Optimal. Leaf size=56 \[ \frac {\sin (x) \cos (x)}{2 \sqrt {-\cos ^2(x)-1}}+\frac {\sqrt {-\cos ^2(x)-1} E\left (\left .x+\frac {\pi }{2}\right |-1\right )}{2 \sqrt {\cos ^2(x)+1}} \]
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Rubi [A] time = 0.03, 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 = {3184, 21, 3178, 3177} \[ \frac {\sin (x) \cos (x)}{2 \sqrt {-\cos ^2(x)-1}}+\frac {\sqrt {-\cos ^2(x)-1} E\left (\left .x+\frac {\pi }{2}\right |-1\right )}{2 \sqrt {\cos ^2(x)+1}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 3177
Rule 3178
Rule 3184
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 = 0.04, size = 43, normalized size = 0.77 \[ \frac {\sin (2 x)-2 \sqrt {\cos (2 x)+3} E\left (x\left |\frac {1}{2}\right .\right )}{2 \sqrt {2} \sqrt {-\cos (2 x)-3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ \frac {2 \, {\left (e^{\left (4 i \, x\right )} + 6 \, e^{\left (2 i \, x\right )} + 1\right )} {\rm integral}\left (\frac {e^{\left (2 i \, x\right )} + 3}{2 \, \sqrt {e^{\left (4 i \, x\right )} + 6 \, e^{\left (2 i \, x\right )} + 1}}, x\right ) - \sqrt {e^{\left (4 i \, x\right )} + 6 \, e^{\left (2 i \, x\right )} + 1} {\left (e^{\left (3 i \, x\right )} + 3 \, e^{\left (i \, x\right )}\right )}}{2 \, {\left (e^{\left (4 i \, x\right )} + 6 \, e^{\left (2 i \, x\right )} + 1\right )}} \]
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 \[ \int \frac {1}{{\left (-\cos \relax (x)^{2} - 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.82, size = 101, normalized size = 1.80 \[ -\frac {\sqrt {\sin ^{4}\relax (x )-2 \left (\sin ^{2}\relax (x )\right )}\, \left (2 i \sqrt {-\left (\sin ^{2}\relax (x )\right )+2}\, \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \EllipticF \left (i \cos \relax (x ), i\right )-i \sqrt {-\left (\sin ^{2}\relax (x )\right )+2}\, \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \EllipticE \left (i \cos \relax (x ), i\right )-\left (\sin ^{2}\relax (x )\right ) \cos \relax (x )\right )}{2 \sqrt {\cos ^{4}\relax (x )-1}\, \sin \relax (x ) \sqrt {-1-\left (\cos ^{2}\relax (x )\right )}} \]
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 \[ \int \frac {1}{{\left (-\cos \relax (x)^{2} - 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (-{\cos \relax (x)}^2-1\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (- \cos ^{2}{\relax (x )} - 1\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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