1}{2} \int \frac {1+\cos ^2(x)}{\sqrt {-1-\cos ^2(x)}} \, dx\\ &=\frac {\cos (x) \sin (x)}{2 \sqrt {-1-\cos ^2(x)}}+\frac {1}{2} \int \sqrt {-1-\cos ^2(x)} \, dx\\ &=\frac {\cos (x) \sin (x)}{2 \sqrt {-1-\cos ^2(x)}}+\frac {\sqrt {-1-\cos ^2(x)} \int \sqrt {1+\cos ^2(x)} \, dx}{2 \sqrt {1+\cos ^2(x)}}\\ &=\frac {\sqrt {-1-\cos ^2(x)} E\left (\left .\frac {\pi }{2}+x\right |-1\right )}{2 \sqrt {1+\cos ^2(x)}}+\frac {\cos (x) \sin (x)}{2 \sqrt {-1-\cos ^2(x)}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 43, normalized size = 0.77 \begin {gather*} \frac {-2 \sqrt {3+\cos (2 x)} E\left (x\left |\frac {1}{2}\right .\right )+\sin (2 x)}{2 \sqrt {2} \sqrt {-3-\cos (2 x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.53, size = 101, normalized size = 1.80
method | result | size |
default | \(-\frac {\sqrt {\sin ^{4}\left (x \right )-2 \left (\sin ^{2}\left (x \right )\right )}\, \left (2 i \sqrt {-\left (\sin ^{2}\left (x \right )\right )+2}\, \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \EllipticF \left (i \cos \left (x \right ), i\right )-i \sqrt {-\left (\sin ^{2}\left (x \right )\right )+2}\, \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \EllipticE \left (i \cos \left (x \right ), i\right )-\left (\sin ^{2}\left (x \right )\right ) \cos \left (x \right )\right )}{2 \sqrt {\cos ^{4}\left (x \right )-1}\, \sin \left (x \right ) \sqrt {-1-\left (\cos ^{2}\left (x \right )\right )}}\) | \(101\) |
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] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 168 vs. \(2 (51) = 102\).
time = 0.11, size = 168, normalized size = 3.00 \begin {gather*} -\frac {{\left ({\left (2 \, \sqrt {2} - 3\right )} e^{\left (4 i \, x\right )} + 6 \, {\left (2 \, \sqrt {2} - 3\right )} e^{\left (2 i \, x\right )} + 2 \, \sqrt {2} - 3\right )} \sqrt {2 \, \sqrt {2} - 3} E(\arcsin \left (\sqrt {2 \, \sqrt {2} - 3} e^{\left (i \, x\right )}\right )\,|\,12 \, \sqrt {2} + 17) + 4 \, {\left ({\left (\sqrt {2} + 3\right )} e^{\left (4 i \, x\right )} + 6 \, {\left (\sqrt {2} + 3\right )} e^{\left (2 i \, x\right )} + \sqrt {2} + 3\right )} \sqrt {2 \, \sqrt {2} - 3} F(\arcsin \left (\sqrt {2 \, \sqrt {2} - 3} e^{\left (i \, x\right )}\right )\,|\,12 \, \sqrt {2} + 17) + \sqrt {e^{\left (4 i \, x\right )} + 6 \, e^{\left (2 i \, x\right )} + 1} {\left (e^{\left (3 i \, x\right )} + 3 \, e^{\left (i \, x\right )}\right )}}{2 \, {\left (e^{\left (4 i \, x\right )} + 6 \, e^{\left (2 i \, x\right )} + 1\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 \frac {1}{\left (- \cos ^{2}{\left (x \right )} - 1\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 = -0.02 \begin {gather*} \int \frac {1}{{\left (-{\cos \left (x\right )}^2-1\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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