3.396 \(\int \frac{\cos (x) (-\cos ^2(x)+2 \sqrt [4]{1+2 \sin (x)})}{(1+2 \sin (x))^{3/2}} \, dx\)

Optimal. Leaf size=55 \[ \frac{1}{12} (2 \sin (x)+1)^{3/2}-\frac{1}{2} \sqrt{2 \sin (x)+1}-\frac{4}{\sqrt [4]{2 \sin (x)+1}}+\frac{3}{4 \sqrt{2 \sin (x)+1}} \]

[Out]

3/(4*Sqrt[1 + 2*Sin[x]]) - 4/(1 + 2*Sin[x])^(1/4) - Sqrt[1 + 2*Sin[x]]/2 + (1 + 2*Sin[x])^(3/2)/12

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Rubi [A]  time = 0.150597, antiderivative size = 55, normalized size of antiderivative = 1., 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.

[In]

Int[(Cos[x]*(-Cos[x]^2 + 2*(1 + 2*Sin[x])^(1/4)))/(1 + 2*Sin[x])^(3/2),x]

[Out]

3/(4*Sqrt[1 + 2*Sin[x]]) - 4/(1 + 2*Sin[x])^(1/4) - Sqrt[1 + 2*Sin[x]]/2 + (1 + 2*Sin[x])^(3/2)/12

Rule 4356

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[d/(b
*c), Subst[Int[SubstFor[1, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a +
b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cos] || EqQ[F, cos])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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*}

Mathematica [A]  time = 0.0771201, 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.

[In]

Integrate[(Cos[x]*(-Cos[x]^2 + 2*(1 + 2*Sin[x])^(1/4)))/(1 + 2*Sin[x])^(3/2),x]

[Out]

-(-3 + Cos[2*x] + 4*Sin[x] + 24*(1 + 2*Sin[x])^(1/4))/(6*Sqrt[1 + 2*Sin[x]])

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Maple [A]  time = 0.436, size = 42, normalized size = 0.8 \begin{align*} -4\,{\frac{1}{\sqrt [4]{1+2\,\sin \left ( x \right ) }}}+{\frac{1}{12} \left ( 1+2\,\sin \left ( x \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{3}{4}{\frac{1}{\sqrt{1+2\,\sin \left ( x \right ) }}}}-{\frac{1}{2}\sqrt{1+2\,\sin \left ( x \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*(-cos(x)^2+2*(1+2*sin(x))^(1/4))/(1+2*sin(x))^(3/2),x)

[Out]

-4/(1+2*sin(x))^(1/4)+1/12*(1+2*sin(x))^(3/2)+3/4/(1+2*sin(x))^(1/2)-1/2*(1+2*sin(x))^(1/2)

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Maxima [A]  time = 0.929482, size = 58, normalized size = 1.05 \begin{align*} \frac{1}{12} \,{\left (2 \, \sin \left (x\right ) + 1\right )}^{\frac{3}{2}} - \frac{16 \,{\left (2 \, \sin \left (x\right ) + 1\right )}^{\frac{1}{4}} - 3}{4 \, \sqrt{2 \, \sin \left (x\right ) + 1}} - \frac{1}{2} \, \sqrt{2 \, \sin \left (x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*(-cos(x)^2+2*(1+2*sin(x))^(1/4))/(1+2*sin(x))^(3/2),x, algorithm="maxima")

[Out]

1/12*(2*sin(x) + 1)^(3/2) - 1/4*(16*(2*sin(x) + 1)^(1/4) - 3)/sqrt(2*sin(x) + 1) - 1/2*sqrt(2*sin(x) + 1)

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Fricas [A]  time = 2.08143, size = 127, normalized size = 2.31 \begin{align*} -\frac{{\left (\cos \left (x\right )^{2} + 2 \, \sin \left (x\right ) - 2\right )} \sqrt{2 \, \sin \left (x\right ) + 1} + 12 \,{\left (2 \, \sin \left (x\right ) + 1\right )}^{\frac{3}{4}}}{3 \,{\left (2 \, \sin \left (x\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*(-cos(x)^2+2*(1+2*sin(x))^(1/4))/(1+2*sin(x))^(3/2),x, algorithm="fricas")

[Out]

-1/3*((cos(x)^2 + 2*sin(x) - 2)*sqrt(2*sin(x) + 1) + 12*(2*sin(x) + 1)^(3/4))/(2*sin(x) + 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*(-cos(x)**2+2*(1+2*sin(x))**(1/4))/(1+2*sin(x))**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (\cos \left (x\right )^{2} - 2 \,{\left (2 \, \sin \left (x\right ) + 1\right )}^{\frac{1}{4}}\right )} \cos \left (x\right )}{{\left (2 \, \sin \left (x\right ) + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*(-cos(x)^2+2*(1+2*sin(x))^(1/4))/(1+2*sin(x))^(3/2),x, algorithm="giac")

[Out]

integrate(-(cos(x)^2 - 2*(2*sin(x) + 1)^(1/4))*cos(x)/(2*sin(x) + 1)^(3/2), x)