Optimal. Leaf size=24 \[ \frac {2 x \sin (x)}{3 \cos ^{\frac {3}{2}}(x)}-\frac {4}{3 \sqrt {\cos (x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {3315} \[ \frac {2 x \sin (x)}{3 \cos ^{\frac {3}{2}}(x)}-\frac {4}{3 \sqrt {\cos (x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3315
Rubi steps
\begin {align*} \int \left (\frac {x}{\cos ^{\frac {5}{2}}(x)}-\frac {x}{3 \sqrt {\cos (x)}}\right ) \, dx &=-\left (\frac {1}{3} \int \frac {x}{\sqrt {\cos (x)}} \, dx\right )+\int \frac {x}{\cos ^{\frac {5}{2}}(x)} \, dx\\ &=-\frac {4}{3 \sqrt {\cos (x)}}+\frac {2 x \sin (x)}{3 \cos ^{\frac {3}{2}}(x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 17, normalized size = 0.71 \[ -\frac {8-4 x \tan (x)}{6 \sqrt {\cos (x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.91, size = 15, normalized size = 0.62 \[ \frac {2 \, {\left (x \sin \relax (x) - 2 \, \cos \relax (x)\right )}}{3 \, \cos \relax (x)^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {x}{3 \, \sqrt {\cos \relax (x)}} + \frac {x}{\cos \relax (x)^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.26, size = 0, normalized size = 0.00 \[ \int \frac {x}{\cos \relax (x )^{\frac {5}{2}}}-\frac {x}{3 \sqrt {\cos \relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {x}{3 \, \sqrt {\cos \relax (x)}} + \frac {x}{\cos \relax (x)^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.14, size = 16, normalized size = 0.67 \[ -\frac {4\,\cos \relax (x)-2\,x\,\sin \relax (x)}{3\,{\cos \relax (x)}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \left (- \frac {3 x}{\cos ^{\frac {5}{2}}{\relax (x )}}\right )\, dx + \int \frac {x}{\sqrt {\cos {\relax (x )}}}\, dx}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________