3.1.11 \(\int \frac {\sin (x)}{\sqrt {\cos ^3(x)}} \, dx\) [11]

Optimal. Leaf size=12 \[ \frac {2 \cos (x)}{\sqrt {\cos ^3(x)}} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3286, 2645, 30} \begin {gather*} \frac {2 \cos (x)}{\sqrt {\cos ^3(x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Cos[x])/Sqrt[Cos[x]^3]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2645

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[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*} \int \frac {\sin (x)}{\sqrt {\cos ^3(x)}} \, dx &=\frac {\cos ^{\frac {3}{2}}(x) \int \frac {\sin (x)}{\cos ^{\frac {3}{2}}(x)} \, dx}{\sqrt {\cos ^3(x)}}\\ &=-\frac {\cos ^{\frac {3}{2}}(x) \text {Subst}\left (\int \frac {1}{x^{3/2}} \, dx,x,\cos (x)\right )}{\sqrt {\cos ^3(x)}}\\ &=\frac {2 \cos (x)}{\sqrt {\cos ^3(x)}}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 12, normalized size = 1.00 \begin {gather*} \frac {2 \cos (x)}{\sqrt {\cos ^3(x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Cos[x])/Sqrt[Cos[x]^3]

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Maple [A]
time = 0.05, size = 11, normalized size = 0.92

method	result	size
derivativedivides	\(\frac {4 \cos \left (x \right )}{\sqrt {\cos \left (3 x \right )+3 \cos \left (x \right )}}\)	\(11\)
default	\(\frac {4 \cos \left (x \right )}{\sqrt {\cos \left (3 x \right )+3 \cos \left (x \right )}}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 3.18, size = 10, normalized size = 0.83 \begin {gather*} \frac {2 \, \cos \left (x\right )}{\sqrt {\cos \left (x\right )^{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*cos(x)/sqrt(cos(x)^3)

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Fricas [A]
time = 0.49, size = 12, normalized size = 1.00 \begin {gather*} \frac {2 \, \sqrt {\cos \left (x\right )^{3}}}{\cos \left (x\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(cos(x)^3)/cos(x)^2

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Sympy [A]
time = 0.24, size = 12, normalized size = 1.00 \begin {gather*} \frac {2 \cos {\left (x \right )}}{\sqrt {\cos ^{3}{\left (x \right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.76, size = 6, normalized size = 0.50 \begin {gather*} \frac {2}{\sqrt {\cos \left (x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/sqrt(cos(x))

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Mupad [B]
time = 0.20, size = 9, normalized size = 0.75 \begin {gather*} \frac {2\,\left |\cos \left (x\right )\right |}{{\cos \left (x\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*abs(cos(x)))/cos(x)^(3/2)

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