3.858 \(\int \frac{1}{\cos ^{\frac{3}{2}}(x) \sqrt{3 \cos (x)+\sin (x)}} \, dx\)

Optimal. Leaf size=19 \[ \frac{2 \sqrt{\sin (x)+3 \cos (x)}}{\sqrt{\cos (x)}} \]

[Out]

(2*Sqrt[3*Cos[x] + Sin[x]])/Sqrt[Cos[x]]

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Rubi [B]  time = 2.24263, antiderivative size = 88, normalized size of antiderivative = 4.63, number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6719, 1063, 8} \[ \frac{2 \cos ^2\left (\frac{x}{2}\right ) \left (-3 \tan ^2\left (\frac{x}{2}\right )+2 \tan \left (\frac{x}{2}\right )+3\right )}{\sqrt{\cos ^2\left (\frac{x}{2}\right ) \left (-3 \tan ^2\left (\frac{x}{2}\right )+2 \tan \left (\frac{x}{2}\right )+3\right )} \sqrt{\cos ^2\left (\frac{x}{2}\right ) \left (1-\tan ^2\left (\frac{x}{2}\right )\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Cos[x/2]^2*(3 + 2*Tan[x/2] - 3*Tan[x/2]^2))/(Sqrt[Cos[x/2]^2*(3 + 2*Tan[x/2] - 3*Tan[x/2]^2)]*Sqrt[Cos[x/2]
^2*(1 - Tan[x/2]^2)])

Rule 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rule 1063

Int[((a_) + (c_.)*(x_)^2)^(p_)*((A_.) + (C_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Si
mp[((a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)*((A*c - a*C)*(2*a*c*e) + c*(A*(2*c^2*d - c*(2*a*f)) + C*(-2*
a*(c*d - a*f)))*x))/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1)), x] + Dist[1/((-4*a*c)*(a*c*e^2 + (c*d - a*f)
^2)*(p + 1)), Int[(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(-2*A*c - 2*a*C)*((c*d - a*f)^2 - (-(a*e))*(c*e
))*(p + 1) + (2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e))
*(p + q + 2) - (2*f*((A*c - a*C)*(2*a*c*e))*(p + q + 2) - (2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(-(c*e*(2*
p + q + 4))))*x - c*f*(2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, c
, d, e, f, A, C, q}, x] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[a*c*e^2 + (c*d - a*f)^2, 0] &&  !( !Intege
rQ[p] && ILtQ[q, -1]) &&  !IGtQ[q, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{1}{\cos ^{\frac{3}{2}}(x) \sqrt{3 \cos (x)+\sin (x)}} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{\frac{3+2 x-3 x^2}{1+x^2}} \sqrt{\frac{1-x^2}{1+x^2}}} \, dx,x,\tan \left (\frac{x}{2}\right )\right )\\ &=\frac{\left (2 \sqrt{3+2 \tan \left (\frac{x}{2}\right )-3 \tan ^2\left (\frac{x}{2}\right )}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+x^2}}{\sqrt{3+2 x-3 x^2} \left (1-x^2\right ) \sqrt{\frac{1-x^2}{1+x^2}}} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{\sqrt{\sec ^2\left (\frac{x}{2}\right )} \sqrt{\cos ^2\left (\frac{x}{2}\right ) \left (3+2 \tan \left (\frac{x}{2}\right )-3 \tan ^2\left (\frac{x}{2}\right )\right )}}\\ &=\frac{\left (2 \cos ^2\left (\frac{x}{2}\right ) \sqrt{3+2 \tan \left (\frac{x}{2}\right )-3 \tan ^2\left (\frac{x}{2}\right )} \sqrt{1-\tan ^2\left (\frac{x}{2}\right )}\right ) \operatorname{Subst}\left (\int \frac{1+x^2}{\sqrt{3+2 x-3 x^2} \left (1-x^2\right )^{3/2}} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{\sqrt{\cos ^2\left (\frac{x}{2}\right ) \left (3+2 \tan \left (\frac{x}{2}\right )-3 \tan ^2\left (\frac{x}{2}\right )\right )} \sqrt{\cos ^2\left (\frac{x}{2}\right ) \left (1-\tan ^2\left (\frac{x}{2}\right )\right )}}\\ &=\frac{2 \cos ^2\left (\frac{x}{2}\right ) \left (3+2 \tan \left (\frac{x}{2}\right )-3 \tan ^2\left (\frac{x}{2}\right )\right )}{\sqrt{\cos ^2\left (\frac{x}{2}\right ) \left (3+2 \tan \left (\frac{x}{2}\right )-3 \tan ^2\left (\frac{x}{2}\right )\right )} \sqrt{\cos ^2\left (\frac{x}{2}\right ) \left (1-\tan ^2\left (\frac{x}{2}\right )\right )}}+\frac{\left (\cos ^2\left (\frac{x}{2}\right ) \sqrt{3+2 \tan \left (\frac{x}{2}\right )-3 \tan ^2\left (\frac{x}{2}\right )} \sqrt{1-\tan ^2\left (\frac{x}{2}\right )}\right ) \operatorname{Subst}\left (\int 0 \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{4 \sqrt{\cos ^2\left (\frac{x}{2}\right ) \left (3+2 \tan \left (\frac{x}{2}\right )-3 \tan ^2\left (\frac{x}{2}\right )\right )} \sqrt{\cos ^2\left (\frac{x}{2}\right ) \left (1-\tan ^2\left (\frac{x}{2}\right )\right )}}\\ &=\frac{2 \cos ^2\left (\frac{x}{2}\right ) \left (3+2 \tan \left (\frac{x}{2}\right )-3 \tan ^2\left (\frac{x}{2}\right )\right )}{\sqrt{\cos ^2\left (\frac{x}{2}\right ) \left (3+2 \tan \left (\frac{x}{2}\right )-3 \tan ^2\left (\frac{x}{2}\right )\right )} \sqrt{\cos ^2\left (\frac{x}{2}\right ) \left (1-\tan ^2\left (\frac{x}{2}\right )\right )}}\\ \end{align*}

Mathematica [A]  time = 0.0668912, size = 19, normalized size = 1. \[ \frac{2 \sqrt{\sin (x)+3 \cos (x)}}{\sqrt{\cos (x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[3*Cos[x] + Sin[x]])/Sqrt[Cos[x]]

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Maple [A]  time = 0.26, size = 16, normalized size = 0.8 \begin{align*} 2\,{\frac{\sqrt{3\,\cos \left ( x \right ) +\sin \left ( x \right ) }}{\sqrt{\cos \left ( x \right ) }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(3*cos(x)+sin(x))^(1/2)/cos(x)^(1/2)

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Maxima [B]  time = 1.63236, size = 196, normalized size = 10.32 \begin{align*} \frac{2 \,{\left (\frac{2 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac{6 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac{2 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + 3\right )}{\left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}^{2}}{\sqrt{\frac{2 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac{3 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 3}{\left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right )}^{\frac{3}{2}}{\left (-\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right )}^{\frac{3}{2}}{\left (\frac{2 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{\sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(2*sin(x)/(cos(x) + 1) - 6*sin(x)^2/(cos(x) + 1)^2 - 2*sin(x)^3/(cos(x) + 1)^3 + 3*sin(x)^4/(cos(x) + 1)^4 +
3)*(sin(x)^2/(cos(x) + 1)^2 + 1)^2/(sqrt(2*sin(x)/(cos(x) + 1) - 3*sin(x)^2/(cos(x) + 1)^2 + 3)*(sin(x)/(cos(
x) + 1) + 1)^(3/2)*(-sin(x)/(cos(x) + 1) + 1)^(3/2)*(2*sin(x)^2/(cos(x) + 1)^2 + sin(x)^4/(cos(x) + 1)^4 + 1))

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Fricas [A]  time = 2.19315, size = 54, normalized size = 2.84 \begin{align*} \frac{2 \, \sqrt{3 \, \cos \left (x\right ) + \sin \left (x\right )}}{\sqrt{\cos \left (x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(3*cos(x) + sin(x))/sqrt(cos(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(sin(x) + 3*cos(x))*cos(x)**(3/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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