3.48 \(\int \frac{1}{\sqrt{a \cos ^3(x)}} \, dx\)

Optimal. Leaf size=42 \[ \frac{2 \sin (x) \cos (x)}{\sqrt{a \cos ^3(x)}}-\frac{2 \cos ^{\frac{3}{2}}(x) E\left (\left .\frac{x}{2}\right |2\right )}{\sqrt{a \cos ^3(x)}} \]

[Out]

(-2*Cos[x]^(3/2)*EllipticE[x/2, 2])/Sqrt[a*Cos[x]^3] + (2*Cos[x]*Sin[x])/Sqrt[a*Cos[x]^3]

________________________________________________________________________________________

Rubi [A]  time = 0.0234349, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3207, 2636, 2639} \[ \frac{2 \sin (x) \cos (x)}{\sqrt{a \cos ^3(x)}}-\frac{2 \cos ^{\frac{3}{2}}(x) E\left (\left .\frac{x}{2}\right |2\right )}{\sqrt{a \cos ^3(x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*Cos[x]^3],x]

[Out]

(-2*Cos[x]^(3/2)*EllipticE[x/2, 2])/Sqrt[a*Cos[x]^3] + (2*Cos[x]*Sin[x])/Sqrt[a*Cos[x]^3]

Rule 3207

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
]])

Rule 2636

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1))/(b*d*(n +
1)), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[2*n]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a \cos ^3(x)}} \, dx &=\frac{\cos ^{\frac{3}{2}}(x) \int \frac{1}{\cos ^{\frac{3}{2}}(x)} \, dx}{\sqrt{a \cos ^3(x)}}\\ &=\frac{2 \cos (x) \sin (x)}{\sqrt{a \cos ^3(x)}}-\frac{\cos ^{\frac{3}{2}}(x) \int \sqrt{\cos (x)} \, dx}{\sqrt{a \cos ^3(x)}}\\ &=-\frac{2 \cos ^{\frac{3}{2}}(x) E\left (\left .\frac{x}{2}\right |2\right )}{\sqrt{a \cos ^3(x)}}+\frac{2 \cos (x) \sin (x)}{\sqrt{a \cos ^3(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0210661, size = 31, normalized size = 0.74 \[ \frac{\sin (2 x)-2 \cos ^{\frac{3}{2}}(x) E\left (\left .\frac{x}{2}\right |2\right )}{\sqrt{a \cos ^3(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*Cos[x]^3],x]

[Out]

(-2*Cos[x]^(3/2)*EllipticE[x/2, 2] + Sin[2*x])/Sqrt[a*Cos[x]^3]

________________________________________________________________________________________

Maple [C]  time = 0.333, size = 191, normalized size = 4.6 \begin{align*} 2\,{\frac{ \left ( \cos \left ( x \right ) +1 \right ) ^{2} \left ( -1+\cos \left ( x \right ) \right ) ^{2}\cos \left ( x \right ) }{\sqrt{a \left ( \cos \left ( x \right ) \right ) ^{3}} \left ( \sin \left ( x \right ) \right ) ^{5}} \left ( i\cos \left ( x \right ) \sin \left ( x \right ) \sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) -i\cos \left ( x \right ) \sin \left ( x \right ) \sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) +i\sin \left ( x \right ) \sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) -i\sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) \sin \left ( x \right ) -\cos \left ( x \right ) +1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(cos(x)+1)^2*(-1+cos(x))^2*(I*cos(x)*sin(x)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*EllipticE(I*(-1+c
os(x))/sin(x),I)-I*cos(x)*sin(x)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*EllipticF(I*(-1+cos(x))/sin(x)
,I)+I*sin(x)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*EllipticE(I*(-1+cos(x))/sin(x),I)-I*(1/(cos(x)+1))
^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*EllipticF(I*(-1+cos(x))/sin(x),I)*sin(x)-cos(x)+1)*cos(x)/(a*cos(x)^3)^(1/2)/
sin(x)^5

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(a*cos(x)^3), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a \cos \left (x\right )^{3}}}{a \cos \left (x\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(a*cos(x)^3)/(a*cos(x)^3), x)

________________________________________________________________________________________

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(1/(a*cos(x)**3)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(a*cos(x)^3), x)