\(\int \frac {\cos ^3(x)}{\sqrt {\sin ^3(x)}} \, dx\) [679]

Optimal result

Integrand size = 13, antiderivative size = 25 \[ \int \frac {\cos ^3(x)}{\sqrt {\sin ^3(x)}} \, dx=-\frac {2 \sin (x)}{\sqrt {\sin ^3(x)}}-\frac {2}{3} \sqrt {\sin ^3(x)} \]

[Out]

-2*sin(x)/(sin(x)^3)^(1/2)-2/3*(sin(x)^3)^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3286, 2644, 14} \[ \int \frac {\cos ^3(x)}{\sqrt {\sin ^3(x)}} \, dx=-\frac {2 \sin (x)}{\sqrt {\sin ^3(x)}}-\frac {2}{3} \sqrt {\sin ^3(x)} \]

[In]

Int[Cos[x]^3/Sqrt[Sin[x]^3],x]

[Out]

(-2*Sin[x])/Sqrt[Sin[x]^3] - (2*Sqrt[Sin[x]^3])/3

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]]

Rule 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 3286

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p])), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rubi steps \begin{align*} \text {integral}& = \frac {\sin ^{\frac {3}{2}}(x) \int \frac {\cos ^3(x)}{\sin ^{\frac {3}{2}}(x)} \, dx}{\sqrt {\sin ^3(x)}} \\ & = \frac {\sin ^{\frac {3}{2}}(x) \text {Subst}\left (\int \frac {1-x^2}{x^{3/2}} \, dx,x,\sin (x)\right )}{\sqrt {\sin ^3(x)}} \\ & = \frac {\sin ^{\frac {3}{2}}(x) \text {Subst}\left (\int \left (\frac {1}{x^{3/2}}-\sqrt {x}\right ) \, dx,x,\sin (x)\right )}{\sqrt {\sin ^3(x)}} \\ & = -\frac {2 \sin (x)}{\sqrt {\sin ^3(x)}}-\frac {2}{3} \sqrt {\sin ^3(x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {\cos ^3(x)}{\sqrt {\sin ^3(x)}} \, dx=\frac {(-7+\cos (2 x)) \sin (x)}{3 \sqrt {\sin ^3(x)}} \]

[In]

Integrate[Cos[x]^3/Sqrt[Sin[x]^3],x]

[Out]

((-7 + Cos[2*x])*Sin[x])/(3*Sqrt[Sin[x]^3])

Maple [A] (verified)

Time = 0.67 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68

method	result	size
derivativedivides	\(-\frac {4 \sin \left (x \right ) \left (\sin \left (x \right )^{2}+3\right )}{3 \sqrt {-\sin \left (3 x \right )+3 \sin \left (x \right )}}\)	\(17\)
default	\(-\frac {4 \sin \left (x \right ) \left (\sin \left (x \right )^{2}+3\right )}{3 \sqrt {-\sin \left (3 x \right )+3 \sin \left (x \right )}}\)	\(17\)

[In]

int(cos(x)^3/(sin(x)^3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {\cos ^3(x)}{\sqrt {\sin ^3(x)}} \, dx=-\frac {2 \, {\left (\cos \left (x\right )^{2} - 4\right )} \sqrt {-{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}}{3 \, {\left (\cos \left (x\right )^{2} - 1\right )}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {\cos ^3(x)}{\sqrt {\sin ^3(x)}} \, dx=- \frac {8 \sin ^{3}{\left (x \right )}}{3 \sqrt {\sin ^{3}{\left (x \right )}}} - \frac {2 \sin {\left (x \right )} \cos ^{2}{\left (x \right )}}{\sqrt {\sin ^{3}{\left (x \right )}}} \]

[In]

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

[Out]

-8*sin(x)**3/(3*sqrt(sin(x)**3)) - 2*sin(x)*cos(x)**2/sqrt(sin(x)**3)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {\cos ^3(x)}{\sqrt {\sin ^3(x)}} \, dx=-\frac {2}{3} \, \sqrt {\sin \left (x\right )^{3}} - \frac {2 \, \sin \left (x\right )}{\sqrt {\sin \left (x\right )^{3}}} \]

[In]

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

[Out]

-2/3*sqrt(sin(x)^3) - 2*sin(x)/sqrt(sin(x)^3)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.52 \[ \int \frac {\cos ^3(x)}{\sqrt {\sin ^3(x)}} \, dx=-\frac {2}{3} \, \sin \left (x\right )^{\frac {3}{2}} - \frac {2}{\sqrt {\sin \left (x\right )}} \]

[In]

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

[Out]

-2/3*sin(x)^(3/2) - 2/sqrt(sin(x))

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^3(x)}{\sqrt {\sin ^3(x)}} \, dx=\int \frac {{\cos \left (x\right )}^3}{\sqrt {{\sin \left (x\right )}^3}} \,d x \]

[In]

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

[Out]

int(cos(x)^3/(sin(x)^3)^(1/2), x)