Optimal. Leaf size=55 \[ \frac {1}{12} (2 \sin (x)+1)^{3/2}-\frac {1}{2} \sqrt {2 \sin (x)+1}-\frac {4}{\sqrt [4]{2 \sin (x)+1}}+\frac {3}{4 \sqrt {2 \sin (x)+1}} \]
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Rubi [A] time = 0.15, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {4356, 14} \[ \frac {1}{12} (2 \sin (x)+1)^{3/2}-\frac {1}{2} \sqrt {2 \sin (x)+1}-\frac {4}{\sqrt [4]{2 \sin (x)+1}}+\frac {3}{4 \sqrt {2 \sin (x)+1}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 4356
Rubi steps
\begin {align*} \int \frac {\cos (x) \left (-\cos ^2(x)+2 \sqrt [4]{1+2 \sin (x)}\right )}{(1+2 \sin (x))^{3/2}} \, dx &=\operatorname {Subst}\left (\int \frac {-1+x^2+2 \sqrt [4]{1+2 x}}{(1+2 x)^{3/2}} \, dx,x,\sin (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {-3+8 x-2 x^4+x^8}{x^3} \, dx,x,\sqrt [4]{1+2 \sin (x)}\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {3}{x^3}+\frac {8}{x^2}-2 x+x^5\right ) \, dx,x,\sqrt [4]{1+2 \sin (x)}\right )\\ &=\frac {3}{4 \sqrt {1+2 \sin (x)}}-\frac {4}{\sqrt [4]{1+2 \sin (x)}}-\frac {1}{2} \sqrt {1+2 \sin (x)}+\frac {1}{12} (1+2 \sin (x))^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 36, normalized size = 0.65 \[ -\frac {4 \sin (x)+24 \sqrt [4]{2 \sin (x)+1}+\cos (2 x)-3}{6 \sqrt {2 \sin (x)+1}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos (x) \left (-\cos ^2(x)+2 \sqrt [4]{1+2 \sin (x)}\right )}{(1+2 \sin (x))^{3/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.80, size = 40, normalized size = 0.73 \[ -\frac {{\left (\cos \relax (x)^{2} + 2 \, \sin \relax (x) - 2\right )} \sqrt {2 \, \sin \relax (x) + 1} + 12 \, {\left (2 \, \sin \relax (x) + 1\right )}^{\frac {3}{4}}}{3 \, {\left (2 \, \sin \relax (x) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.65, size = 43, normalized size = 0.78 \[ \frac {1}{12} \, {\left (2 \, \sin \relax (x) + 1\right )}^{\frac {3}{2}} - \frac {16 \, {\left (2 \, \sin \relax (x) + 1\right )}^{\frac {1}{4}} - 3}{4 \, \sqrt {2 \, \sin \relax (x) + 1}} - \frac {1}{2} \, \sqrt {2 \, \sin \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.48, size = 31, normalized size = 0.56 \[\frac {\sin ^{2}\relax (x )-2 \sin \relax (x )-12 \left (1+2 \sin \relax (x )\right )^{\frac {1}{4}}+1}{3 \sqrt {1+2 \sin \relax (x )}}\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 43, normalized size = 0.78 \[ \frac {1}{12} \, {\left (2 \, \sin \relax (x) + 1\right )}^{\frac {3}{2}} - \frac {16 \, {\left (2 \, \sin \relax (x) + 1\right )}^{\frac {1}{4}} - 3}{4 \, \sqrt {2 \, \sin \relax (x) + 1}} - \frac {1}{2} \, \sqrt {2 \, \sin \relax (x) + 1} \]
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 \[ -\int -\frac {\cos \relax (x)\,\left (2\,{\left (2\,\sin \relax (x)+1\right )}^{1/4}-{\cos \relax (x)}^2\right )}{{\left (2\,\sin \relax (x)+1\right )}^{3/2}} \,d x \]
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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