Optimal. Leaf size=55 \[ \frac {\cos (x)}{15 \sqrt {7 \cos ^2(x)-2}}-\frac {\cos (x)}{15 \left (7 \cos ^2(x)-2\right )^{3/2}}+\frac {\cos (x)}{10 \left (7 \cos ^2(x)-2\right )^{5/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4357, 192, 191} \[ \frac {\cos (x)}{15 \sqrt {7 \cos ^2(x)-2}}-\frac {\cos (x)}{15 \left (7 \cos ^2(x)-2\right )^{3/2}}+\frac {\cos (x)}{10 \left (7 \cos ^2(x)-2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 4357
Rubi steps
\begin {align*} \int \frac {\sin (x)}{\left (5 \cos ^2(x)-2 \sin ^2(x)\right )^{7/2}} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{\left (-2+7 x^2\right )^{7/2}} \, dx,x,\cos (x)\right )\\ &=\frac {\cos (x)}{10 \left (-2+7 \cos ^2(x)\right )^{5/2}}+\frac {2}{5} \operatorname {Subst}\left (\int \frac {1}{\left (-2+7 x^2\right )^{5/2}} \, dx,x,\cos (x)\right )\\ &=\frac {\cos (x)}{10 \left (-2+7 \cos ^2(x)\right )^{5/2}}-\frac {\cos (x)}{15 \left (-2+7 \cos ^2(x)\right )^{3/2}}-\frac {2}{15} \operatorname {Subst}\left (\int \frac {1}{\left (-2+7 x^2\right )^{3/2}} \, dx,x,\cos (x)\right )\\ &=\frac {\cos (x)}{10 \left (-2+7 \cos ^2(x)\right )^{5/2}}-\frac {\cos (x)}{15 \left (-2+7 \cos ^2(x)\right )^{3/2}}+\frac {\cos (x)}{15 \sqrt {-2+7 \cos ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 37, normalized size = 0.67 \[ \frac {\cos (x) (56 \cos (2 x)+49 \cos (4 x)+67)}{15 \sqrt {2} (7 \cos (2 x)+3)^{5/2}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin (x)}{\left (5 \cos ^2(x)-2 \sin ^2(x)\right )^{7/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.98, size = 51, normalized size = 0.93 \[ \frac {{\left (98 \, \cos \relax (x)^{5} - 70 \, \cos \relax (x)^{3} + 15 \, \cos \relax (x)\right )} \sqrt {7 \, \cos \relax (x)^{2} - 2}}{30 \, {\left (343 \, \cos \relax (x)^{6} - 294 \, \cos \relax (x)^{4} + 84 \, \cos \relax (x)^{2} - 8\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.67, size = 30, normalized size = 0.55 \[ \frac {{\left (14 \, {\left (7 \, \cos \relax (x)^{2} - 5\right )} \cos \relax (x)^{2} + 15\right )} \cos \relax (x)}{30 \, {\left (7 \, \cos \relax (x)^{2} - 2\right )}^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 44, normalized size = 0.80
method | result | size |
derivativedivides | \(\frac {\cos \relax (x )}{10 \left (-2+7 \left (\cos ^{2}\relax (x )\right )\right )^{\frac {5}{2}}}-\frac {\cos \relax (x )}{15 \left (-2+7 \left (\cos ^{2}\relax (x )\right )\right )^{\frac {3}{2}}}+\frac {\cos \relax (x )}{15 \sqrt {-2+7 \left (\cos ^{2}\relax (x )\right )}}\) | \(44\) |
default | \(\frac {\cos \relax (x )}{10 \left (-2+7 \left (\cos ^{2}\relax (x )\right )\right )^{\frac {5}{2}}}-\frac {\cos \relax (x )}{15 \left (-2+7 \left (\cos ^{2}\relax (x )\right )\right )^{\frac {3}{2}}}+\frac {\cos \relax (x )}{15 \sqrt {-2+7 \left (\cos ^{2}\relax (x )\right )}}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 43, normalized size = 0.78 \[ \frac {\cos \relax (x)}{15 \, \sqrt {7 \, \cos \relax (x)^{2} - 2}} - \frac {\cos \relax (x)}{15 \, {\left (7 \, \cos \relax (x)^{2} - 2\right )}^{\frac {3}{2}}} + \frac {\cos \relax (x)}{10 \, {\left (7 \, \cos \relax (x)^{2} - 2\right )}^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.66, size = 28, normalized size = 0.51 \[ \frac {\cos \relax (x)\,\left (98\,{\cos \relax (x)}^4-70\,{\cos \relax (x)}^2+15\right )}{30\,{\left (7\,{\cos \relax (x)}^2-2\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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