Optimal. Leaf size=73 \[ -\frac {1}{6} \cos (x) \left (2 \cos ^2(x)+1\right )^{5/2}-\frac {5}{24} \cos (x) \left (2 \cos ^2(x)+1\right )^{3/2}-\frac {5}{16} \cos (x) \sqrt {2 \cos ^2(x)+1}-\frac {5 \sinh ^{-1}\left (\sqrt {2} \cos (x)\right )}{16 \sqrt {2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 67, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3190, 195, 215} \[ -\frac {1}{6} \cos (x) (\cos (2 x)+2)^{5/2}-\frac {5}{24} \cos (x) (\cos (2 x)+2)^{3/2}-\frac {5}{16} \cos (x) \sqrt {\cos (2 x)+2}-\frac {5 \sinh ^{-1}\left (\sqrt {2} \cos (x)\right )}{16 \sqrt {2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 195
Rule 215
Rule 3190
Rubi steps
\begin {align*} \int \left (1+2 \cos ^2(x)\right )^{5/2} \sin (x) \, dx &=-\operatorname {Subst}\left (\int \left (1+2 x^2\right )^{5/2} \, dx,x,\cos (x)\right )\\ &=-\frac {1}{6} \cos (x) (2+\cos (2 x))^{5/2}-\frac {5}{6} \operatorname {Subst}\left (\int \left (1+2 x^2\right )^{3/2} \, dx,x,\cos (x)\right )\\ &=-\frac {5}{24} \cos (x) (2+\cos (2 x))^{3/2}-\frac {1}{6} \cos (x) (2+\cos (2 x))^{5/2}-\frac {5}{8} \operatorname {Subst}\left (\int \sqrt {1+2 x^2} \, dx,x,\cos (x)\right )\\ &=-\frac {5}{16} \cos (x) \sqrt {2+\cos (2 x)}-\frac {5}{24} \cos (x) (2+\cos (2 x))^{3/2}-\frac {1}{6} \cos (x) (2+\cos (2 x))^{5/2}-\frac {5}{16} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+2 x^2}} \, dx,x,\cos (x)\right )\\ &=-\frac {5 \sinh ^{-1}\left (\sqrt {2} \cos (x)\right )}{16 \sqrt {2}}-\frac {5}{16} \cos (x) \sqrt {2+\cos (2 x)}-\frac {5}{24} \cos (x) (2+\cos (2 x))^{3/2}-\frac {1}{6} \cos (x) (2+\cos (2 x))^{5/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 61, normalized size = 0.84 \[ \frac {1}{96} \left (-2 \sqrt {\cos (2 x)+2} (92 \cos (x)+23 \cos (3 x)+2 \cos (5 x))-15 \sqrt {2} \log \left (\sqrt {2} \cos (x)+\sqrt {\cos (2 x)+2}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (1+2 \cos ^2(x)\right )^{5/2} \sin (x) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.95, size = 108, normalized size = 1.48 \[ -\frac {1}{48} \, {\left (32 \, \cos \relax (x)^{5} + 52 \, \cos \relax (x)^{3} + 33 \, \cos \relax (x)\right )} \sqrt {2 \, \cos \relax (x)^{2} + 1} + \frac {5}{256} \, \sqrt {2} \log \left (2048 \, \cos \relax (x)^{8} + 2048 \, \cos \relax (x)^{6} + 640 \, \cos \relax (x)^{4} + 64 \, \cos \relax (x)^{2} - 8 \, {\left (128 \, \sqrt {2} \cos \relax (x)^{7} + 96 \, \sqrt {2} \cos \relax (x)^{5} + 20 \, \sqrt {2} \cos \relax (x)^{3} + \sqrt {2} \cos \relax (x)\right )} \sqrt {2 \, \cos \relax (x)^{2} + 1} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.67, size = 55, normalized size = 0.75 \[ -\frac {1}{48} \, {\left (4 \, {\left (8 \, \cos \relax (x)^{2} + 13\right )} \cos \relax (x)^{2} + 33\right )} \sqrt {2 \, \cos \relax (x)^{2} + 1} \cos \relax (x) + \frac {5}{32} \, \sqrt {2} \log \left (-\sqrt {2} \cos \relax (x) + \sqrt {2 \, \cos \relax (x)^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.08, size = 56, normalized size = 0.77
method | result | size |
derivativedivides | \(-\frac {5 \cos \relax (x ) \left (1+2 \left (\cos ^{2}\relax (x )\right )\right )^{\frac {3}{2}}}{24}-\frac {\cos \relax (x ) \left (1+2 \left (\cos ^{2}\relax (x )\right )\right )^{\frac {5}{2}}}{6}-\frac {5 \arcsinh \left (\cos \relax (x ) \sqrt {2}\right ) \sqrt {2}}{32}-\frac {5 \cos \relax (x ) \sqrt {1+2 \left (\cos ^{2}\relax (x )\right )}}{16}\) | \(56\) |
default | \(-\frac {5 \cos \relax (x ) \left (1+2 \left (\cos ^{2}\relax (x )\right )\right )^{\frac {3}{2}}}{24}-\frac {\cos \relax (x ) \left (1+2 \left (\cos ^{2}\relax (x )\right )\right )^{\frac {5}{2}}}{6}-\frac {5 \arcsinh \left (\cos \relax (x ) \sqrt {2}\right ) \sqrt {2}}{32}-\frac {5 \cos \relax (x ) \sqrt {1+2 \left (\cos ^{2}\relax (x )\right )}}{16}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.96, size = 55, normalized size = 0.75 \[ -\frac {1}{6} \, {\left (2 \, \cos \relax (x)^{2} + 1\right )}^{\frac {5}{2}} \cos \relax (x) - \frac {5}{24} \, {\left (2 \, \cos \relax (x)^{2} + 1\right )}^{\frac {3}{2}} \cos \relax (x) - \frac {5}{32} \, \sqrt {2} \operatorname {arsinh}\left (\sqrt {2} \cos \relax (x)\right ) - \frac {5}{16} \, \sqrt {2 \, \cos \relax (x)^{2} + 1} \cos \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.13, size = 43, normalized size = 0.59 \[ -\frac {5\,\sqrt {2}\,\mathrm {asinh}\left (\sqrt {2}\,\cos \relax (x)\right )}{32}-\frac {\sqrt {2}\,\sqrt {{\cos \relax (x)}^2+\frac {1}{2}}\,\left (\frac {4\,{\cos \relax (x)}^5}{3}+\frac {13\,{\cos \relax (x)}^3}{6}+\frac {11\,\cos \relax (x)}{8}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________