Integrand size = 18, antiderivative size = 19 \[ \int \frac {1}{\cos ^{\frac {3}{2}}(x) \sqrt {3 \cos (x)+\sin (x)}} \, dx=\frac {2 \sqrt {3 \cos (x)+\sin (x)}}{\sqrt {\cos (x)}} \]
Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\cos ^{\frac {3}{2}}(x) \sqrt {3 \cos (x)+\sin (x)}} \, dx=\frac {2 \sqrt {3 \cos (x)+\sin (x)}}{\sqrt {\cos (x)}} \]
Leaf count is larger than twice the leaf count of optimal. \(100\) vs. \(2(19)=38\).
Time = 0.87 (sec) , antiderivative size = 100, normalized size of antiderivative = 5.26, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4902, 2058, 7270, 2136, 24}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\cos ^{\frac {3}{2}}(x) \sqrt {\sin (x)+3 \cos (x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (x)^{3/2} \sqrt {\sin (x)+3 \cos (x)}}dx\) |
| \(\Big \downarrow \) 4902 |
| \(\displaystyle 2 \int \frac {1}{\left (1-\tan ^2\left (\frac {x}{2}\right )\right ) \sqrt {\frac {-3 \tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+3}{\tan ^2\left (\frac {x}{2}\right )+1}} \sqrt {\frac {1-\tan ^2\left (\frac {x}{2}\right )}{\tan ^2\left (\frac {x}{2}\right )+1}}}d\tan \left (\frac {x}{2}\right )\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {2 \sqrt {1-\tan ^2\left (\frac {x}{2}\right )} \int \frac {\sqrt {\tan ^2\left (\frac {x}{2}\right )+1}}{\left (1-\tan ^2\left (\frac {x}{2}\right )\right )^{3/2} \sqrt {\frac {-3 \tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+3}{\tan ^2\left (\frac {x}{2}\right )+1}}}d\tan \left (\frac {x}{2}\right )}{\sqrt {\frac {1-\tan ^2\left (\frac {x}{2}\right )}{\tan ^2\left (\frac {x}{2}\right )+1}} \sqrt {\tan ^2\left (\frac {x}{2}\right )+1}}\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle \frac {2 \sqrt {-3 \tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+3} \sqrt {1-\tan ^2\left (\frac {x}{2}\right )} \int \frac {\tan ^2\left (\frac {x}{2}\right )+1}{\sqrt {-3 \tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+3} \left (1-\tan ^2\left (\frac {x}{2}\right )\right )^{3/2}}d\tan \left (\frac {x}{2}\right )}{\sqrt {\frac {-3 \tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+3}{\tan ^2\left (\frac {x}{2}\right )+1}} \sqrt {\frac {1-\tan ^2\left (\frac {x}{2}\right )}{\tan ^2\left (\frac {x}{2}\right )+1}} \left (\tan ^2\left (\frac {x}{2}\right )+1\right )}\) |
| \(\Big \downarrow \) 2136 |
| \(\displaystyle \frac {2 \sqrt {-3 \tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+3} \sqrt {1-\tan ^2\left (\frac {x}{2}\right )} \left (\frac {1}{8} \int 0d\tan \left (\frac {x}{2}\right )+\frac {\sqrt {-3 \tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+3}}{\sqrt {1-\tan ^2\left (\frac {x}{2}\right )}}\right )}{\sqrt {\frac {-3 \tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+3}{\tan ^2\left (\frac {x}{2}\right )+1}} \sqrt {\frac {1-\tan ^2\left (\frac {x}{2}\right )}{\tan ^2\left (\frac {x}{2}\right )+1}} \left (\tan ^2\left (\frac {x}{2}\right )+1\right )}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {2 \left (-3 \tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+3\right )}{\sqrt {\frac {-3 \tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+3}{\tan ^2\left (\frac {x}{2}\right )+1}} \sqrt {\frac {1-\tan ^2\left (\frac {x}{2}\right )}{\tan ^2\left (\frac {x}{2}\right )+1}} \left (\tan ^2\left (\frac {x}{2}\right )+1\right )}\) |
(2*(3 + 2*Tan[x/2] - 3*Tan[x/2]^2))/(Sqrt[(3 + 2*Tan[x/2] - 3*Tan[x/2]^2)/ (1 + Tan[x/2]^2)]*Sqrt[(1 - Tan[x/2]^2)/(1 + Tan[x/2]^2)]*(1 + Tan[x/2]^2) )
3.9.58.3.1 Defintions of rubi rules used
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ (r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))] Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
Int[(Px_)*((a_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_ ), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[P x, x, 2]}, Simp[(a + c*x^2)^(p + 1)*((d + e*x + f*x^2)^(q + 1)/((-4*a*c)*(a *c*e^2 + (c*d - a*f)^2)*(p + 1)))*((A*c - a*C)*(2*a*c*e) + ((-a)*B)*(2*c^2* d - c*(2*a*f)) + c*(A*(2*c^2*d - c*(2*a*f)) - B*(-2*a*c*e) + C*(-2*a*(c*d - a*f)))*x), x] + Simp[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 - B*c*e - a*C* f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e) + ((-a)*B)*(2*c ^2*d - c*((Plus[2])*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a*c*e) + ((-a )*B)*(2*c^2*d + (-c)*((Plus[2])*a*f)))*(p + q + 2) - (2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*((-c)*e*(2*p + q + 4)))*x - c*f*(2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x]] /; Free Q[{a, c, d, e, f, q}, x] && PolyQ[Px, x, 2] && LtQ[p, -1] && NeQ[a*c*e^2 + (c*d - a*f)^2, 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q, 0]
Int[u_, x_Symbol] :> With[{w = Block[{$ShowSteps = False, $StepCounter = Nu ll}, Int[SubstFor[1/(1 + FreeFactors[Tan[FunctionOfTrig[u, x]/2], x]^2*x^2) , Tan[FunctionOfTrig[u, x]/2]/FreeFactors[Tan[FunctionOfTrig[u, x]/2], x], u, x], x]]}, Module[{v = FunctionOfTrig[u, x], d}, Simp[d = FreeFactors[Tan [v/2], x]; 2*(d/Coefficient[v, x, 1]) Subst[Int[SubstFor[1/(1 + d^2*x^2), Tan[v/2]/d, u, x], x], x, Tan[v/2]/d], x]] /; CalculusFreeQ[w, x]] /; Inve rseFunctionFreeQ[u, x] && !FalseQ[FunctionOfTrig[u, x]]
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
Time = 5.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84
| method | result | size |
| default | \(\frac {2 \sqrt {3 \cos \left (x \right )+\sin \left (x \right )}}{\sqrt {\cos \left (x \right )}}\) | \(16\) |
Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\cos ^{\frac {3}{2}}(x) \sqrt {3 \cos (x)+\sin (x)}} \, dx=\frac {2 \, \sqrt {3 \, \cos \left (x\right ) + \sin \left (x\right )}}{\sqrt {\cos \left (x\right )}} \]
\[ \int \frac {1}{\cos ^{\frac {3}{2}}(x) \sqrt {3 \cos (x)+\sin (x)}} \, dx=\int \frac {1}{\sqrt {\sin {\left (x \right )} + 3 \cos {\left (x \right )}} \cos ^{\frac {3}{2}}{\left (x \right )}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (15) = 30\).
Time = 0.34 (sec) , antiderivative size = 145, normalized size of antiderivative = 7.63 \[ \int \frac {1}{\cos ^{\frac {3}{2}}(x) \sqrt {3 \cos (x)+\sin (x)}} \, dx=\frac {2 \, {\left (\frac {2 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {6 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {2 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + 3\right )} {\left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}^{2}}{\sqrt {\frac {2 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {3 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 3} {\left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right )}^{\frac {3}{2}} {\left (-\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right )}^{\frac {3}{2}} {\left (\frac {2 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {\sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + 1\right )}} \]
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/(s qrt(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))
\[ \int \frac {1}{\cos ^{\frac {3}{2}}(x) \sqrt {3 \cos (x)+\sin (x)}} \, dx=\int { \frac {1}{\sqrt {3 \, \cos \left (x\right ) + \sin \left (x\right )} \cos \left (x\right )^{\frac {3}{2}}} \,d x } \]
Time = 26.86 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\cos ^{\frac {3}{2}}(x) \sqrt {3 \cos (x)+\sin (x)}} \, dx=\frac {2\,\sqrt {3\,\cos \left (x\right )+\sin \left (x\right )}}{\sqrt {\cos \left (x\right )}} \]